Scalar Curvature Splittings II: Removal of Singularities
Joachim Lohkamp

Mathematisches Institut, Universität Münster, Einsteinstraße 62, Germany
_e-mail<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
    1. 1.1 Statement and Discussion of Results
    2. 1.2 Overview of the Argument
    3. 1.3 Area Minimizers in $\Lambda$-Bumped Spaces
  2. 2 Constructions on Cones
    1. 2.1 Deformations on Product Cones
    2. 2.2 Removal of Intersections with $\Sigma_{C}$
    3. 2.3 Tightness of Cavities
    4. 2.4 Local Bumps and Shields
  3. 3 From Local to Global Deformations
    1. 3.1 $\mathbf{ID}$-Maps and Ball Covers
    2. 3.2 Transfer of Local Bumps
    3. 3.3 Global Bumps and Alignments

##  1 Introduction

We resume our discussion of general inductive scalar curvature splittings,
involving singular minimal hypersurfaces, from the first part [L1]. We also
adopt the notations, concepts and results from [L1]. In this paper we
establish splitting schemes with built-in regularizations to treat problems in
scalar curvature geometry and general relativity in arbitrary dimensions.

###  1.1 Statement and Discussion of Results

The splitting approach involves the use of singular minimal hypersurfaces. In
[L1] we have seen that any singular compact area minimizer $(H^{n},d_{H})$ in
a $\operatorname{scal}>0$-manifold $M^{n+1}$ admits a conformal deformation to
some minimal factor geometry $(H^{n},d_{\mathcal{S}})$ that shares many
properties with $(H^{n},d_{H})$ like the Ahlfors regularity, the validity of
Poincaré inequalities and the presence of, in this case,
$\operatorname{scal}>0$-tangent cones equipped with their minimal factor
geometry. These geometries are amenable to surgery style arguments to
eliminate singular sets stepwise. The basic building blocks we get are
_splitting with boundary_ theorems.

Theorem 1 Let $H^{n}$, $n\geq 2$, be a compact area minimizer, with singular
set $\Sigma$, in a $\operatorname{scal}>0$-manifold $M^{n+1}$. Then there are
arbitrarily small neighborhoods $U$ of $\Sigma$, so that $H\setminus U$ is
conformal to some $\boldsymbol{\operatorname{scal}>0}$-manifold
$(X_{U},g_{X})$ with minimal boundary $\partial X_{U}=\partial U$.

For $\Sigma_{H}=\emptyset$ we choose $U=\emptyset$, $\partial U=\emptyset$. If
$\Sigma\neq\emptyset$, then it suffices if $M^{n+1}$ has
$\operatorname{scal}\geq 0$. We actually get a smooth (generally non-complete)
open $\operatorname{scal}>0$-manifold $Y_{U}^{n}$ that extends $X_{U}$, i.e.
$\overline{X_{U}}\subset Y_{U}$, so that $\partial X_{U}\subset Y_{U}$ is
_locally area minimizing_.

Inductive Removal of Singularities Theorem 1 replaces the singular set
$\Sigma_{H}$ of $(H^{n},d_{\mathcal{S}})$ by a minimal boundary. The point is
that although, in general, the boundary $\partial X_{U}=\partial U$ is also
singular, its singular set has a _lower dimension_ than $\Sigma_{H}$. The
minimality of $\partial X_{U}$ then allows us to iteratively shift singular
problems to lower dimensions before they disappear in dimension $7$. We
describe some basic implementations of this idea.
The straightforward way to apply Theorem 1 is to use $\overline{X_{U}^{n}}$ as
a replacement for $(H^{n},d_{\mathcal{S}})$, for small $U$. Following the
classical splitting recipe, we consider a local area minimizer
$L^{n-1}\subset\overline{X_{U}^{n}}$. Since $X_{U}^{n}$ is not complete we
observe that $L^{n-1}$ satisfies an additional constraint: $L^{n-1}$ is an
_area minimizer with obstacle_ $\partial U$ that keeps $L^{n-1}$ in
$\overline{X_{U}^{n}}$. This is a standard situation in geometric measure
theory, cf. [Gi, Th. 1.20, Rm. 1.22], but it is rather uncommon in the context
of scalar curvature splittings.

  * •

However, since $\partial U\subset Y_{U}$ is an unconstrained area minimizer by
itself, the strict maximum principle [Si] applies and it shows, componentwise,
that either $L^{n-1}\equiv\partial U$ or $L^{n-1}\cap\partial U=\emptyset$.
This means $L^{n-1}$ always is an _unconstrained_ area minimizer in
$Y_{U}^{n}$.

  * •

$L^{n-1}$ could be singular, but when $L^{n-1}$ is compact we can apply
Theorem 1 to $L^{n-1}\subset Y_{U}^{n}$, in dimension $n-1$, since $Y_{U}^{n}$
is a _smooth_ $\operatorname{scal}>0$-manifold. In this sense Theorem 1 allows
us to stay in the smooth category. Then we are ready for the next loop in the
dimensional descent. Once we reach $n=7$ all further area minimizers are
regular.

Applications  We indicate some typical applications (addressed in separate
accounts). We start from a compact $\operatorname{scal}>0$-manifold $M^{n+1}$
and classes $\alpha[1],\dots,\alpha[m]\in H^{1}(M^{n+1},\mathbb{Z})$, $m\leq
n-1$. The scheme above gives us the means to represent
$\alpha[k]\cap\cdots\cap\alpha[1]\cap[M]$, $k\leq m$, by area minimizers
$L^{n+1-k}$ in _smooth_ $\operatorname{scal}>0$-manifolds of dimension
$n+2-k$, provided we broaden our playground and include _complete spaces with
periodic ends_ and _coarse homologies_ , cf. [R] and [L5] for a survey (also
of underlying techniques and applications).
To explain the idea, we let $H^{n}$ represent $\alpha[1]\cap[M]$ and apply
Theorem 1 to get a conformal deformation to a $\operatorname{scal}>0$-geometry
with minimal boundary $\partial U$ for some neighborhood $U$ of $\Sigma_{H}$.
When $L^{n-1}=L^{n-1}(U)\equiv\partial U$, the class
$\alpha[2]\cap\alpha[1]\cap[M]$ is trivial, since $L^{n-1}$ is null-homologous
in $H^{n}\setminus U$. For non-trivial $\alpha[2]\cap\alpha[1]\cap[M]$, and
small $U$, we have two possible scenarios for $L^{n-1}$, (i) and (ii),
illustrated in Fig. 1.

Figure 1: $\Sigma_{H}$ has an empty intersection (i) with _some_ hypersurface
in $\alpha[2]\cap\alpha[1]\cap[M]$ or not (ii).

  1. (i)

In many cases one can choose a setup so that the class contains hypersurfaces
disjoint from $\Sigma$ and thus from $\overline{U}$, when $U$ is small enough.
Then we find some representing _compact_ and unconstrained area minimizer
$L^{n-1}\subset H^{n}\setminus\overline{U}$. A typical situation where this
happens is that where $M^{n+1}$ has _geometrically large_ components, like
nearly flat torus summands. Examples are the _positive mass/energy theorems_
where we get such components from a compactification of asymptotically flat
ends [L6, Ch. 6], [L7].

  2. (ii)

Otherwise the minimal factor geometry $(H^{n},d_{\mathcal{S}})$ allows us to
get _intrinsically complete_ local area minimizers $L^{n-1}(U)$ asymptotically
approaching $\partial U$. For shrinking $U$, the $L^{n-1}(U)$ either
approximate or represent $\alpha[2]\cap\alpha[1]\cap[M]$ in coarse homology,
annihilating the infinite spinning of $L^{n-1}$ towards the compact $\partial
U$. We apply Theorem 1 to $\partial U$ and periodically transfer the outcome
to the end of $L^{n-1}$. (The deformation in Theorem 1 are assembled from
local deformations and we can combine the transferred deformations from
$\partial U$ with those on the compact part of $L^{n-1}$ away from these
ends.) As a concrete application, we note that this matches the scheme in
[GL1, Ch. 12] and it shows that _enlargeable manifolds_ cannot admit
$\operatorname{scal}>0$-metrics, cf. [L8].

Another way of applying this scheme is that of creating a smooth replacement
$H_{\circ}^{n}$ for the singular $H^{n}$ that admits a
$\operatorname{scal}>0$-metric. This does not change the scope of the method,
but it fits into the layout of some classical arguments for the lower
dimensional cases.

Variations and Extensions We extend Theorem 1 in two directions. We start with
the case of manifolds and area minimizers with already given boundaries. This
scenario is used, for instance, in arguments in the style of classical
comparison geometry, [GL1], Ch. 12:

Theorem 2 Let $H^{n}$ be a compact area minimizer with boundary $\partial
H\cap\Sigma=\emptyset$ in a $\operatorname{scal}>0$-manifold $M^{n+1}$. Then
there are arbitrarily small neighborhoods $U\cap\partial H=\emptyset$ of
$\Sigma$, so that $H\setminus U$ is conformal to some
$\boldsymbol{\operatorname{scal}>0}$-manifold $X_{U}$ with disjoint boundary
components $\partial X_{0}$ diffeomorphic to $\partial H$ and minimal
$\partial X_{1}$.

The methods equally apply to larger classes of _almost_ minimizers, like
$\mu$-bubbles, levels sets of various geometric flows or horizons of black
holes from Lorentzian geometry. The admissible classes $\cal{H}$ and $\cal{G}$
of almost minimizers are discussed and specified in [L1, Ch. 1.2] and [L4, Ch.
3.1]. We have the following generalization of Theorem 1.

Theorem 3 Let $H^{n}\subset M^{n+1}$ be a compact almost minimizer with
singular set $\Sigma_{H}$ with $\mathcal{S}$-adapted conformal Laplacian
$L_{H}$. Then there are arbitrarily small neighborhoods $U$ of $\Sigma$, so
that $H\setminus U$ is conformal to a
$\boldsymbol{\operatorname{scal}>0}$-manifold $X_{U}$ with minimal boundary
$\partial X_{U}$.

The $\mathcal{S}$-adaptedness of $L_{H}$ means that the principal eigenvalue
of $\langle A\rangle^{-2}\cdot L_{H}$ is positive and, thus, $H$ can be
conformally deformed into a $\operatorname{scal}>0$-minimal factor
$(H,d_{\mathcal{S}})$ by an eigenfunction of $\langle A\rangle^{-2}\cdot
L_{H}$. There is also a boundary version similar to Theorem 2.

###  1.2 Overview of the Argument

Basic Strategy The deformations in our Theorems are similar to those in
classical $\operatorname{scal}>0$-preserving surgeries [GL2], [SY]: for a
$\operatorname{scal}>0$-manifold $M$ and a compact submanifold $N\subset M$ of
codimension $\geq 3$, there are small neighborhoods $U\subset V$ of $N$ so
that $M\setminus U$ can be conformally deformed within $V$ to a
$\operatorname{scal}>0$-metric so that $\partial U$ is a local area minimizer.
This variant is taken from [L5, Ch. 4]. For the subsequent surgery/gluing
processes in [GL2], [SY], which we do not need in our context, one appends
(non-conformal) deformations of $V\setminus U$ to transform $\partial U$ into
a totally geodesic boundary.
In our case, the assumptions in our Theorems mean that the singular (almost)
minimizer $H^{n}\subset M^{n+1}$ admits a conformal deformation to a still
singular but well-behaved $\operatorname{scal}>0$-minimal factor geometry
$(H,d_{\mathcal{S}})$ from [L1]. We take $(H,d_{\mathcal{S}})$ as our initial
$\operatorname{scal}>0$-geometry and let $\Sigma\subset(H,d_{\mathcal{S}})$
play the rôle of $N\subset M$ in the classical smooth theory above. We
construct small neighborhoods $U\subset V$ of $\Sigma$ and a conformal
deformation, this time of $d_{\mathcal{S}}$, supported in $V\setminus\Sigma$,
to another $\operatorname{scal}>0$-metric on $H\setminus U$ so that $\partial
U$ is a local area minimizer. Since $\Sigma$ can be rather complicated and
$H\setminus\Sigma$ degenerates towards $\Sigma$, the construction of this
deformation is broken into simpler _local bump_ deformations of
$(H,d_{\mathcal{S}})$. Each of these local bumps creates a cavity oriented
along some part of $\Sigma$. This gives rise to a local area minimizer $L$,
stretched over the bump, that bounds an open subset $L^{+}\subset H$
containing a non-empty controllable subset $S_{L}\subset\Sigma$, see the right
image of Fig. 2. This is part of a bootstrapping where we use $L$ to show that
the bump _shields_ $S_{L}$ against intersections not only with $L$ but with
further area minimizers in $H$. The class of such minimizers is large enough
to see that we can combine local bumps to deformations shielding all of
$\Sigma$, proving the Theorems.

Figure 2: In this illustration, the initial minimal factors
$(C,d_{\mathcal{S}})$ are entirely flat planes. The singularities are marked
as bluish spots and lines. After bump deformations we get local area
minimizers $L^{n-1}$ = the red rubber bands. In other words, the deformations
shield (parts of) $\Sigma$ from intrusions of $L^{n-1}$.

A Glance at Technical Details We work with the metric measure space
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ associated to the metric completion
$(H,d_{\mathcal{S}})$ of $(H\setminus\Sigma,\Psi_{H}^{4/(n-2)}\cdot g_{H})$,
where $\Psi_{H}>0$ is a $C^{2,\alpha}$-(super)solution, for an
$\alpha\in(0,1)$, with _minimal growth_ towards $\Sigma_{H}$, of

(1) $L_{H,\lambda}(\phi):=L_{H}(\phi)-\lambda\cdot\langle
A\rangle^{2}\cdot\phi:=-\Delta\phi+\frac{n-2}{4(n-1)}\cdot\operatorname{scal}_{H}\cdot\phi-\lambda\cdot\langle
A\rangle^{2}\cdot\phi=0\,\mbox{ on }H\setminus\Sigma,$

for some subcritical eigenvalue $\lambda\in(0,\lambda^{\langle
A\rangle}_{H})$, where $\lambda^{\langle A\rangle}_{H}$ is the principal
eigenvalue of ${\langle A\rangle}^{-2}\cdot L_{H}$, to get a well-controlled
geometry $(H,d_{\mathcal{S}})$. Equation (1) and the value of $\lambda$ remain
invariant under scalings. Under blow-up in any $p\in\Sigma_{H}$, $\Psi_{H}$
induces the unique# solution111 _unique #_ is our abbreviation for _unique up
to multiplication by a positive constant_. $\Psi_{\circ}>0$ of
$L_{C,\lambda}\,\phi=0$ with minimal growth towards $\Sigma_{C}$ on each of
its tangent cones $C$ [L1, Th. 1.5 and 1.6]. From this, the completion
$(C,d_{\mathcal{S}})$ of $(C,\Psi_{\circ}^{4/(n-2)}\cdot g_{C})$ is again a
cone and it is a $\operatorname{scal}>0$_-tangent cone_ of
$(H,d_{\mathcal{S}})$: for large $\tau\gg 1$, it locally approximates
$(H,\tau\cdot d_{\mathcal{S}})$ similarly to ordinary tangent cones for the
original geometry $(H,\tau\cdot d_{H})$. This allows us to construct local
bumps on $(H,d_{\mathcal{S}})$ from cone reduction arguments we indicate in
the following.

Dimension 7: Isolated Singularities We start in dimension $7$ since all almost
minimizers are regular in lower dimensions. The singular set $\Sigma_{H}$ of a
compact singular almost minimizer $H^{7}\subset M^{8}$ contains at most
finitely many singular points $p_{1},...,p_{m}$ and in each
$p_{i}\in\Sigma_{H}$ we have tangent cones $C_{i}^{7}$ approximating
$\tau\cdot H$ near $B_{1}(p_{i})\subset\tau\cdot H$ for large $\tau\gg 1$.
That is, for each $i$, there is a canonical $C^{2,\alpha}$-diffeomorphism
$\mathbf{ID}_{i}:B_{1}(0)\setminus B_{r}(0)\cap C_{i}^{7}\rightarrow
B_{1}(p_{i})\setminus B_{r}(p_{i})\subset\tau\cdot H$, up to minor
modifications near the boundary. $\mathbf{ID}_{i}|_{B_{1}(0)\setminus
B_{r}(0)}$ is nearly an isometry when $\tau$ is large enough, cf. [L1, Ch.
1.3.D.2.], and this allows us the transfer local deformations between $C_{i}$
and $\tau\cdot H$. The $C_{i}^{7}$ are singular only in $0$ and there is some
$\lambda(H)\in(0,\lambda^{\langle A\rangle}_{H})$ so that for
$(\omega,r)\in(\partial B_{1}(0)\cap
C_{i}^{7})\times\mathbb{R}^{>0}=C_{i}^{7}\setminus\\{0\\}$, cf. [L4, Prop.
4.6]:

(2) $\textstyle\Psi_{\circ}(\omega,r)=\psi(\omega)\cdot
r^{\alpha_{\circ}},\mbox{ with
}\textstyle{0>\alpha_{\circ}\geq-(1-\sqrt{\frac{2}{3}})\cdot\frac{7-2}{2}>-\frac{7-2}{2}}$

and for the antagonist of $\Psi_{\circ}$, the solution $\Psi_{\bullet}$ with
_maximal growth_ towards $\\{0\\}$:

(3) $\textstyle\Psi_{\bullet}(\omega,r)=\psi(\omega)\cdot
r^{\alpha_{\bullet}},\mbox{ with
}\textstyle{-\frac{7-2}{2}>-(1+\sqrt{\frac{2}{3}})\cdot\frac{7-2}{2}\geq\alpha_{\bullet}\geq-(7-2)}.$

For balls $B_{\varrho}(0)\subset C_{i}^{7}$, $\varrho>0$, the mean curvature
of $\partial B_{\varrho}$ is negative222Our sign convention is that $\partial
B_{1}(0)\subset(\mathbb{R}^{n},g_{Eucl})$ has _negative_ mean curvature
relative to $\mathbb{R}^{n}\setminus\overline{B_{1}(0)}$.. The critical
exponent $\beta$ for a conformal deformation $(r^{\beta})^{4/7-2}\cdot
g_{C_{i}^{7}}$ to flip the sign of the mean curvature is
$\beta=-\frac{7-2}{2}$ where this deformation yields a
$\operatorname{scal}>0$-cylinder making $\partial B_{\varrho}$ totally
geodesic. Thus the mean curvature of $\partial B_{\varrho}$ relative to
$\Psi_{\circ}^{4/7-2}\cdot g_{C_{i}^{7}}$ is still negative, but relative to
$\Psi_{\bullet}^{4/7-2}\cdot g_{C_{i}^{7}}$ it is positive. We merge these
metrics to a new $\operatorname{scal}>0$-metric, the bumped metric
$g_{\Omega}=(\Phi_{\Delta}\cdot\Psi_{\circ})^{4/7-2}\cdot g_{C_{i}^{7}}$, for
some $C^{2,\alpha}$-regular $\Phi_{\Delta}\geq 1$, cf. the left image of Fig.
2, so that:

  * •

there is a radius $r\in(0,1)$ so that
$\operatorname{supp}|\Phi_{\Delta}-1|\subset\overline{B_{1}(0)}\setminus
B_{r}(0)$,

  * •

there is a $\varrho\in(r,1)$ so that $L^{n-1}:=\partial B_{\varrho}(0)$ is
locally area minimizing relative to $g_{\Omega}$.

We call
$\Delta_{C_{i}^{7}}:=\big{(}(\Phi_{\Delta}-1)\cdot\Psi_{\circ}\big{)}^{4/7-2}\cdot
g_{C_{i}^{7}}$ a local bump pseudo metric333In this paper a pseudo metric is a
symmetric _positive semi-definite_ bilinear form.. Now we use the maps
$\mathbf{ID}_{i}$, for $\tau$ large enough, and transplant such local bumps
from tangent cones to the balls $B_{1}(p_{i})$ in $(H,\tau\cdot
d_{\mathcal{S}})$ adding the $\mathbf{ID}_{i}$-push-forward of the pseudo
metric from $C_{i}$ to $\tau\cdot H$ and get Theorem 1 in dimension $7$.

Dimensions $\boldsymbol{\geq}8$: Localization via Ahlfors regularity and
Isoperimetry As before we want use local bumps. The practical hurdle is that
for a general area minimizer $H^{n}$, $n\geq 8$, the singular set is no longer
a discrete set. A typical example is a product cone $C^{n-1}\times\mathbb{R}$
singular in $\\{0\\}\times\mathbb{R}$ with an $\mathbb{R}$-invariant bump
pseudo metric that shields $\Sigma_{C^{n-1}\times\mathbb{R}}$, cf. the middle
of Fig. 2. These deformations have _non-compact_ support whereas tangent cones
smoothly approximate $(H,\tau\cdot d_{\mathcal{S}})$ only on subsets with
_compact closure_ in $C^{n}\setminus\Sigma_{C}$. This obstructs the transfer
of such deformations from $(C,d_{\mathcal{S}})$ to $(H,\tau\cdot
d_{\mathcal{S}})$ as a whole.
We trim these pseudo metrics on $C^{n}\setminus\Sigma_{C}$ to deformations
with compact support keeping $\operatorname{scal}>0$ to define a local bump.
(This is indicated in the right image of Fig. 2. The proper definition needs a
bit more work illustrated in Fig. 5. We give a separate overview in the
beginning of Ch. 2.) The trade-off is that the (support of) local bumps no
longer topologically separate small neighborhoods of the singularities from
large regular complementary parts of $H$. In view of the singular nature of
the underlying space this raises the question whether such a trimming ruins
the shielding effect of the bumps.
At this point the Ahlfors regularity and the isoperimetric inequality for
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ [L1, Th. 1.11] enter the game. They
compensate this shortcoming and keep the area minimizer $L$ from entering the
cavity surrounded by local bumps. This shielding property of local bumps is
stable enough to survive the transfer from cones to the more general $H$. We
use a suitable covering scheme to place such bumps along $\Sigma_{H}$ while
keeping their support disjoint. As a direct consequence of this disjointness
and the stable shielding property of local bumps the resulting global bump on
$H$ shields all of $\Sigma_{H}$.

Organization of the Paper We focus on Theorem 3 and assume
$\Sigma_{H}\neq\emptyset$. Theorems 1 and 2 are special cases.
In Ch. 1.3, we specify the admissible class of bumped metrics we use in this
paper and we discuss results from [L1] and [L4] that we may apply to such
metrics.
In Ch. 2.1, we construct simple bump pseudo metrics on product cones
$\mathbb{R}^{n-k}\times C^{k}$ shielding the axis
$\mathbb{R}^{n-k}\times\\{0\\}$. Then, in Ch. 2.2-2.4, we apply localizing
modifications to this basic pseudo metric and assemble local bumps on
$\mathbb{R}^{n-k}\times C^{k}$ shielding a cube $Q^{n-k}$ in
$\mathbb{R}^{n-k}\times\\{0\\}$.
In Ch. 3.1-3.2, we prepare the transfer of local bumps to general (almost)
minimizers and, in Ch. 3.3 we place families of local bumps so that the union
of the shielded subsets covers $\Sigma_{H}$.

Acknowledgement The author thanks Matthias Kemper for valuable suggestions
which helped us to improve the exposition.

###  1.3 Area Minimizers in $\Lambda$-Bumped Spaces

We summarize some of the results from [L2]-[L4] and extend them to the class
of controllably bumped metrics we work with. Throughout the paper we exploit
particularities of the conformal Laplacians $L_{H}$ and ${\langle
A\rangle}^{-2}\cdot L_{H}$ for _subcritical_ but positive eigenvalues where we
have a good control over the analysis near the singular set.
For the sake of consistency we oftentimes state results for all almost
minimizers $H\in{\cal{G}}$. As in [L1] this comes with the caveat that for
regular $H\in{\cal{G}}^{c}$ or $H=\mathbb{R}^{n}\in{\cal{H}}^{\mathbb{R}}_{n}$
some statements become trivial/void because there is no singular set or no
subcritical eigenfunction.

Minimal Factors The singular almost minimizer $H\in{\cal{G}}^{c}$ in Theorem 3
has a positive principal eigenvalue $\lambda^{\langle A\rangle}_{H}>0$. This
means that for any $\lambda<\lambda^{\langle A\rangle}_{H}$ and
$L_{H,\lambda}:=L_{H}-\lambda\cdot\langle A\rangle^{2}$ there are positive
solutions of $L_{H,\lambda}\,\phi=0$, cf. [L3]. The coefficients are locally
Lipschitz regular, hence any such solution is $C^{2,\alpha}$-regular for an
$\alpha\in(0,1)$. For some subcritical eigenvalue
$\lambda\in(0,\lambda^{\langle A\rangle}_{H})$, we set, following [L1]:

###### Definition 1.1

_(Minimal Factor Metrics)_ For $H\in{\cal{G}}_{n}$, let $\Psi>0$ be a
$C^{2,\alpha}$-supersolution of $L_{H,\lambda}\phi=0$ on
$H\setminus\Sigma_{H}$ so that in the case

  * •

$H\in{\cal{G}}^{c}_{n}$: $\Psi$ is a solution in a neighborhood of $\Sigma$
with minimal growth towards $\Sigma$.

  * •

$H\in{\cal{H}}^{\mathbb{R}}_{n}$: $\Psi$ is a solution on
$H\setminus\Sigma_{H}$ with minimal growth towards $\Sigma$.

We call the $\operatorname{scal}>0$-metrics $\Psi^{4/(n-2)}\cdot g_{H}$ on
$H\setminus\Sigma$ the minimal factor metrics.

For non-totally geodesic $H\in{\cal{H}}^{\mathbb{R}}_{n}$ there is a unique#
such solution $\Psi_{\circ}$ of $L_{H,\lambda}\,\phi=0$ for subcritical
$\lambda>0$ [L3, Theorem 3]. To any singular $H\in{\cal{G}}^{c}$ we assign one
fixed supersolution $\Psi_{H}$.

Choice of Subcritical Eigenvalues Now we select a subcritical eigenvalue
$\lambda^{\bullet}_{H}$ of ${\langle A\rangle}^{-2}\cdot L_{H}$ to gain the
needed control over the growth rate of a supersolution $\Psi$, as in Def. 1.1,
towards $\Sigma_{H}$. To this end, we consider tangent cones $C^{n}$ obtained
from blow-ups around a singular point of $H^{n}$. The blow-up process can be
iterated in singular points outside $0$. After each blow-up the resulting
minimal cone acquires an additional Euclidean factor. After ${n-k}$ blow-up
steps we reach a singular cone $C^{n}=\mathbb{R}^{n-k}\times
C^{k}\subset\mathbb{R}^{n+1}$, $k\geq 7$. When $\Sigma_{C^{k}}=\\{0\\}$ the
blow-up process terminates.
From [L4, Theorem 3] we know that $\Psi$ induces the unique# solution
$\Psi_{\circ}=\Psi_{\circ}[C^{n}]$ of $(L_{C^{n}})_{\lambda}\phi=0$ on
$C^{n}\setminus\Sigma_{C}$ with minimal growth towards $\Sigma_{C}$. This
uniqueness implies that $\Psi_{\circ}$ shares the
$\mathbb{R}^{n-k}$-translation symmetry with the underlying space
$\mathbb{R}^{n-k}\times C^{k}$. This also means that $(\mathbb{R}^{n-k}\times
C^{k},d_{\mathcal{S}})$ is again a cone invariant under
$\mathbb{R}^{n-k}$-translation and this means that
$\Psi_{\circ}|_{\\{0\\}\times C^{k}}$ satisfies the equation
$(L_{C^{k},n})_{\lambda}\Psi_{\circ}(\omega,r)|_{\\{0\\}\times C^{k}}=0$ where
we have set

(4) $L_{C^{k},n}:=-\Delta-\frac{n-2}{4(n-1)}\cdot|A|^{2}\mbox{ for }n\geq
k\mbox{ and }(L_{C^{k},n})_{\lambda}=L_{C^{k},n}-\lambda\cdot\langle
A\rangle^{2}.$

$L_{C^{k},n}$ is a dimensionally shifted conformal Laplacian on $C^{k}$.
$\Delta$, $|A|^{2}=-\operatorname{scal}$ and $\langle A\rangle^{2}$ are
defined relative to $C^{k}$. The dimensional shift comes from using
$\frac{n-2}{4(n-1)}$ in place of $\frac{k-2}{4(k-1)}$. From $\langle
A\rangle(\omega,\rho)=\langle A\rangle(\omega,1)\cdot\rho^{-1}$ for
$x=(\omega,\rho)\in S_{C^{k}}\setminus\Sigma_{C^{k}}\times\mathbb{R}^{>0}\cong
C^{k}\setminus\\{0\\}$, where $S_{C^{k}}:=\partial B_{1}(0)\cap
C^{k}\setminus\Sigma_{C}$, we have

(5) $(L_{C^{k},n})_{\lambda}\,v=-\frac{\partial^{2}v}{\partial
r^{2}}-\frac{k-1}{r}\cdot\frac{\partial v}{\partial r}+\frac{1}{r^{2}}\cdot
L_{\lambda}^{\times}\,v,\mbox{ where }$ (6)
${L_{\lambda}^{\times}(v)(\omega):=-\Delta_{S_{C^{k}}}v(\omega)+\big{(}-\frac{k-2}{4(k-1)}\cdot|A|^{2}(\omega,1)-\lambda\cdot\langle
A\rangle^{2}(\omega,1)\big{)}\cdot v(\omega)}.$

We recall [L4, Prop. 4.6]:

###### Proposition 1.2

_(Growth Estimates)_ Let $C^{k}\subset\mathbb{R}^{k+1}$ be a singular area
minimizing cone and $n\geq k\geq 7$ where $C^{k}$ can also be singular outside
$\\{0\\}$. Then there is a constant $\lambda^{*}_{k}>0$ such that for any
$n\geq k$ and $\lambda\in(0,\lambda^{*}_{k}]$ there are two distinguished
solutions of $(L_{C^{k},n})_{\lambda}\phi=0$:

(7)
$\Psi_{\circ}[C^{k}](\omega,\rho)=\psi(\omega)\cdot\rho^{\alpha_{\circ}}\mbox{
and
}\Psi_{\bullet}[C^{k}](\omega,\rho)=\psi(\omega)\cdot\rho^{\alpha_{\bullet}},\mbox{
for }(\omega,\rho)\in S_{C^{k}}\times\mathbb{R}^{>0}$

$\Psi_{\circ}$ and $\Psi_{\bullet}$ are the unique# solutions with minimal
growth towards $\Sigma_{C^{k}}$ and $\widehat{\Sigma_{C^{k}}}\setminus\\{0\\}$
and we have some $\vartheta^{*}_{\lambda}>0$ depending only on $\lambda$ and
$k$, but not on $n$, so that

(8) $\textstyle
0>-\vartheta^{*}_{\lambda}>\alpha_{\circ}\geq-(1-\sqrt{\frac{2}{3}})\cdot\frac{k-2}{2}>-\frac{k-2}{2}>-(1+\sqrt{\frac{2}{3}})\cdot\frac{k-2}{2}\geq\alpha_{\bullet}>\vartheta^{*}_{\lambda}-(k-2)$

We interpret $\Psi_{\circ}$, $\Psi_{\bullet}$ as solutions
$\Psi_{\circ}[\mathbb{R}^{n-k}\times C^{k}](x,y)=\Psi_{\circ}(x)$ and
$\Psi_{\bullet}[\mathbb{R}^{n-k}\times C^{k}](x,y)=\Psi_{\bullet}(x)$ of
$(L_{\mathbb{R}^{n-k}\times C^{k}})_{\lambda}\phi=0$ for
$(x,y)\in\mathbb{R}^{n-k}\times C^{k}\setminus\Sigma_{\mathbb{R}^{n-k}\times
C^{k}}$.

The spherical component $\psi(\omega)$ and the radial growth rates
$\alpha_{\circ}$ of $\Psi_{\circ}$ and $\alpha_{\bullet}$ of $\Psi_{\bullet}$
in (7) are related through a spherical eigenvalue equation [L4, Theorem 4.4]
to is the (non-weighted) principal eigenvalue
$\mu_{C^{k},L^{\times}_{\lambda}}$ of the associated operator
$L_{\lambda}^{\times}$ on $S_{C^{k}}$ for $\psi(\omega)>0$ with
$L_{\lambda}^{\times}\,\psi=\mu_{C^{k},L^{\times}_{\lambda}}\cdot\psi,\mbox{
on }S_{C^{k}}\setminus\Sigma_{S_{C^{k}}}.$ In particular we have
$\alpha_{\circ}+\alpha_{\bullet}=-(k-2)$.

We also recall from [L4, Lemma 3.9 and Theorem 4.5] that for any tangent cone
$C$ in some singular point of $H$, we have $\lambda^{\langle
A\rangle}_{H}\leq\lambda^{\langle A\rangle}_{C}$ and there is a constant
$\widetilde{\lambda}_{k}^{*}>0$ so that $\lambda^{\langle
A\rangle}_{C^{k}}\geq\widetilde{\lambda}_{k}^{*}$ for any singular area
minimizing cone $C^{k}$. This suggests the following choice for a subcritical
eigenvalue that respects all constraints and yields the uniform growth
estimates of (8). Let $H^{n}\in{\cal{G}}^{c}_{n}$ be a singular almost
minimizer so that $\langle A\rangle^{-2}\cdot L_{H}$ has principal eigenvalue
$\lambda^{\langle A\rangle}_{H}>0$. Then we define our standard eigenvalue

(9) $\lambda^{\bullet}_{H}:=1/2\cdot\min\\{\lambda^{\langle
A\rangle}_{H},\lambda^{*}_{7},..,\lambda^{*}_{n},\widetilde{\lambda}_{7}^{*},..,\widetilde{\lambda}_{n}^{*}\\}>0.$

For the rest of this paper, on a given $H^{n}$, we choose
$\lambda:=\lambda^{\bullet}_{H}>0$ and a fixed supersolution of
$(L_{H^{n}})_{\lambda}\phi=0$ as in Def. 1.1, also written as $\Psi_{\circ}$,
and we use this $\lambda$ and the induced unique# solutions $\Psi_{\circ}$ of
$(L_{C^{n}})_{\lambda}\phi=0$ on $C^{n}\setminus\Sigma_{C}$ on any of its
(iterated) tangent cones $C^{n}$. Since we keep these choices unchanged, we
generally do not mention $\lambda$ and $\Psi_{\circ}$ explicitly.

Controlled Bumps The metric completion $(H,d_{\mathcal{S}})$ of
$(H\setminus\Sigma,\Psi_{\circ}^{4/(n-2)}\cdot g_{H})$ can be augmented to a
metric measure space $(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$. As indicated in
Ch. 1.2 above, our plan is to add disjointly supported local bump deformations
to $(H,d_{\mathcal{S}})$. This means that all bump deformations in this paper
remain $\boldsymbol{\Lambda}$-equivalent to $(H,d_{\mathcal{S}})$, in the
sense of $\boldsymbol{\Lambda}$-bumped spaces in the following result. From
this, many of the results for $(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$, in
[L1], remain valid for the bumped cases. We define, for some fixed bumping
constant $\Lambda\geq 1$, the class of admissible $\Lambda$-bounded conformal
deformation functions

(10)
${\boldsymbol{{\cal{C}}[H^{n},\Lambda]}}:=\\{f:H^{n}\rightarrow[1,\Lambda^{(n-2)/2}]\,|\,f\in
C^{0}(H^{n})\cap C^{2,\alpha}(H^{n}\setminus\Sigma_{H}),\mbox{ for any
}\alpha\in(0,1)\\}.$

In the following we simplify the notations and only use the index $\Lambda$
for quantities actually depending on $\mathbf{G}\in{\cal{C}}[H,\Lambda]$ since
the estimates only depend on $\Lambda$.

###### Theorem 1.3

_( $\boldsymbol{\Lambda}$-Bumped Spaces)_ For $\Lambda\geq 1$,
$H^{n}\in{\cal{G}}_{n}$, and $\mathbf{G}\in{\cal{C}}[H,\Lambda]$ we have:

  * •

$(\mathbf{G}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{H}$ on $H\setminus\Sigma$ can
be completed to a geodesic metric space $(X^{n},d_{X})$. The space
$(X^{n},d_{X})$ is homeomorphic to $(H,d_{H})$ with singular set
$\Sigma_{X}\cong\Sigma_{H}$ of Hausdorff dimension $\leq n-7$ relative to
$(X^{n},d_{X})$. From this we can write $(X^{n},d_{X})$ as
$(H,d^{\Lambda}_{\mathcal{S}})$.

  * •

The volume $(\mathbf{G}\cdot\Psi_{\circ})^{2\cdot n/(n-2)}\cdot\mu_{H}$ on
$H\setminus\Sigma_{H}$ extends to a Borel measure
$\mu^{\Lambda}_{\mathcal{S}}(E)$ defined by
$\int_{E\setminus\Sigma_{H}}(\mathbf{G}\cdot\Psi_{\circ})^{2\cdot
n/(n-2)}\cdot d\mu_{H}$, for any Borel set $E\subset H$, and we get the metric
measure space $(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ and
we call it a $\boldsymbol{\Lambda}$-bumped space originating from
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$

  * •

$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ is Ahlfors
$\boldsymbol{n}$-regular: there are constants $A,B(H,\Psi_{\circ},\Lambda)>0$
so that:

(11) $A\cdot
r^{n}\leq\mu^{\Lambda}_{\mathcal{S}}(B_{r}(q),d_{\mathcal{S}})\leq B\cdot
r^{n},\mbox{ for any }r\in[0,\mathrm{diam}(H,d^{\Lambda}_{\mathcal{S}}))\mbox{
and any }q\in H.$

For $H\in{\cal{H}}^{\mathbb{R}}_{n}$ the constants only depend on the
dimension and $\Lambda$.

  * •

For any $H\in{\cal{G}}$, there is constant $C_{0}(H,\Psi_{\circ},\Lambda)>0$,
depending only on $n$ and $\Lambda$ for $H\in{\cal{H}}^{\mathbb{R}}_{n}$, so
that for any ball $B\subset(H,d_{\mathcal{S}})$ we have a Poincaré inequality:
for any function $u$ on $H$, integrable on bounded balls, and any upper
gradient $w$ of $u$:

(12) $\textstyle\int_{B}|u-u_{B}|\,d\mu^{\Lambda}_{\mathcal{S}}\leq
C_{0}\cdot\mathrm{diam}(B)\cdot\int_{B}w\,d\mu^{\Lambda}_{\mathcal{S}}.$

The assertions readily follow from their counterparts for minimal factor
geometries in [L1] and the $\Lambda$-equivalence of the original
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ and the bumped spaces
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$. From (11) and
(12) we also get an isoperimetric inequality
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ and the following growth estimates.
Alternatively, one can use the $\Lambda$-equivalence of the metrics to get
such applications from the estimates we have for
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ from [L1, Cor. 3.16 and Prop. 3.17].

###### Corollary 1.4

For $(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$, some open
subset $\Omega\subset H$ and an oriented minimal boundary
$L^{n-1}\subset\Omega$ bounding an open set $L^{+}\subset\Omega$, there are
$\kappa,\kappa^{+}(H,\Psi_{\circ},\Lambda)>0$, so that for $p\in L$:

(13) $\kappa\cdot r^{n-1}\leq\mu^{\Lambda,n-1}_{\mathcal{S}}(L\cap
B_{r}(p))\,\mbox{ and }\,\kappa^{+}\cdot
r^{n}\leq\mu^{\Lambda}_{\mathcal{S}}(L^{+}\cap B_{r}(p)),$

for $r\in[0,(A/B)^{1/n}\cdot\mathrm{dist}(p,\partial\Omega)/4)$ and where
$0<A<B$ are the Ahlfors constants. For $H\in{\cal{H}}^{\mathbb{R}}_{n}$,
$\kappa,\kappa^{+}>0$ depend only on $n$ and $\Lambda$.

From (11) and (12) the BV-theory of Ambrosio [A] and Miranda [M] applies to
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$. The coarea
formula [M, Prop. 4.2] and the Ahlfors $n$-regularity yield growth estimates
for $\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial B_{r}(q))$ in terms of a density
result for the Lebesgue measure $\mu_{1}$ on $\mathbb{R}$, cf. [M, Ch. 5.2]:

###### Corollary 1.5

For any $q\in H$, we define the set of radii where
$\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial B_{r}(q))$ exceeds $c\cdot r^{n-1}$
by $J(q,a,b,c):=\\{r\in[a,b]\,|\,\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial
B_{r}(q))>c\cdot r^{n-1}\\}$, $0\leq a<b$ and $c>0$. We have

(14) $\mu_{1}(J(q,a,b,c))\leq\big{(}B\cdot b-A\cdot a+(n-1)\cdot
B\cdot(b-a)\big{)}/c.$

That is, for large $c>0$ it is increasingly likely that
$\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial B_{r}(q))\leq c\cdot r^{n-1}$ since,
choosing a small $\varepsilon>0$ and $c=n\cdot B/\varepsilon$, we have
$\mu_{1}(J(q,0,b,c))\leq\varepsilon\cdot b=\varepsilon\cdot\mu_{1}([0,b])$.
Using balls with radii $\notin J(q,0,b,c)$ we may exploit the low Hausdorff
dimension of $\Sigma\subset(H,d^{\Lambda}_{\mathcal{S}})$ to get:

###### Corollary 1.6

For $H^{n}\in{\cal{G}}_{n}$ and any $\varepsilon>0$ and $R>0$ there is a
locally finite cover of $\Sigma_{H}$ by balls $B_{r_{i}}(p_{i})$,
$p_{i}\in\Sigma_{H}$, $r_{i}\in(0,R)$, $i\in I$, in
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ so that for $c\geq
2\cdot n\cdot B$:

(15) $\textstyle\sum_{i\in I}\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial
B_{r_{i}}(p_{i}))\leq c\cdot\sum_{i\in I}r_{i}^{n-1}<\varepsilon.$

Similarly, 1.5 shows the otherwise non-obvious finiteness of the perimeter
when we look for area minimizers in
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$. The lower semi-
continuity of perimeters [M, Prop. 3.6] and the compactness of the BV-function
space in the $L^{1}$-function space [M, Prop. 3.7] yield the existence of
oriented minimal boundaries in
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ as in [Gi, Theorem
1.20].

###### Proposition 1.7

_(Plateau Problems in
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$)_ Let
$\Omega\subset H$ be a bounded open and orientable set and let $A\subset H$ be
a possibly empty Caccioppoli set, i.e., a set of locally finite perimeter.
Then there exists a set $E\subset H$ coinciding with $A$ outside $\Omega$ and
such that

(16) $\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial
E\cap\Omega)\leq\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial F\cap\Omega)$

for any Borel set $F\subset H$ with $F=A$ outside $\Omega$. We say that $E$
has $\Omega$-boundary value $A$, and for open $E\cap\Omega$, we call $\partial
E$ a Plateau solution with $\boldsymbol{\Omega}$-boundary value
$\boldsymbol{\partial A}$.

##  2 Constructions on Cones

In this chapter we construct local bumps on product cones. We start with a
more detailed description of the key steps of these deformations and their
intended use.

  1. (1)

Product Bumps In Ch. 2.1 we are starting from some singular cone
$C^{n}=\mathbb{R}^{n-k}\times C^{k}$ with its $\operatorname{scal}>0$-minimal
factor metric $\Psi_{\circ}^{4/(n-2)}\cdot g_{C^{n}}$. We emphasize that
$C^{k}$ can also be singular outside $\\{0\\}$. We make an elementary
deformation $(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$, with
$\mathbf{F}(x,y)=f(|y|)$, for some smooth $f\geq 1$ with
$\operatorname{supp}|f-1|\subset[r,1]$, for some $1>r>0$.
$(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ keeps
$\operatorname{scal}>0$ and it turns $\mathbb{R}^{n-k}\times\partial
B_{\varrho}(0)$, for some $\varrho\in(r,1)$, into a local area minimizer.
$(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ is the source
material for all further steps where we trim and rearrange this deformation to
build larger bump structures.

  2. (2)

Regular Support For the first trimming process, in Ch. 2.2, we define a cut-
off function $\phi_{\beta}:C\setminus\Sigma_{C}\rightarrow[0,1]$ with
$\phi_{\beta}=0$ on $\delta_{\langle A\rangle}^{-1}(\mathbb{R}^{\leq\beta})$
and $\phi_{\beta}=1$ on $\delta_{\langle
A\rangle}^{-1}(\mathbb{R}^{\geq\delta(\beta)})$, for some
$\delta(\beta)\rightarrow 0$, for $\beta\rightarrow 0$. We set
$\mathbf{F}_{\beta}=\phi_{\beta}\cdot(\mathbf{F}-1)+1$ for usage in the new
metric $(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ and
observe that
$\operatorname{supp}|\mathbf{F}_{\beta}-1|\cap\Sigma_{C}=\emptyset$. We think
of $\delta_{\langle A\rangle}^{-1}(\mathbb{R}^{\leq\beta})$, where we have
trimmed $\mathbf{F}-1$ to zero, as a _tunnel_ that surrounds the
singularities. This tunnel passes (and removes) part of the support of
$(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ reaching from
$\mathbb{R}^{n-k}\times\partial B_{1}(0)$ to $\mathbb{R}^{n-k}\times\partial
B_{r}(0)$, cf. Fig. 3. We choose $\delta_{\langle A\rangle}$ in place of the
metric distance since it commutes with the convergence of the underlying
spaces as needed to transfer the bumps between different spaces and it gives
us a sufficiently good control to carry this out keeping
$\operatorname{scal}(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}>0$.

  3. (3)

Tightness of Regular Bumps For
$(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ the boundary
$\mathbb{R}^{n-k}\times\partial B_{\varrho}(0)$ is no longer minimal, but in
Ch. 2.3 we show that for small $\beta>0$, there is an area minimizer
${\cal{T}}^{\beta}$, in this metric which is Hausdorff close to
$\mathbb{R}^{n-k}\times\partial B_{\varrho}(0)$. In particular,
${\cal{T}}^{\beta}\cap\delta_{\langle A\rangle}^{-1}(\mathbb{R}^{\leq\beta})$
remains close to $\delta_{\langle
A\rangle}^{-1}(\mathbb{R}^{\leq\beta})\cap\mathbb{R}^{n-k}\times\partial
B_{\varrho}(0)$ and we will see that this is true for any area minimizer
relative to $(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$
within a certain threshold distance from $\mathbb{R}^{n-k}\times\partial
B_{\varrho}(0)$. In this sense, the tunnel is _tight_ and
$(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$ still
_shields_ a neighborhood of $\mathbb{R}^{n-k}\times\\{0\\}$ against intrusions
of these area minimizers.

  4. (4)

Compact Support In Ch. 2.4 we apply an additional cut-off, now along the
$\mathbb{R}^{n-k}$-axis, to
$(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$, to make the
support compact, cf. the right image of Fig. 2. We call the resulting
deformation of $\Psi_{\circ}^{4/(n-2)}\cdot g_{C^{n}}$ an _(n-k)-bump element_
and they are the elementary particles of our bump construction. In this step
we can preserve $\operatorname{scal}>0$, but it is difficult to prevent
minimizers from shrinking along the $\mathbb{R}^{n-k}$-axis.

  5. (5)

Tight Configurations To resolve this issue, we inductively place ($m$)-bump
elements along the ($m$)-dimensional faces of the unit cube
$Q_{1}^{n-k}\times\\{0\\}\subset\mathbb{R}^{n-k}\times C^{k}$. (We get an
($m$)-bump element writing
$C^{n}=\mathbb{R}^{m}\times(\mathbb{R}^{n-k-m}\times C^{k})$.) In this
process, the cut-off ends of ($m$)-elements belong to subsets shielded from
($q$)-elements for $q<m$, cf. Fig. 5. The tightness (3) ensures that these
bump configurations shield $Q_{1}^{n-k}$ against intrusion of area minimizers.
We call them _local bumps_. They deform $\Psi_{\circ}^{4/(n-2)}\cdot
g_{C^{n}}$ to $\operatorname{scal}>0$-metrics containing open sets $U\supset
Q_{1}^{n-k}\times\\{0\\}$ with locally minimal boundary $\partial U$.

  6. (6)

Stable Shielding The bump elements in local bumps (and in all larger
configurations we build until we reach global bumps) have disjoint supports,
but the generated minimizing boundaries intersect the support of other bump
elements and they change whenever we add further bump elements. In general,
any such addition produces several new locally minimizing boundaries. Some of
them jump far away and they become useless for our purposes, but the central
tightness result in Ch. 2.3 also shows that there is always at least one
minimizing boundary only slightly perturbed compared to the initial one.

###  2.1 Deformations on Product Cones

The main building blocks for our deformations are blends of the two extremal
metrics associated to $\Psi_{\circ}$ and $\Psi_{\bullet}$ on
$\mathbb{R}^{n-k}\times C^{k}$:

(17) $g^{\circ}_{C^{n}}:=\Psi^{4/(n-2)}_{\circ}\cdot g_{C^{n}}\,\mbox{ and
}\,g^{\bullet}_{C^{n}}:=\Psi^{4/(n-2)}_{\bullet}\cdot g_{C^{n}}.$

We compute the mean curvature of distance tubes of
$\mathbb{R}^{n-k}\times\\{0\\}$ relative to these two metrics.

###### Lemma 2.1

_(Bending Effects)_ We consider $u=f(\omega)\cdot r^{\beta}$, for $\omega\in
S_{C^{k}}:=\partial B_{1}(0)\cap C^{k}$,
$r(x):=\mathrm{dist}(x,\mathbb{R}^{n-k}\times\\{0\\})$, for
$x\in\mathbb{R}^{n-k}\times C^{k}$ and some positive $f\in
C^{2,\alpha}(S_{C^{k}}\setminus\Sigma_{S_{C^{k}}},\mathbb{R})$. The mean
curvature $tr\,A_{\mathcal{T}_{\varrho}}$ in regular points of the boundary
$\mathcal{T}_{\varrho}$ of distance tubes $\mathbb{R}^{n-k}\times
B_{\varrho}(0)\cap C^{k}\subset C^{n}$ of $\mathbb{R}^{n-k}\times\\{0\\}$, of
radius $\varrho>0$ relative to $g_{C^{n}}$, is given by

(18) $u^{4/(n-2)}\cdot tr\,A_{\mathcal{T}_{\varrho}}(u^{4/(n-2)}\cdot
g_{C^{n}})=-\left((k-1)+2\cdot(n-1)\cdot\beta/(n-2)\right)\cdot\varrho^{-1}.$

In particular, for the two $\operatorname{scal}>0$-metrics $g^{\circ}_{C^{n}}$
and $g^{\bullet}_{C^{n}}$ on $C^{n}\setminus\Sigma_{C^{n}}$, we get for any
$\varrho>0$: $tr\,A_{\mathcal{T}_{\varrho}}(g^{\circ}_{C^{n}})<0$ and
$tr\,A_{\mathcal{T}_{\varrho}}(g^{\bullet}_{C^{n}})>0$. For the superposition,
we get some $\varrho_{0}(\alpha_{\circ},n,k)>0$ which continuously depends on
$\alpha_{\circ}$ so that

(19)
$tr\,A_{\mathcal{T}_{\varrho}}\big{(}(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
g_{C^{n}}\big{)}>0\mbox{ for }\varrho<\varrho_{0},=0\mbox{ for
}\varrho=\varrho_{0}\mbox{ and }<0\mbox{ for }\varrho>\varrho_{0}.$

$\mathcal{T}_{\varrho_{0}}$ is an area minimizer in the metric measure space
associated to the completion of
$(C^{n}\setminus\Sigma_{C^{n}},(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
g_{C^{n}})$. For any bounded open set $\Omega\subset C^{n}$,
$\mathcal{T}_{\varrho_{0}}\cap\Omega$ is the unique oriented minimal boundary
with $\Omega$-boundary value $\mathcal{T}_{\varrho_{0}}$ bounding an open set
containing $\mathbb{R}^{n-k}\times\\{0\\}$.

Proof The second fundamental form $A_{L}(g)$ of a submanifold $L$ with respect
to a metric $g$ transforms under conformal deformations $g\rightarrow
u^{4/(n-2)}\cdot g$ as follows [Be, 1.163, p. 60]:

(20) $\textstyle A_{L}(u^{4/(n-2)}\cdot
g)(v,w)=A_{L}(g)(v,w)-\frac{2}{n-2}\cdot{\cal{N}}(\nabla u/u)\cdot g(v,w),$

where ${\cal{N}}(\nabla u/u)$ is the normal component of $\nabla u/u$.
$C^{k}\subset\mathbb{R}^{k+1}$ is a cone and
$\mathbb{R}^{n-k}\times\\{z\\}\subset\mathcal{T}_{\varrho}$, $z\in C^{k}$, is
totally geodesic. Thus we have
$tr\,A_{\mathcal{T}_{\varrho}}(g)=-(k-1)/\varrho$. The trace of the second
summand is $2/(n-2)\cdot\beta/\varrho$ multiplied by $n-1$. This yields:

$\Psi^{4/(n-2)}_{\circ}\cdot
tr\,A_{\mathcal{T}_{\varrho}}(\Psi^{4/(n-2)}_{\circ}\cdot
g_{C^{n}})=-\left((k-1)+2\cdot(n-1)\cdot\alpha_{\circ}/(n-2)\right)\cdot\varrho^{-1}<-(k-2)/2\cdot\varrho^{-1}<0,$
$\Psi^{4/(n-2)}_{\bullet}\cdot
tr\,A_{\mathcal{T}_{\varrho}}(\Psi^{4/(n-2)}_{\bullet}\cdot
g_{C^{n}})=-\left((k-1)+2\cdot(n-1)\cdot\alpha_{\bullet}/(n-2)\right)\cdot\varrho^{-1}>(k-2)/2\cdot\varrho^{-1}>0,$

since $k\geq 7$ and, in particular, $k\geq 3$. For
$(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot g_{C^{n}}$ we get

(21) $(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
tr\,A_{\mathcal{T}_{\varrho}}((\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
g_{C^{n}})=$ (22)
$-(k-1)/\varrho+2(n-1)/(n-2)\cdot(\alpha_{\circ}\cdot\varrho^{\alpha_{\circ}-1}+\alpha_{\bullet}\cdot\varrho^{\alpha_{\bullet}-1})/(\varrho^{\alpha_{\circ}}+\varrho^{\alpha_{\bullet}}).$

The mean curvature is constant on each $\mathcal{T}_{\varrho}$. To understand
its sign, we recall that

  * •

$\alpha_{\circ}+\alpha_{\bullet}=-(k-2)$ and $2\cdot\alpha_{\circ}+1+(k-2)>0$,

  * •

$|2(n-1)/(n-2)\cdot\alpha_{\circ}|<k-1$ and
$|2(n-1)/(n-2)\cdot\alpha_{\bullet}|>k-1$.

We multiply (22) by $(\varrho^{\alpha_{\circ}}+\varrho^{\alpha_{\bullet}})$
and $\varrho^{\alpha_{\circ}+1+(k-2)}$ and get:

$-((k-1)+2(n-1)/(n-2)\cdot\alpha_{\circ})\cdot\varrho^{2\cdot\alpha_{\circ}+1+(k-2)}-((k-1)+2(n-1)/(n-2)\cdot\alpha_{\bullet}).$

From this we see that there is a unique $\varrho_{0}(\alpha_{\circ},n,k)>0$
with

$tr\,A_{\mathcal{T}_{\varrho}}\big{(}(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
g_{C^{n}}\big{)}>0\mbox{ for $\varrho<\varrho_{0}$, $=0$ for
$\varrho=\varrho_{0}$ and $<0$ for $\varrho>\varrho_{0}$}.$

Hence, the radial projection in the $C^{k}$-factor
$\pi_{s,\varrho}:\mathcal{T}_{s}\rightarrow\mathcal{T}_{\varrho_{0}}$ strictly
decreases the hypersurface area element for any $\varrho\neq\varrho_{0}$. From
this we see that $\mathcal{T}_{\varrho_{0}}$ is the unique oriented minimal
boundary with boundary value $\mathcal{T}_{\varrho_{0}}$ bounding an open set
containing $\mathbb{R}^{n-k}\times\\{0\\}$.
This carries over to the metric completion of
$(C^{n}\setminus\Sigma_{C^{n}},(\Psi_{\circ}+\Psi_{\bullet})^{4/(n-2)}\cdot
g_{C^{n}})$ since the Hausdorff dimension of
$\Sigma_{C}\setminus\mathbb{R}^{n-k}\times\\{0\\}$ is still $\leq n-7$ from
Theorem 1.3. $\Box$

We use these estimates in our basic bump deformation of the minimal factor
geometry $(C^{n},d_{\mathcal{S}})$. The construction uses two particularities
of $(C^{n},d_{\mathcal{S}})$ while we approach
$\mathbb{R}^{n-k}\times\\{0\\}$.

  * •

A first transition from $\Psi_{\circ}$ to $\Psi_{\bullet}$ uses, in the guise
of $tr\,A_{\mathcal{T}_{\varrho}}(g^{\bullet}_{C^{n}})>0$, that the
codimension of the singular set is $\geq 3$ matching the condition in the
$\operatorname{scal}>0$-preserving surgeries of [GL2], [SY]. This deformation
yields a local area minimizer $\mathcal{T}_{\varrho}$ with
$\mathcal{T}_{\varrho}\cap\mathbb{R}^{n-k}\times\\{0\\}$.

  * •

Then we append a second transition back from $\Psi_{\bullet}$ to
$\Psi_{\circ}$ and return to the geometry $(C,d_{\mathcal{S}})$ near
$\mathbb{R}^{n-k}\times\\{0\\}\subset\Sigma_{C}$. This step has no classical
counterpart in the mentioned $\operatorname{scal}>0$-preserving surgeries. It
uses the scaling invariance of the $\operatorname{scal}>0$-cone
$(C,d_{\mathcal{S}})$.

###### Proposition 2.2

_(Basic Deformations)_ For any singular cone $C^{n}=\mathbb{R}^{n-k}\times
C^{k}\in{\cal{SC}}_{n}$ there are a smooth function
$\mathbf{f}[C]:\mathbb{R}^{>0}\rightarrow\mathbb{R}^{\geq 1}$ and a core
radius $r[n]\in(0,1/100)$, such that for $z=(x,y)\in
C^{n}\setminus\Sigma_{C^{n}}=\mathbb{R}^{n-k}\times(C^{k}\setminus\Sigma_{C^{k}})$
and $\mathbf{F}(x,y):=\mathbf{f}(|y|)$, the metric

$g^{n,n-k}_{C^{n}}:=(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}=(\mathbf{f}(|y|)\cdot\Psi_{\circ}(x,y))^{4/(n-2)}\cdot
g_{C^{n}}\,\mbox{ has the following properties: }$

  1. (i)

$\mathbf{F}=\mathbf{F}[C]$ depends continuously on $C\in\mathcal{SC}_{n}$: for
a converging sequence $C_{i}\rightarrow C$ in $\cal{SC}$ we have a compact
$C^{2,\alpha}$-convergence
$\mathbf{F}[C_{i}]\circ\mathbf{ID}\rightarrow\mathbf{F}[C]$ on
$C\setminus\Sigma_{C}$.

  2. (ii)

$\operatorname{supp}|\mathbf{f}-1|\subset[r,1]$,
$\mathbf{f}\leq\Lambda^{(n-2)/2}$, for some $\Lambda[n]\geq 1$, i.e.
$\mathbf{F}\in{\cal{C}}[H,\Lambda[n]]$,

  3. (iii)

$L_{C^{n}}(\mathbf{F}\cdot\Psi_{\circ})\geq\lambda/4\cdot\langle
A\rangle^{2}\cdot\mathbf{F}\cdot\Psi_{\circ}$ and, in particular,
$\operatorname{scal}(g^{n,n-k}_{C^{n}})>0$,

  4. (iv)

there are constants $\vartheta_{i}[C]\in(r,1)$, $i=1,2$, with
$\vartheta_{1}<\vartheta_{2}$ and some
$\varrho\in(\vartheta_{1},\vartheta_{2})$, all depending continuously on $C$,
so that $\mathcal{T}_{\varrho}=\mathbb{R}^{n-k}\times\partial B_{\varrho}(0)$
is a local area minimizer. For any bounded open set $\Omega\subset C^{n}$,
$\mathcal{T}_{\varrho}\cap\Omega$ is the unique oriented minimal boundary of
some Caccioppoli set in
$\mathbb{R}^{n-k}\times\overline{B_{\vartheta_{4}}(0)}\setminus
B_{\vartheta_{3}}(0)$ with $\Omega$-boundary value $\mathcal{T}_{\varrho}$.

We call
$\Delta^{n,n-k}:=\big{(}(\mathbf{F}-1)\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot
g_{C^{n}}$ the associated bump pseudo metric and $\mathbb{R}^{n-k}\times
B^{k}_{r[n]}(0)\subset\mathbb{R}^{n-k}\times C^{k}$ its core.

Proof We deform $\Psi_{\circ}$ on $C$ consecutively in 3 steps while we
gradually approach the axis $\mathbb{R}^{n-k}\times\\{0\\}$. To simplify the
definition of the deformations we work also with large radii, but since the
setup is scaling invariant we can finally rescale the steps to compose them
seamlessly.

Step 1 ($\Psi_{\circ}\rightarrowtail\Psi_{\bullet}$) We first pass from
$\Psi_{\circ}$ to $\Psi_{\circ}+\Psi_{\bullet}$. We choose a cut-off $\phi\in
C^{\infty}(\mathbb{R},[0,1])$ with $\phi\equiv 1$ on $\mathbb{R}^{\leq 0}$,
$\phi\equiv 0$ on $\mathbb{R}^{\geq 1}$ and $\phi^{\prime}\leq 0$. For large
$D>0$, we set $h_{D}(z):=\phi_{D}(\rho):=\phi(\rho-D)$, where
$\rho=\mathrm{dist}(z,\mathbb{R}^{n-k}\times\\{0\\})$ and get (usually
omitting to write the coordinate $z$)

(23)
$(\Psi_{\circ}(z)+h_{D}(z)\cdot\Psi_{\bullet}(z))^{(n+2)/(n-2)}\cdot\operatorname{scal}\left((\Psi_{\circ}(z)+h_{D}(z)\cdot\Psi_{\bullet}(z))^{4/(n-2)}\cdot
g_{C^{n}}\right)=$
$\textstyle-(\Delta\Psi_{\circ}+h_{D}\cdot\Delta\Psi_{\bullet})+\frac{(n-2)}{4(n-1)}\cdot\operatorname{scal}(g_{C^{n}})\cdot(\Psi_{\circ}+h_{D}\cdot\Psi_{\bullet})-(\Delta
h_{D}\cdot\Psi_{\bullet}+2\cdot\langle\nabla
h_{D},\nabla\Psi_{\bullet}\rangle)$ (24) $=\lambda\cdot\langle
A\rangle^{2}\cdot(\Psi_{\circ}+h_{D}\cdot\Psi_{\bullet})-(\Delta
h_{D}\cdot\Psi_{\bullet}+2\cdot\langle\nabla
h_{D},\nabla\Psi_{\bullet}\rangle)$ (25)
$\geq\psi_{C}(\omega)\cdot\Big{(}\lambda\cdot\rho^{-2}\cdot(\rho^{\alpha_{\circ}}+\phi_{D}\cdot\rho^{\alpha_{\bullet}})-(\phi^{\prime\prime}_{D}+(k-1)/\rho\cdot\phi^{\prime}_{D})\cdot\rho^{\alpha_{\bullet}}-2\cdot\alpha_{\bullet}\cdot\phi^{\prime}_{D}\cdot\rho^{\alpha_{\bullet}-1}\Big{)},$

from $\langle A\rangle(z)\geq 1/\mathrm{dist}(z,\Sigma)$ and
$|\phi^{\prime}_{D}|,|\phi^{\prime\prime}_{D}|\leq c(\phi)$, for some
$c(\phi)>0$, independent of $D$. For large
$D(n,k,\alpha_{\circ},\alpha_{\bullet})>0$, independent of the chosen cone the
expression (25) is _positive_ , since $\lambda>0$,
$\alpha_{\circ}-\alpha_{\bullet}\geq 2\cdot\sqrt{2/3}\cdot\frac{k-2}{2}\geq 3$
and, hence,
$\rho^{\alpha_{\circ}-2}/\rho^{\alpha_{\bullet}}\rightarrow\infty$, for
$\rho\rightarrow\infty$.

Now we pass from $\Psi_{\circ}+\Psi_{\bullet}$ to $\Psi_{\bullet}$. For $E>0$,
$h_{E}(z):=\phi_{E}(\rho):=1-\phi(\rho/E-1)$ we have

$(h_{E}(z)\cdot\Psi_{\circ}(z)+\Psi_{\bullet}(z))^{(n+2)/(n-2)}\cdot\operatorname{scal}\left((h_{E}(z)\cdot\Psi_{\circ}(z)+\Psi_{\bullet}(z))^{4/(n-2)}\cdot
g_{C^{n}}\right)\geq$ (26)
$\psi_{C}(\omega)\cdot\left(\lambda\cdot\rho^{-2}\cdot(\phi_{E}\cdot\rho^{\alpha_{\circ}}+\rho^{\alpha_{\bullet}})-(\phi^{\prime\prime}_{E}+(k-1)/\rho\cdot\phi^{\prime}_{E})\cdot\rho^{\alpha_{\circ}}-2\cdot\alpha_{\circ}\cdot\phi^{\prime}_{E}\cdot\rho^{\alpha_{\circ}-1}\right).$

For small $E(n,k,\alpha_{\circ},\alpha_{\bullet})>0$ this is _positive_.
Namely, $|\phi^{\prime}_{E}|\cdot E,|\phi^{\prime\prime}_{E}|\cdot E^{2}\leq
c^{*}(\phi)$, for some $c^{*}(\phi)>0$ and $\alpha_{\circ}-\alpha_{\bullet}>0$
and, hence, for $\rho\leq E$:
$\rho^{\alpha_{\bullet}-2}/(\rho^{\alpha_{\circ}}\cdot
E^{-2})\geq\rho^{\alpha_{\bullet}-2}/\rho^{\alpha_{\circ}-2}\rightarrow\infty$,
for $\rho\rightarrow 0$. Summarizing we have some $D>10$, an $E\in(0,1/10)$
and a smooth function $\Psi>0$ so that
$\operatorname{scal}(\Psi^{4/(n-2)}\cdot g_{C^{n}})>0$ on
$C^{n}\setminus\Sigma,$ with $\Psi\equiv\Psi_{\circ}$ for $\rho\geq D+1$ and
$\Psi\equiv\Psi_{\bullet}$ for $\rho\leq E.$ We observe that $D$ and $E$ can
be chosen depending continuously on $\alpha_{\circ}$, $\alpha_{\bullet}$ and,
hence, on $C$.

Step 2 ($\Psi_{\bullet}\rightarrowtail c\cdot\Psi_{\circ}$) Now we pass from
$\Psi_{\bullet}$ back to $c\cdot\Psi_{\circ}$, for some $c>0$. Since

  * •

$\psi_{C}(\omega)\cdot\rho^{\alpha_{\bullet}}$ and
$c\cdot\psi_{C}(\omega)\cdot\rho^{\alpha_{\circ}}$ solve
$L_{C^{k},n}\phi=\lambda\cdot\langle A\rangle^{2}\cdot\phi$

  * •

$(E/8)^{\alpha_{\bullet}-\alpha_{\circ}}\cdot\rho^{\alpha_{\circ}}|_{\rho=E}=\rho^{\alpha_{\bullet}}|_{\rho=E}$
and
$(E/8)^{\alpha_{\bullet}-\alpha_{\circ}}\cdot(\rho^{\alpha_{\circ}})^{\prime}|_{\rho=E/8}>(\rho^{\alpha_{\bullet}})^{\prime}|_{\rho=E/8}$,

we can find a smooth interpolation $F>0$, continuously depending on
$\alpha_{\circ}$, $\alpha_{\bullet}$, such that
$F\leq\min\\{(E/8)^{\alpha_{\bullet}-\alpha_{\circ}}\cdot\rho^{\alpha_{\circ}},\rho^{\alpha_{\bullet}}\\}$,
$F(\rho)=E^{\alpha_{\bullet}-\alpha_{\circ}}\cdot\rho^{\alpha_{\circ}}$ for
$\rho<99/100\cdot(E/8)$, $F(\rho)=\rho^{\alpha_{\bullet}}$ for
$\rho>101/100\cdot(E/8)$, with

$L_{C^{k},n}(\psi_{C}(\omega)\cdot
F(r))=-\psi_{C}(\omega)\cdot\left(F^{\prime\prime}+(k-1)/\rho\cdot
F^{\prime}\right)+L_{\lambda}^{\times}\psi(\omega)\cdot F(r)=$ (27)
$\psi_{C}(\omega)\cdot\left(-(F^{\prime\prime}+(k-1)/\rho\cdot
F^{\prime})+\mu_{C^{k},L_{\lambda}^{\times}}\cdot
F\right)\geq\lambda/2\cdot\langle A\rangle^{2}\cdot\psi_{C}(\omega)\cdot
F(r).$

Step 3 ($c\cdot\Psi_{\circ}\rightarrowtail\Psi_{\circ}$) In Step 2 we returned
to $E^{\alpha_{\bullet}-\alpha_{\circ}}\cdot\Psi_{\circ}$. To reach
$\Psi_{\circ}$, we now show how to pass from $c\cdot\Psi_{\circ}$, for
$c:=E^{\alpha_{\bullet}-\alpha_{\circ}}>1$, to $\Psi_{\circ}$ while keeping
$\operatorname{scal}>0$. For $d\in\mathbb{Z}$, we set
$h^{*}_{d}(z):=\phi^{*}_{d}(\rho):=\phi(2^{d}\cdot(\rho-2^{-d}))$. For
$\zeta\in(-1/2,0)$, we have:

$((1+\zeta\cdot
h^{*}_{d})\cdot\Psi_{\circ})^{(n+2)/(n-2)}\cdot\operatorname{scal}\left(((1+\zeta\cdot
h^{*}_{d})\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}\right)$ (28)
$=\lambda\cdot\langle A\rangle^{2}\cdot(1+\zeta\cdot
h^{*}_{d})\cdot\Psi_{\circ}-\zeta\cdot\big{(}\Delta
h^{*}_{d}\cdot\Psi_{\circ}+2\cdot\langle\nabla
h^{*}_{d},\nabla\Psi_{\circ}\rangle\big{)}$

Now we write
$\Psi_{\circ}(\omega,\rho)=\psi(\omega)\cdot\rho^{\alpha_{\circ}}$ and recall
that $\langle A\rangle(\omega,\rho)=\langle A\rangle(\omega,1)\cdot\rho^{-1}$
and, since $|\delta_{\langle A\rangle}(p)-\delta_{\langle A\rangle}(q)|\leq
d_{H}(p,q)$, we have $\langle A\rangle(\omega,1)\geq 1$. Thus (28) is lower
bounded by

(29)
$\,\psi(\omega)\cdot\Big{(}\lambda\cdot\rho^{-2}\cdot(1+\zeta\cdot\phi^{*}_{d})-|\zeta|\cdot|(\phi^{*}_{d})^{\prime\prime}+(k-1)/\rho\cdot(\phi^{*}_{d})^{\prime}|-2\cdot|\zeta\cdot\alpha_{\circ}\cdot(\phi^{*}_{d})^{\prime}|\cdot\rho^{-1}\Big{)}\cdot\rho^{\alpha_{\circ}}.$

From (8) we have negative upper and lower bounds on $\alpha_{\circ}$,
independent of $C$, and, hence, when $|\zeta|\leq\zeta_{0}(d)$, for some small
$\zeta_{0}(d)>0$, (28) is _positive_. $\zeta_{0}(d)$ can actually be chosen
independently of $d$, i.e. $\zeta_{0}=\zeta_{0}(n,k)$. Namely, since
$\Psi_{\circ}$ is unique#, scaling by $2^{d}$ transforms

(30) $(1+\zeta\cdot h^{*}_{d})^{4/(n-2)}\cdot\Psi_{\circ}^{4/(n-2)}\cdot
g_{C^{n}}\mbox{ into }(1+\zeta\cdot
h^{*}_{0})^{4/(n-2)}\cdot\Psi_{\circ}^{4/(n-2)}\cdot g_{C^{n}},$

up to multiplication by a positive constant. (More explicitly, the scaling
effects $(2^{-d}\cdot\rho)^{-2}=2^{2\cdot d}\cdot\rho^{-2}$,
$(\phi^{*}_{d})^{\prime}(2^{-d}\cdot z)\cdot(2^{-d}\cdot\rho)^{-1}=2^{2\cdot
d}\cdot(\phi^{*}_{1})^{\prime}(z)\cdot\rho^{-1}$ and
$(\phi^{*}_{d})^{\prime\prime}(2^{-d}\cdot z)=2^{2\cdot
d}\cdot(\phi^{*}_{1})^{\prime\prime}(z)$ are all the same.) The interiors of
$\operatorname{supp}|\nabla h^{*}_{d_{1}}|$ and $\operatorname{supp}|\nabla
h^{*}_{d_{2}}|$ are disjoint when $d_{1}\neq d_{2}$ and, hence, we can
iteratively multiply by $1+\zeta\cdot h^{*}_{d}$ keeping
$\operatorname{scal}>0$ when $|\zeta|\leq\zeta_{0}(n,k)$:

(31) $\textstyle\operatorname{scal}((\prod^{j}_{d=1}(1+\zeta\cdot
h^{*}_{d})\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}})>0,\mbox{ for any }j\geq
1.$

In turn, we observe that for any fixed $\zeta\in(-1/2,0)$ and $\eta>0$ there
is some $j>0$ so that $\textstyle\prod^{j}_{d=1}(1+\zeta\cdot
h^{*}_{d})\leq\eta$ on $\mathbb{R}^{\leq 2^{-j}}$.
Therefore, there are some $\zeta<0$ and $j\geq 1$ so that (31) holds,
$\prod^{j}_{d=1}(1+\zeta\cdot h^{*}_{d})=1$, for $\rho\geq 1$, and $=1/c$, for
$\rho\leq 2^{-j}$. Since $c=E^{\alpha_{\bullet}-\alpha_{\circ}}>1$ is upper
bounded, there is a large $j(i,n,k)$, independent of $C$, so that we can
choose $\zeta<0$ depending continuously on $c$ and thus on $C$.

Conclusion We scale the constructions in steps 1-3 individually. In step 1 and
2 we rescale $C^{n}$ by $(D+1)^{-1}$ and multiply $\Psi_{\circ}$ by
$(D+1)^{-\alpha_{\circ}}$. Then the deformation starts from $\mathcal{T}_{1}$.
To append step 3, we rescale $C^{n}$ by $E/10$ and multiply $\Psi_{\circ}$ by
$(E/10)^{-\alpha_{\circ}}$. This defines $\mathbf{F}(C)\cdot\Psi_{\circ}(C)$
as (i). The various radii and $\Lambda$ depend on $n$ and $k$, but since there
are only finitely many $k\leq n$ we can drop $k$ choosing common estimates.
From the compactness of $\mathcal{SC}_{n}$ we find some lower bound $r(n)>0$
so that $\operatorname{supp}|\mathbf{f}(C)-1|\subset[r,1]$ and some upper
bound $f\leq\Lambda^{(n-2)/2}$ for any $C\in\mathcal{SC}_{n}$ as claimed in
(ii). We observe that each step still works, with adjusted parameters and
function $\mathbf{F}$, when we replace the dominating term
$\lambda\cdot\langle A\rangle^{2}\cdot\phi$ for $\lambda/2\cdot\langle
A\rangle^{2}\cdot\phi$ in (24), (26) and (28). Then we finally add
$\lambda/4\cdot\langle A\rangle^{2}\cdot\mathbf{F}\cdot\Psi_{\circ}$ and get
$L_{C^{n}}(\mathbf{F}\cdot\Psi_{\circ})\geq\lambda/4\cdot\langle
A\rangle^{2}\cdot\mathbf{F}\cdot\Psi_{\circ}$ and (iii). (iv) follows from
Step 1 and Lemma 2.1. $\Box$

###  2.2 Removal of Intersections with $\Sigma_{C}$

The _metric distance_ $\mathrm{dist}(\cdot,\Sigma)$ has allowed us to find the
area minimizer $\mathcal{T}_{\varrho}$ in Prop. 2.2 and we have
$\operatorname{supp}g^{n,n-k}_{\Delta}\cap\mathbb{R}^{n-k}\times\\{0\\}=\emptyset$.
In general, $C^{k}$ is singular also outside $\\{0\\}$ and
$\operatorname{supp}g^{n,n-k}_{\Delta}\cap\mathbb{R}^{n-k}\times(\Sigma_{C^{k}}\cap\overline{B_{1}(0)}\setminus
B_{r}(0))\neq\emptyset$. An important case is that of $\Delta^{n,m}$, for
$0\leq m<n-k$, on $\mathbb{R}^{n-k}\times C^{k}$ considered as
$\mathbb{R}^{m}\times(\mathbb{R}^{n-k-m}\times C^{k})$, cf. Fig. 3.
We use the _$\mathcal{S}$ -distance_ $\delta_{\langle A\rangle}=1/\langle
A\rangle$ to trim such bump pseudo metrics so that their support no longer
intersects $\Sigma_{C}$ while keeping $\operatorname{scal}>0$. This is a
preparation to be able to combine and transfer local bumps to general
hypersurfaces. Since $\delta_{\langle A\rangle}(z)$ is merely Lipschitz
regular, we also use its Whitney smoothing $\delta_{\langle A\rangle^{*}}$.
$\delta_{\langle A\rangle^{*}}$ has the same coarse geometric properties as
$\delta_{\langle A\rangle}$, cf. [L1, Appendix B]. In _normal coordinates_
$\partial/\partial x_{i}$, $i=1,...,n$, in $z\in C\setminus\Sigma_{C}$, we
have:

(32) $1/c_{*}\cdot\delta_{\langle A\rangle}(z)\leq\delta_{\langle
A\rangle^{*}}(z)\leq c_{*}\cdot\delta_{\langle
A\rangle}(z)\quad\mbox{and}\quad|\partial^{\beta}\delta_{\langle
A\rangle^{*}}/\partial x^{\beta}|(z)\leq c_{**}(\beta)\cdot\delta_{\langle
A\rangle}^{1-|\beta|}(z)$

for constants $c_{*}(n)\geq 1$, $c_{**}(n,\beta)>0$. $\beta$ is a multi-index
for derivatives. In particular,

(33) $|\nabla\delta_{\langle A\rangle^{*}}|(z)\leq c_{\nabla}\mbox{ and
}|\Delta\delta_{\langle A\rangle^{*}}|(z)\leq c_{\Delta}\cdot\delta_{\langle
A\rangle}^{-1}(z)=c_{\Delta}\cdot\langle A\rangle(z),\mbox{ for
}c_{\nabla}(n),c_{\Delta}(n)\geq 1.$

Now we introduce the secondary or $\beta$-trimming deformation.

###### Proposition 2.3

_( $\boldsymbol{\beta}$-Trimming)_ There is a $\beta_{0}(n)\in(0,1)$ so that
for $\beta\in[0,\beta_{0})$ there is a smooth
$\mathbf{F}_{\beta}(x,y):=\mathbf{f}_{\beta}(y)$ depending continuously on $C$
in $C^{2,\alpha}$-norm, with $\mathbf{F}\geq\mathbf{F}_{\beta}\geq 1$ and
$\mathbf{F}_{\beta}\in{\cal{C}}[C^{n},\Lambda[n]]$ and some
$\delta(\beta,n)>\beta$, with $\delta(\beta)\rightarrow 0$ for
$\beta\rightarrow 0$, so that:

  1. (i)

$g^{n,n-k\,|\,\beta}_{C^{n}}=(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}$ equals $g^{[n,n-k]}_{C^{n}}\mbox{ for }\delta_{\langle
A\rangle^{*}}(z)\geq\delta$, $g^{\circ}_{C^{n}}\mbox{ for }\delta_{\langle
A\rangle^{*}}(z)\leq\beta$, i.e. the $\boldsymbol{\beta}$-trimmed bump
$\Delta^{n,n-k\,|\,\beta}:=\big{(}(\mathbf{F}_{\beta}-1)\cdot\Psi_{\circ}(x,y)\big{)}^{4/(n-2)}\cdot
g_{C^{n}}$ is supported on a subset of $\delta^{-1}_{\langle
A\rangle^{*}}(\mathbb{R}^{\geq\beta})$ and
$g^{n,n-k\,|\,0}_{C^{n}}=g^{n,n-k}_{C^{n}}$,

  2. (ii)

$L_{C^{n}}(\mathbf{F}_{\beta}\cdot\Psi_{\circ})\geq\lambda/8\cdot\langle
A\rangle^{2}\cdot\mathbf{F}_{\beta}\cdot\Psi_{\circ}$, in particular,
$g^{n,n-k\,|\,\beta}_{C^{n}}$ has $\operatorname{scal}>0$.

We call _$\delta_{\langle
A\rangle^{*}}^{-1}(\mathbb{R}^{\leq\beta})\cap\operatorname{supp}g^{n,n-k}_{C^{n}}$_
a $\boldsymbol{\beta}$-tunnel.

Proof of 2.3 The idea of the $\beta$-trimming process is the same as in Step 3
of 2.2 where we have deformed $c\cdot\Psi_{\circ}$, for some $c>1$, to
$\Psi_{\circ}$ near $\mathbb{R}^{n-k}\times\\{0\\}$. That process can be
interpreted as a trimming of the support of $(c-1)\cdot\Psi_{\circ}$. Here we
trim $\Delta^{n,n-k}$ towards the remainder of $\Sigma_{C}$, i.e., we
gradually deform $\mathbf{F}-1$ to zero keeping $\operatorname{scal}>0$. We
exploit again that we are using the unique# solution $\Psi_{\circ}>0$ of the
equation
$-\Delta\Psi_{\circ}-\frac{n-2}{4(n-1)}\cdot|A|^{2}\cdot\Psi_{\circ}=\lambda\cdot\langle
A\rangle^{2}\cdot\Psi_{\circ}$.

Figure 3: Illustration of a $\boldsymbol{\beta}$-trimming near $\Sigma_{C}$,
here for $n-k=0$, to get bumps supported in $C\setminus\Sigma_{C}$. In Ch. 2.3
we show that for small $\beta>0$ the resulting $\boldsymbol{\beta}$-tunnels
are still tight enough to shield the core.

For a cut-off function $\phi\in C^{\infty}(\mathbb{R},[0,1])$ with $\phi\equiv
1$ on $\mathbb{R}^{\leq 0}$, $\phi\equiv 0$ on $\mathbb{R}^{\geq 1}$, we
define $h^{\diamondsuit}_{d}(z):=\phi^{\diamondsuit}_{d}(\delta_{\langle
A\rangle^{*}}(z)):=\phi(2^{d}\cdot(\delta_{\langle A\rangle^{*}}(z)-2^{-d}))$,
for $d\in\mathbb{Z}$ and $z\in C\setminus\Sigma_{C}$. We collect some basic
properties $h^{\diamondsuit}_{d}$ where we denote the interior of a set $A$ by
$A^{\circ}$:

  * •

$\operatorname{supp}|\nabla h^{\diamondsuit}_{d}|\subset\delta_{\langle
A\rangle^{*}}^{-1}([2^{-d},2^{1-d}])$ and $\operatorname{supp}|\nabla
h^{\diamondsuit}_{d_{1}}|^{\circ}\cap\operatorname{supp}|\nabla
h^{\diamondsuit}_{d_{2}}|^{\circ}=\emptyset$, for $d_{1}\neq d_{2}$,

  * •

from (33) and $|\phi^{\prime}|,|\phi^{\prime\prime}|\leq c_{\phi}$, for some
$c_{\phi}\geq 1$, we get for $z\in C\setminus\Sigma_{C}$:

(34) $|\nabla h^{\diamondsuit}_{d}|(z)\leq n\cdot 2^{d}\cdot c_{\nabla}\cdot
c_{\phi}\,,\,|\Delta h^{\diamondsuit}_{d}|(z)\leq n\cdot
c_{\phi}\cdot\big{(}2^{2\cdot d}\cdot c^{2}_{\nabla}+2^{d}\cdot
c_{\Delta}\cdot\langle A\rangle(z)\big{)}.$

For $\mathbf{P}_{i}^{j}(\zeta):=\prod^{j}_{d=i}(1+\zeta\cdot
h^{\diamondsuit}_{d})$, where $\zeta<0$ with norm $|\zeta|<1/2$, we have

(35) $\textstyle|\nabla\mathbf{P}_{i}^{j}(\zeta)|(z)\leq|\zeta|\cdot n\cdot
2^{a}\cdot c_{\nabla}\cdot c_{\phi},\mbox{ on }\operatorname{supp}|\nabla
h^{\diamondsuit}_{a}|,$ (36)
$\textstyle|\Delta\mathbf{P}_{i}^{j}(\zeta))|(z)\leq|\zeta|\cdot n\cdot
c_{\phi}\cdot\big{(}2^{2\cdot a}\cdot c^{2}_{\nabla}+2^{a}\cdot
c_{\Delta}\cdot\langle A\rangle(z)\big{)},\mbox{ on
}\operatorname{supp}|\nabla h^{\diamondsuit}_{a}|,$

and for any integers $i\leq a$ and any $\eta>0$ there is an
$j(i,\eta,\zeta)>a$ with

(37) $\textstyle 0<\mathbf{P}_{i}^{j}(\zeta)(z)\leq\eta,\mbox{ for }z\in
C\setminus\Sigma_{C}\mbox{ with }\delta_{\langle A\rangle^{*}}(z)\leq 2^{-j}.$

Step 1 (Cut-Off adapted to $\boldsymbol{\delta_{\langle A\rangle^{*}}}$) For
$\zeta\in(-1/2,0)$ and integers $i<j$, we define

(38) $\textstyle
g^{n,n-k}_{C^{n}}(i,j):=\big{(}(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot
g_{C^{n}}.$

We note that
$g^{n,n-k}_{C^{n}}(i,j)=(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}(z)\cdot
g_{C^{n}}$ for $\delta_{\langle A\rangle^{*}}(z)\geq 2^{1-i}$ and that
$g^{n,n-k}_{C^{n}}(i,j)$ asymptotically approaches
$\Psi_{\circ}^{4/(n-2)}\cdot g_{C^{n}}$ towards $\Sigma$, from (37). We claim
that there is an $\zeta_{0}(n,k)\in(0,1/2)$ so that, for any $\zeta<0$ with
norm $|\zeta|\leq\zeta_{0}(n,k)$ and any pair $i<j$, we have
$\operatorname{scal}(g^{n,n-k}_{C^{n}}(i,j))>0$ on $C\setminus\Sigma_{C}$. As
before, we start from the $\operatorname{scal}$-transformation law:

$\textstyle\big{(}(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\big{)}^{\frac{n+2}{n-2}}\cdot\operatorname{scal}(g^{n,n-k}_{C^{n}}(i,j))=$
$\textstyle-\Delta\big{(}(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\big{)}-\frac{(n-2)}{4(n-1)}\cdot|A|^{2}\cdot\big{(}(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\big{)}$

Using the brackets $[$, $\llbracket$ and $[[$ to group the terms the second
line of this equation becomes

$\textstyle[-\Delta\mathbf{P}_{i}^{j}(\zeta)\cdot\mathbf{F}\cdot\Psi_{\circ}]+[-2\cdot\langle\nabla(\mathbf{P}_{i}^{j}(\zeta),\nabla(\mathbf{F}\cdot\Psi_{\circ})\rangle]+[[-\mathbf{P}_{i}^{j}(\zeta)\cdot\Delta(\mathbf{F}\cdot\Psi_{\circ})]]+...$
$[-\Delta(1-\mathbf{P}_{i}^{j}(\zeta)\cdot\Psi_{\circ}]+[-2\cdot\langle\nabla(1-\mathbf{P}_{i}^{j}(\zeta),\nabla\Psi_{\circ}\rangle]+\llbracket-(1-\mathbf{P}_{i}^{j}(\zeta))\cdot\Delta\Psi_{\circ}\rrbracket+...$
$\textstyle[[-\frac{(n-2)}{4(n-1)}\cdot|A|^{2}\cdot\mathbf{P}_{i}^{j}(\zeta)\cdot\mathbf{F}\cdot\Psi_{\circ}]]+\llbracket-\frac{(n-2)}{4(n-1)}\cdot|A|^{2}\cdot\big{(}1-\mathbf{P}_{i}^{j}(\zeta)\big{)}\cdot\Psi_{\circ}\rrbracket$

  * •

For the sum of the $[$-terms we use
$\nabla(\mathbf{F}\cdot\Psi_{\circ})=\mathbf{F}\cdot\nabla\Psi_{\circ}+\Psi_{\circ}\cdot\nabla\mathbf{F}$
and (34)-(36). The definition of $\mathbf{F}$ in Prop. 2.2 shows that
$\mathbf{F},|\nabla\mathbf{F}|,|\Delta\mathbf{F}|\leq c_{\mathbf{F}}$ on
$C\setminus\Sigma_{C}$, for some $c_{\mathbf{F}}(n,k)>0$. This gives us the
following lower estimate on $\delta_{\langle
A\rangle^{*}}^{-1}([2^{-a},2^{1-a}])$:

(39) $-|\zeta|\cdot n\cdot c_{\phi}\cdot\big{(}2^{2\cdot a}\cdot
c^{2}_{\nabla}+2^{a}\cdot c_{\Delta}\cdot\langle A\rangle(z)\big{)}\cdot
c_{\mathbf{F}}\cdot\Psi_{\circ}-2\cdot n\cdot|\zeta|\cdot 2^{a}\cdot
c_{\nabla}\cdot c_{\phi}\cdot c_{\mathbf{F}}\cdot\Psi_{\circ}-...$ $2\cdot
n\cdot|\zeta|\cdot 2^{a}\cdot c_{\nabla}\cdot c_{\phi}\cdot
c_{\mathbf{F}}\cdot|\nabla\Psi_{\circ}|-|\zeta|\cdot n\cdot
c_{\phi}\cdot\big{(}2^{2\cdot a}\cdot c^{2}_{\nabla}+2^{a}\cdot
c_{\Delta}\cdot\langle A\rangle(z)\big{)}\cdot\Psi_{\circ}-2\cdot
n\cdot|\zeta|\cdot 2^{a}\cdot c_{\nabla}\cdot
c_{\phi}\cdot|\nabla\Psi_{\circ}|.$

  * •

The sum of the $[[$ and $\llbracket$-terms can be lower estimated from
2.2(iii) and $\mathbf{F}\geq 1$:

(40) $\lambda\cdot\langle
A\rangle^{2}\cdot\big{(}1-\mathbf{P}_{i}^{j}(\zeta)\big{)}\cdot\Psi_{\circ}+\lambda/4\cdot\langle
A\rangle^{2}\cdot\mathbf{P}_{i}^{j}(\zeta)\cdot\mathbf{F}\cdot\Psi_{\circ}\geq\lambda/4\cdot\langle
A\rangle^{2}\cdot\Psi_{\circ}.$

Step 2 (Elliptic Estimates relative to $\boldsymbol{\delta_{\langle
A\rangle^{*}}}$) To compare (39) with (40), we estimate $|\nabla\Psi_{\circ}|$
in terms of $|\Psi_{\circ}|$. $\operatorname{supp}|\nabla
h^{\diamondsuit}_{a}|\subset\delta_{\langle
A\rangle^{*}}^{-1}([2^{-a},2^{1-a}])$ is geometrically well-controlled: for
$a=1$ we have $\delta_{\langle
A\rangle^{*}}^{-1}([1/2,1])\subset\delta_{\langle
A\rangle}^{-1}([c^{-1}_{*}/2,c_{*}])$ and thus

(41) $|A|(z)\leq 2\cdot c_{*},\mbox{ for any }z\in\delta_{\langle
A\rangle^{*}}^{-1}([1/2,1])\mbox{ from }\langle A\rangle\geq|A|.$

From Gauß equations this yields a uniform control over the geometry of
$\delta_{\langle A\rangle^{*}}^{-1}([1/2,1])$ even on the neighborhood
$V:=\bigcup\\{B_{1/(2\cdot\langle A\rangle(p)}(p)\,|\,p\in\delta_{\langle
A\rangle}^{-1}([c_{a}^{-1}/2,c_{a}])\\}$ from [L1, Lemma B.2]:

(42) $|\langle A\rangle(p)/\langle A\rangle(q)-1|\leq\langle A\rangle(p)\cdot
d_{H}(p,q),\mbox{ for }q\in B_{1/(2\cdot\langle A\rangle(p)}(p).$

We get for any $\epsilon>0$ an $r(n,\epsilon)>0$ so that for
$x\in\delta_{\langle A\rangle^{*}}^{-1}([1/2,1])$ the exponential map
$\exp_{x}:B_{r(n,\epsilon)}(0)\rightarrow C\setminus\Sigma_{C}$ is a local
diffeomorphism with
$|\exp^{*}_{x}(g_{C})-g_{T_{x}C}|_{C^{5}(B_{r(n,\eta)}(0))}\leq\epsilon$.
$g_{T_{x}C}$ is the flat metric on the tangent space $T_{x}C$ in $x$.
The coefficients of
$-\Delta\Psi_{\circ}-\frac{n-2}{4(n-1)}\cdot|A|^{2}\cdot\Psi_{\circ}=\lambda\cdot\langle
A\rangle^{2}\cdot\Psi_{\circ}$ are uniformly bounded in $C^{\alpha}$-norm on
any $B_{r(n,\eta)}(x)$, $\alpha\in(0,1)$, hence we get uniform constants for
the Harnack inequality and elliptic estimates for $\Psi_{\circ}$ on
$B_{r(n,\eta)}(x)$ independent of $C$, i.e., there is some $c_{\Box}(n,k)>0$
with

(43) $|\nabla\Psi_{\circ}|\leq c_{\Box}\cdot\Psi_{\circ}\mbox{ on }V\mbox{
and, thus, }|\nabla\Psi_{\circ}|\leq c_{\Box}\cdot
2^{a}\cdot\Psi_{\circ}\mbox{ on }2^{-a}\cdot V,$

where we have also used that $\Psi_{\circ}>0$ is the unique# solution on
$C\setminus\Sigma_{C}$. Since $\delta_{\langle A\rangle_{c\cdot
H}}=c\cdot\delta_{\langle A\rangle}$, for any $c>0$, and thus $c\cdot
c^{-1}_{*}\cdot\delta_{\langle A\rangle^{*}}\leq\delta_{\langle
A\rangle^{*}_{c\cdot H}}\leq c\cdot c_{*}\cdot\delta_{\langle A\rangle^{*}}$,
we have

(44) $\delta_{\langle
A\rangle^{*}}^{-1}([2^{-a},2^{1-a}])\subset\delta_{\langle
A\rangle}^{-1}([c^{-1}_{*}\cdot 2^{-a},c_{*}\cdot 2^{1-a}])=2^{-a}\cdot V.$

From (43) we find some $\zeta_{0}(1)\in(0,1/2)$ so that for $\zeta<0$ with
$|\zeta|\leq\zeta_{0}$, the norm of (39) is smaller that
$\lambda/16\cdot\langle A\rangle^{2}\cdot\Psi_{\circ}$ on $V$. As in 2.2, we
observe that, due to the uniqueness of $\Psi_{\circ}$, we have, under scaling
by $2^{a}$, that, up to a constant multiple

(45) $\Psi_{\circ}\mbox{ and }h^{\diamondsuit}_{a}\mbox{ on }2^{-a}\cdot
V\mbox{ transform into }\Psi_{\circ}\mbox{ and }h^{\diamondsuit}_{1}\mbox{ on
}V.$

From (44), (43) and $\langle A\rangle_{2^{-a}\cdot H}=2^{a}\cdot\langle
A\rangle_{H}$ we therefore see that, for $a>0$ and $|\zeta|\leq\zeta_{0}$, the
norm of (39) is smaller than $\lambda/100\cdot\langle
A\rangle^{2}\cdot\Psi_{\circ}$ on $2^{-a}\cdot V$, i.e. $\zeta_{0}$ can be
chosen to depend on $n$ and $k$ and not on $a$. This shows that for $\zeta<0$
with $|\zeta|\leq\zeta_{0}$ and $0<i\leq j$ on $C\setminus\Sigma_{C}$:

(46) $\textstyle
L_{C^{n}}(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\geq\lambda/6\cdot(\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}.$

Step 3 (Iterated Cut-Offs) We use this $\zeta_{0}(n,k)$ and some $\zeta<0$
with small enough $|\zeta|\leq\zeta_{0}$ and, from (37), some sufficiently
large $j$ so that the trimmed pseudo metric

(47)
$\textstyle\big{(}((1-h^{\diamondsuit}_{j})\cdot\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1)\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot
g_{C^{n}}$

has $\operatorname{scal}>0$ on $C\setminus\Sigma_{C}$. Namely, we consider the
transformation under scaling by $2^{j}$:

(48) $\textstyle\mathbf{P}_{i}^{j}(\zeta)\mbox{ on }2^{-j}\cdot V\mbox{
transform into }\mathbf{P}_{i-(j-1)}^{1}(\zeta)(1+\zeta\cdot
h^{\diamondsuit}_{d})\mbox{ on }V$

and this uniformly $C^{3}$-converges to zero on $V$, when
$j\rightarrow\infty$. From this a suitably large $j(i,n,k)$ can be chosen so
that the metric (47) has $\operatorname{scal}>0$ on $C\setminus\Sigma_{C}$.
To choose the parameter $\delta(\beta)>0$, we start with $i=1$ and $j(1)$ and
set $\beta_{0}(n,k):=2^{-j(1)}$. For any $\beta\in(0,\beta_{0}(n,k))$, we
therefore have some $\delta(\beta)\leq 1$ as claimed. For
$2^{-j(i+1)}\leq\beta<2^{-j(i)}$, we choose $\delta(\beta):=2^{-i}$ and we see
that $\delta(\beta)\rightarrow 0$ for $\beta\rightarrow 0$. In turn, we
associate these $i$ and $j(i+1)$ to any $\beta$ with
$2^{-j(i+1)}\leq\beta<2^{-j(i)}$ and we set

(49)
$\textstyle\mathbf{F}_{\beta}:=((1-h^{\diamondsuit}_{j})\cdot\mathbf{P}_{i}^{j}(\zeta)\cdot(\mathbf{F}-1)+1).$

As before, we can drop the dependency on $k$ to simplify our notations. From
the discussion above we see that for $\mathbf{F}_{\beta}$ we have
$L_{C^{n}}(\mathbf{F}_{\beta}\cdot\Psi_{\circ})\geq\lambda/8\cdot\langle
A\rangle^{2}\cdot\mathbf{F}_{\beta}\cdot\Psi_{\circ}$ and it satisfies the
other asserted properties directly from the construction. $\Box$

###  2.3 Tightness of Cavities

We show that there are area minimizers $\mathcal{T}^{\beta}$ relative to the
trimmed bump metric
$g^{n,n-k\,|\,\beta}_{C^{n}}=(\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}$ and, as a part of the argument, we also prove that
$\mathcal{T}^{\beta}$ Hausdorff converges to $\mathcal{T}_{\varrho}$, for
$\beta\rightarrow 0$, where $\mathcal{T}_{\varrho}$ is the minimizer relative
to $g^{[n,n-k]}_{C^{n}}$ of Prop. 2.2.
In our assembly of global bumps, we superpose disjointly supported families of
trimmed bumps to $(C,d_{\mathcal{S}})$ and later to $(H,d_{\mathcal{S}})$, for
$H\in{\cal{G}}^{c}$, cf. Fig. 5. Therefore we consider a more general
$\Lambda$-bumped situation and include the following type of additional
deformations:

(50)
$(\mathbf{G}_{\beta}\cdot\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}\mbox{ with
}\operatorname{supp}|\mathbf{G}_{\beta}-1|\cap\operatorname{supp}|\mathbf{F}_{\beta}-1|=\emptyset\mbox{
and }\mathbf{G}_{\beta}\in{\cal{C}}[C^{n},\Lambda[n]].$

The $\mathbf{G}_{\beta}$ are the placeholders for further local bumps. They do
_not_ need to be translation $\mathbb{R}^{n-k}$-invariant and we note from
$\operatorname{supp}|\mathbf{G}_{\beta}-1|\cap\operatorname{supp}|\mathbf{F}_{\beta}-1|=\emptyset$
that
$\mathbf{G}_{\beta}\cdot\mathbf{F}_{\beta}\in{\cal{C}}[C^{n},\Lambda[n]]$. We
denote the associated hypersurface and volume measures by
$\mu^{\Lambda,n-1}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}]$ and
$\mu^{\Lambda}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}]$, where
$[\beta,\mathbf{G}_{\beta}]=[0,1]$ means the case of the untrimmed metric
$(\mathbf{F}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}$.

Local versus Global Minimizers Now we consider local area minimizers in the
metric measure space
$(C,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$, of Theorem 1.3,
associated to
$(\mathbf{G}_{\beta}\cdot\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}$. The attribute _local_ means that these minimizers are area
minimizing under, not quantitatively specified, small perturbations. The
hypersurfaces $T_{\varrho}$ in Prop. 2.2 or the minimal boundaries in our main
Theorems belong to this class. In some cases we can specify an open test set
$\Omega\subset(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ so
that the area minimizer becomes a _global_ area minimizer when compared to all
other hypersurfaces within $\Omega$ and with the same (Plateau) boundary data
along $\partial\Omega$. In the BV-approach, $\partial\Omega$ also serves as an
_obstacle_ that keeps interior points of these minimizers to stay in
$\overline{\Omega}$, cf. [Gi, Rm. 1.22]. We choose the auxiliary

(51) $\Omega:=Q_{\ell}^{n-k}\times
C^{k},Q_{\ell}^{n-k}:=(-\ell/2,\ell/2)^{n-k},\ell\geq 1\mbox{ and
}O_{\rho}:=\mathbb{R}^{n-k}\times B_{\rho}(0),\rho>0,$

to get existence results and, in a second step, we check that the minimizers
actually avoid $\partial\Omega$ away from their Plateau boundary. Concretely,
for small $\beta$ we get the existence of global minimizers in a suitably
chosen
$\Omega\subset(C,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ close
to the unique $T_{\varrho}$ we have for $\beta=0$. For $\beta>0$, and even in
$\Omega$, such global minimizers need not to be unique, but we observe a
stability property: all competing global area minimizers, relative to
$\Omega$, stay close to $T_{\varrho}$.

###### Proposition 2.4

_(Tightness of $\beta$-Tunnels)_ There is some $\beta^{*}[n,\ell]\in(0,1)$, so
that for $\beta\in[0,\beta^{*}]$, $\mathbf{F}_{\beta}$ from Prop. 2.3, and any
$\mathbf{G}_{\beta}\in{\cal{C}}[C^{n},\Lambda[n]]$ with
_$\operatorname{supp}|\mathbf{G}_{\beta}-1|\cap\operatorname{supp}|\mathbf{F}_{\beta}-1|=\emptyset$_
, we have for $\vartheta_{i}[n,C]\in(r,1)$,
$\varrho\in(\vartheta_{1},\vartheta_{2})$ from Prop. 2.2 and
$\zeta=1/4\cdot\min\\{|\varrho-\vartheta_{1}|,|\varrho-\vartheta_{2}|\\}$:

  1. (i)

there is an open Caccioppoli set $U[n-k,\ell,\mathbf{G}_{\beta}]\subset C^{n}$
with $\overline{O_{\varrho-\zeta}}\subset U\subset O_{\varrho+\zeta}$ so that
$\mathcal{T}[n-k,\ell,\mathbf{G}_{\beta}]:=\partial U$ and
$\mathcal{T}\cap\Omega$ is area minimizing with $\Omega$-boundary value
$\mathcal{T}_{\varrho}$, relative to
$(C,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ associated to
$(\mathbf{G}_{\beta}\cdot\mathbf{F}_{\beta}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}$,

  2. (ii)

for any such oriented minimal boundary $\mathcal{T}_{\bullet}:=\partial
U_{\bullet}\subset\overline{O_{\varrho+2\cdot\zeta}}\setminus
O_{\varrho-2\cdot\zeta}$ with the obstacle
$\partial(O_{\varrho+2\cdot\zeta}\setminus O_{\varrho-2\cdot\zeta})$, we
already have $\mathcal{T}_{\bullet}\subset
O_{\varrho+\zeta}\setminus\overline{O_{\varrho-\zeta}}$. We write
$\mathcal{P}[n-k,\ell,\mathbf{G}_{\beta}]$ for the class of all such Plateau
solutions $\mathcal{T}_{\bullet}$.

In the case $k=n$, $\ell$ is a void parameter, but we keep writing it for a
consistent notation. In turn, for known dimension of the Euclidean factor we
also drop writing $n-k$.

Proof We consider the auxiliary _obstacle problem_ for open Caccioppoli sets
$U$

(52) $\mathbb{R}^{n-k}\times B_{\vartheta_{1}}(0)\subset\mathbb{R}^{n-k}\times
B_{\varrho-2\cdot\zeta}(0)\subset U\subset\mathbb{R}^{n-k}\times
B_{\varrho+2\cdot\zeta}(0)\subset\mathbb{R}^{n-k}\times B_{\vartheta_{2}}(0).$

To be able to use convergence arguments, we ensure compactness properties for
the obstacles. For $n=k$ any such $\partial
U\subset\overline{B_{\vartheta_{2}}(0)}\setminus B_{\vartheta_{1}}(0)$ is
compact. For $n>k$ we choose the compact set
$\overline{Q_{\ell}^{n-k}}\times\overline{B_{\vartheta_{2}}(0)}\setminus
B_{\vartheta_{1}}(0)\subset\Omega$. We let $\mathcal{T}\subset\Omega$ be an
oriented minimal boundary $\mathcal{T}=\partial U$ with $\Omega$-boundary
value $\mathcal{T}_{\varrho}$ solving the obstacle problem (52). We observe
that $Q_{\ell}^{n-k}$ is an intersection of Euclidean halfspaces and thus
$\partial\Omega$ is locally outer minimizing, that is, its area increases
under local outward deformations, cf. the argument of 2.8. That is, a _free_
Plateau solution with $\Omega$-boundary value $\mathcal{T}_{\varrho}$ does not
leave $\overline{\Omega}$. We claim that $\mathcal{T}$ is also a free solution
in the radial direction if $\beta>0$ is small enough and we even show that
$\mathcal{T}[\ell,\mathbf{G}_{\beta}]\subset
O_{\varrho+\zeta}\setminus\overline{O_{\varrho-\zeta}}$.

Step 1 We define the $\Omega$_-flat norm_ as the
$\mu^{\Lambda}_{\mathcal{S}}[0,1]$-volume of the difference set
$(U\,\Delta\,O_{\varrho})\cap\Omega$. In this step we show that for the
oriented minimal boundaries $\mathcal{T}=\partial U$,

(53)
$\mathcal{T}[\ell,\mathbf{G}_{\beta}]\rightarrow\mathcal{T}_{\varrho}\mbox{ in
}\Omega\mbox{-flat norm, for }\beta\rightarrow 0,$

uniformly for all singular area minimizing cones $C^{n}=\mathbb{R}^{n-k}\times
C^{k}$ and for all admissible $\mathbf{G}_{\beta}$. From the naturality of
$\langle A\rangle_{H}$ on $\cal{G}$ and its properness on
$H\setminus\Sigma_{H}$, for any non-totally geodesic $H\in\cal{G}$, we get
some constant $\nu(\beta,\Omega)>0$, independent of $\mathbb{R}^{n-k}\times
C^{k}$, so that

(54) $\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1]\big{(}\delta^{-1}_{\langle
A\rangle^{*}}(\mathbb{R}^{\leq\delta(\beta)})\cap\mathcal{T}_{\varrho}\cap\Omega\big{)}\leq\nu\mbox{
and }\nu(\beta,\Omega)\rightarrow 0\mbox{ for }\beta\rightarrow 0.$

The minimality of the $\mathcal{T}$ and of $\mathcal{T}_{\varrho}$ and
$\mathbf{G}_{\beta}\geq 1$ show that

(55)
$\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1]\big{(}\mathcal{T}_{\varrho}\cap\Omega\big{)}\leq\mu^{\Lambda,n-1}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}]\big{(}\mathcal{T}\cap\Omega\big{)}\leq\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1]\big{(}\delta^{-1}_{\langle
A\rangle^{*}}(\mathbb{R}^{\geq\delta})\cap\mathcal{T}_{\varrho}\cap\Omega\big{)}+\Lambda^{n-1}\cdot\nu.$

Since
$\mu^{\Lambda,n-1}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}](\mathcal{T}\cap\Omega)\geq\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1](\mathcal{T}\cap\Omega)$,
we infer that for $\beta\rightarrow 0$,

(56)
$\mu^{\Lambda,n-1}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}](\mathcal{T}[\ell,\mathbf{G}_{\beta}]\cap\Omega)\mbox{
and
}\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1](\mathcal{T}[\ell,\mathbf{G}_{\beta}]\cap\Omega)\rightarrow\mu^{\Lambda,n-1}_{\mathcal{S}}[0,1](\mathcal{T}_{\varrho}\cap\Omega).$

From this, the BV-compactness relative to $\mu^{\Lambda}_{\mathcal{S}}[0,1]$,
[M, Prop. 3.7], shows that for any sequence $\beta_{i}\rightarrow 0$, for
$i\rightarrow\infty$, there is a $\Omega$-flat norm converging subsequence of
the $\mathcal{T}[\ell,\mathbf{G}_{\beta_{i}}]\cap\Omega$. From the lower
semicontinuity of the perimeter [M, Prop. 3.6] and the uniqueness of
$\mathcal{T}_{\varrho}$ we infer that the limit is $\mathcal{T}_{\varrho}$.
Another loop of the argument shows the convergence of the entire sequence
$\mathcal{T}[\ell,\mathbf{G}_{\beta_{i}}]\cap\Omega$,

(57)
$\mu^{\Lambda}_{\mathcal{S}}[0,1]\big{(}(U[\ell,\mathbf{G}_{\beta_{i}}]\,\Delta\,O_{\varrho})\cap\Omega\big{)}\rightarrow
0\mbox{ for }i\rightarrow\infty.$

From compactness results for minimal factor geometries on cones and from the
continuous dependence of $\mathbf{F}_{\beta}$ and $\varrho$ on $C$, this
convergence is uniform for all $C^{n}=\mathbb{R}^{n-k}\times C^{k}$ and
$\mathbf{G}_{\beta}$.

Step 2 We apply the $\mu^{\Lambda}_{\mathcal{S}}$-growth estimate of Corollary
1.4 (13) to the difference set to upgrade the _flat norm_ convergence from
step 1 to _Hausdorff_ convergence.
If
$\overline{U[\ell,\mathbf{G}_{\beta_{i}}]}\cap\mathbb{R}^{n-k}\times\partial
B_{\varrho+\zeta}(0)\neq\emptyset$ for all $i$, then there is a
$p_{i}\in(U[\ell,\mathbf{G}_{\beta_{i}}]\,\Delta\,O_{\varrho})\cap\mathbb{R}^{n-k}\times\partial
B_{\varrho+\zeta/2}(0)$. We have
$B_{\zeta/4}(p_{i})\cap\mathbb{R}^{n-k}\times\partial
B_{\varrho}(0)=\emptyset$, $p_{i}\in\overline{U[\ell,\mathbf{G}_{\beta_{i}}]}$
and the $\mathcal{T}[\ell,\mathbf{G}_{\beta_{i}}]$ are (global) area
minimizers relative to the test set $B_{\zeta/4}(p_{i})\subset\Omega$. Thus,
Cor. 1.4 (13) shows

(58)
$\kappa^{+}\cdot(\zeta/8)^{n}\leq\mu^{\Lambda}_{\mathcal{S}}[\beta,\mathbf{G}_{\beta}](U[\ell,\mathbf{G}_{\beta_{i}}]\cap
B_{\zeta/8}(p_{i}))\leq\Lambda[n]^{n-1}\cdot\mu^{\Lambda}_{\mathcal{S}}[0,1]\big{(}(U[\ell,\mathbf{G}_{\beta_{i}}]\,\Delta\,O_{\varrho})\cap\Omega\big{)}.$

This contradicts (57). For
$\overline{U[\ell,\mathbf{G}_{\beta_{i}}]}\cap\mathbb{R}^{n-k}\times\partial
B_{\varrho-\zeta}(0)=\emptyset$ we argue similarly. Since only
$1\leq\mathbf{G}_{\beta}\leq\Lambda$ for $\mathbf{G}_{\beta}$ were used,
compactness arguments for the space of all singular cones
$C^{n}=\mathbb{R}^{n-k}\times C^{k}$ with their minimal factor geometry show
that there is some $\beta^{*}[n,\ell]>0$, independent of $C^{n}$ and
$\mathbf{G}_{\beta}$, so that for any $\beta\in[0,\beta^{*}]$:
$\mathcal{T}[\ell,\mathbf{G}_{\beta}]\subset
O_{\varrho+\zeta}\setminus\overline{O_{\varrho-\zeta}}$ proving claim (i). The
argument also implies (ii) since it applies to all such area minimizers.
$\Box$

###  2.4 Local Bumps and Shields

The $\beta$-trimmed bump pseudo metrics $\Delta^{n,n-k\,|\,\beta}$ are
supported away from $\Sigma_{C}$. This is one of the requirements to transfer
such bumps from $C$ to other hypersurfaces. Since tangent cone approximations
generally only work on bounded subsets of the cone, we additionally need to
trim the bump pseudo metrics so that their support becomes _compact_ in
$C\setminus\Sigma_{C}$.
For this we inductively use the $\Delta^{n,q\,|\,\beta}$, for all $q$ starting
from $q=0$ up to $q=n-k$, on
$C^{n}=\mathbb{R}^{q}\times(\mathbb{R}^{n-k-q}\times
C^{k})=\mathbb{R}^{n-k}\times C^{k}$. All of the $\Delta^{n,q\,|\,\beta}$ are
defined on the same cone $\mathbb{R}^{n-k}\times C^{k}$, but for $q<n-k$, the
definition of $\Delta^{n,q\,|\,\beta}$ ignores the
$\mathbb{R}^{n-k-q}$-symmetry and treats $\mathbb{R}^{n-k-q}\times C^{k}$ as
the cone factor. $\Delta^{n,0\,|\,\beta}$ already has compact support.

Figure 4: The illustration shows $g^{n,0\,|\,\beta}_{C^{n}}$ for
$\Sigma_{C^{n}}=\mathbb{R}^{n-k}\times\\{0\\}$ where the $\beta$-trimming
ignores the $\mathbb{R}^{n-k}$-translation symmetry of $C^{n}$. By contrast,
$g^{n,n-k\,|\,\beta}_{C^{n}}$ looks like the middle image of Fig. 2 without
$\beta$-tunnels.

###### Definition 2.5

_(Bump Elements)_ For $q=0,...n-k$, $\ell\geq 1$ and $\phi\in
C^{\infty}(\mathbb{R},[0,1])$ with $\phi\equiv 1$ on $\mathbb{R}^{\leq 0}$,
$\phi\equiv 0$ on $\mathbb{R}^{\geq 1}$, we set
$\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}:=\phi\left(\mathrm{dist}\left(z,Q_{\ell}^{q}\times\mathbb{R}^{n-k-q}\times
C^{k}\right)\right)\cdot(\mathbf{F}_{\beta}-1)+1$:

  * •

$\Delta^{n,q\,|\,\beta,\ell}:=\big{(}\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}-1\big{)}^{4/(n-2)}\cdot
g_{C^{n}}$ is the (q)-bump element aligned to $Q_{\ell}^{q}$,

  * •

$\big{(}\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot
g_{C^{n}}$ is the associated bumped metric,

  * •

_$\operatorname{cut}\Delta^{n,q\,|\,\beta,\ell}:=\operatorname{supp}\Delta^{n,q\,|\,\beta,\ell}\cap\\{z\in
C^{n}\,|\,0 <\mathrm{dist}\left(z,Q_{\ell}^{q}...\right)<1\\}$_ is the cut-off
region.

For our purposes we can think of $\mathrm{dist}$ as a _smooth_ function. As in
Ch. 2.2, we may choose Whitney smoothings of $\mathrm{dist}$ whenever needed.

###### Remark 2.6

($\boldsymbol{\operatorname{scal}}$ on
$\boldsymbol{\operatorname{cut}\Delta}$) On
$C\setminus(\operatorname{cut}\Delta\cup\Sigma_{C})$ the bumped metric
coincides with $g^{n,n-k\,|\,\beta}_{C^{n}}$ respectively with
$\Psi_{\circ}^{4/(n-2)}\cdot g_{C^{n}}$. In both case we have
$L_{C^{n}}(\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ})\geq\lambda/8\cdot\langle
A\rangle^{2}\cdot\mathbf{F}^{\phi,Q_{\ell}^{m}}_{\beta}\cdot\Psi_{\circ}$. In
turn, we anticipate that $\operatorname{cut}\Delta$ will be inductively
shielded by other bump elements, i.e. it belongs to the finally _deleted_
neighborhood $U$ of $\Sigma$ with minimal boundary $\partial U$ of $H\setminus
U$ we described in our theorems. This makes knowing $\operatorname{scal}$ on
$\operatorname{cut}\Delta$ dispensable. Even so, to avoid exceptional subsets
in our further discussion we indicate how to ensure that on
$\operatorname{cut}\Delta$:

(59)
$L_{C^{n}}(\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ})\geq\lambda/10\cdot\langle
A\rangle^{2}\cdot\mathbf{F}^{\phi,Q_{\ell}^{m}}_{\beta}\cdot\Psi_{\circ}\mbox{
and, hence,
}\operatorname{scal}(\big{(}\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot
g_{C^{n}})>0.$

We observe that $\mathbf{F}_{\beta}\cdot\Psi_{\circ}$ and $\Psi_{\circ}$
satisfy the relation $L_{C^{n}}F\geq\lambda/8\cdot\langle A\rangle^{2}\cdot
F$. When we replace $\phi$ by $\phi_{\tau}$, $\tau>0$, with
$\phi_{\tau}(t):=\phi(\tau\cdot t)$, we have
$|\phi^{\prime}_{\tau}|,|\phi^{\prime\prime}_{\tau}|\rightarrow 0$, for
$\tau\rightarrow 0$. Thus the terms involving $|\phi^{\prime}_{\tau}|$ and
$|\phi^{\prime\prime}_{\tau}|$ are majorized by $\phi_{\tau}\cdot
L_{C^{n}}(\mathbf{F}_{\beta}\cdot\Psi_{\circ})+(1-\phi_{\tau})\cdot
L_{C^{n}}\Psi_{\circ}$ and we get, for small $\tau>0$, that
$L_{C^{n}}(\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ})\geq\lambda/10\cdot\langle
A\rangle^{2}\cdot\mathbf{F}^{\phi,Q_{\ell}^{q}}_{\beta}\cdot\Psi_{\circ}$. For
fixed $\phi$ this $\tau$ depends only on $n$ and $\beta$. To avoid non-
essential parameters we henceforth assume that $\tau=1$. $\Box$

Local Bumps For $0\leq q\leq n-k$,
$\operatorname{cut}\Delta^{n,q\,|\,\beta,\ell}$ is compact. For $q=0$, this
leaves $\Delta^{n,0\,|\,\beta}$ unchanged, but for the sake of consistency we
keep the parameter $\ell$. For (0)-bump elements, Prop. 2.4 shows that for
small $\beta>0$ there are minimizing boundaries ${\cal{T}}^{\beta}=\partial U$
for an open $U$ with $O_{r}\subset U\subset O_{1}$. For $m>0$ we inductively
and disjointly place scaled bump elements along all lower dimensional faces of
$Q^{n-k}_{1}$, so that the cut-off regions of the ($q$)-bump elements belong
to the union of the ($\phi$-trimmings of the) cores $\mathbb{R}^{q}\times
B^{n-k-q}_{r[n]}(0)\subset\mathbb{R}^{q}\times(\mathbb{R}^{n-k-q}\times
C^{k})$, from 2.2, of the already assigned ($q$)-bump elements, for $q<m$, cf.
Fig. 5. For each ($q$)-dimensional face $Q^{q,f}_{1}\subset Q^{n-k}_{1}$,
$f\in F_{q}$ for some index set $F_{q}$, we choose an affine transformation

(60) $T[\eta,f]:\mathbb{R}^{n-k}\times C^{k}\rightarrow\mathbb{R}^{n-k}\times
C^{k}\mbox{ with }(x,y)\mapsto\eta\cdot(x,A_{f}(y)+v_{f}),$

where $\eta\in(0,\ell^{-1}],A_{f}\in O(n-k),v_{f}\in\mathbb{R}^{n-k},$ that
maps the model ($q$)-cube $Q_{\ell}^{q}$ to $Q^{q,f}_{1}$ so that
$T[\ell^{-1},f](Q_{\ell}^{q})=Q^{q,f}_{1}$. From Prop.1.2 we have
$\Psi_{\circ}(\omega,\rho)=\psi(\omega)\cdot\rho^{\alpha_{\circ}}$ and hence

(61) $T[\eta,f]_{*}(\Psi_{0})=\Psi_{0}\circ
T[\eta,f]^{-1}=\eta^{-\alpha_{\circ}}\cdot\Psi_{0}.$

The parameters $\beta$, $\ell$ and $\eta$ are determined inductively in the
following construction. The choices anticipate additions of ($q$)-bump
elements for $q\geq 1$, as shown in Fig. 5, and of local bumps when we form
larger configurations finally reaching global bumps. To this end, we recall
from 2.4 that any area minimizer
$\mathcal{T}_{\bullet}\in\mathcal{P}[q,\ell,\mathbf{G}_{\beta}]$ with
$\Omega$-boundary value $\mathcal{T}_{\varrho}$, for
$\Omega_{q}=Q_{\ell}^{q}\times\mathbb{R}^{n-k-q}\times C^{k}$ and
$O_{\rho}=\mathbb{R}^{n-k}\times B_{\rho}(0),\rho>0$, satisfies

(62)
$\mathcal{T}_{\bullet}\cap\Omega\subset(O_{\varrho+\zeta}\setminus\overline{O_{\varrho-\zeta}})\cap\Omega\subset(O_{1}\setminus\overline{O_{r}})\cap\Omega,$

where $r=r[n]\in(0,1)$ is the core radius of
$\big{(}\mathbf{F}\cdot\Psi_{\circ}\big{)}^{4/(n-2)}\cdot g_{C^{n}}$ in Prop.
2.2.

0-faces

For $\eta_{0}=1/10$, we place $\eta_{0}$-scaled (0)-bump elements
$\Delta^{n,0\,|\,\beta,\ell}$ in each 0-dimensional face (=vertex)
$v_{f}:=p_{f}$ of $Q^{n-k}_{1}$. For $\beta^{*}[n]\in(0,1)$ from 2.4,
$\beta_{0}\in(0,\beta^{*}]$, $\xi\in[r[n],1]$, we set

$\textstyle\boldsymbol{\square}^{\beta_{0},\eta_{0},\ell_{0}}:=\sum_{f\in
F_{0}}T[\eta_{0},f]_{*}\big{(}\Delta^{n,0\,|\,\beta_{0},\ell_{0}}\big{)}\mbox{,
}\,\mathbb{O}^{\eta_{0},\ell_{0}}_{0,\xi}:=\bigcup_{f\in
F_{0}}O^{\eta_{0},\ell_{0}}_{f,0,\xi}:=\bigcup_{f\in
F_{0}}T[\eta_{0},f]\big{(}O_{\xi}\big{)}$

for the void parameter $\ell_{0}=1$. We observe that

$\operatorname{supp}T[\eta_{0},f_{1}]_{*}\big{(}\Delta^{n,0\,|\,\beta_{0},\ell_{0}}\big{)}\cap\operatorname{supp}T[\eta_{0},f_{2}]_{*}\big{(}\Delta^{n,0\,|\,\beta_{0},\ell_{0}}\big{)}=\emptyset\mbox{
for }f_{1}\neq f_{2}\in F_{0}.$

q-faces

We continue inductively for $q$ with $1\leq q\leq n-k$. We choose a large
$\ell_{q}\geq 1$, some
$\eta_{q}\in(0,\min\\{\eta_{0}\cdot\beta_{0}/10,...,\eta_{q-1}\cdot\beta_{q-1}/10,\ell_{q}^{-1}\\})$
and an $\beta_{q}\in[0,\beta^{*}[n,\ell_{q}]]$, so that after placing the
$\eta_{q}$-scaled ($q$)-bump elements $\Delta^{n,q\,|\,\beta_{q},\ell_{q}}$
along the ($q$)-faces of the cube $Q^{n-k}_{1}$, the support of any two
($q$)-bump elements and of any ($q$)-bump element and the already placed
($m$)-bump element, $0\leq m<q$ are _disjoint_ , cf. (i) and (ii) below. The
cut-off region of each ($q$)-bump element belongs to the union of the cores of
the ($m$)-bump elements for $0\leq m<q$, cf. (iii).

  1. (i)

$\operatorname{supp}T[\eta_{q},f_{1}]_{*}(\Delta^{n,q\,|\,\beta_{q},\ell_{q}})\cap\operatorname{supp}T[\eta_{q},f_{2}]_{*}(\Delta^{n,q\,|\,\beta_{q},\ell_{q}})=\emptyset$,
for $f_{1}\neq f_{2}\in F_{q}$,

  2. (ii)

$\operatorname{supp}T[\eta_{q},f]_{*}(\Delta^{n,q\,|\,\beta_{q},\ell_{q}})\cap\bigcup_{k=0,..q-1}\operatorname{supp}\boldsymbol{\square}^{\beta_{k},\eta_{k},\ell_{k}}=\emptyset$,
for $f\in F_{q}$,

  3. (iii)

$\bigcup_{f=F_{q}}T[\eta_{q},f](\operatorname{cut}\Delta^{n,q\,|\,\beta_{q},\ell_{q}})\subset\bigcup^{q-1}_{m=0}\mathbb{O}^{\eta_{m},\ell_{m}}_{m,r[n]}$,

and for the induction step we set for $q$ the following unions of bump
elements and cylinders:

(63) $\displaystyle\boldsymbol{\square}^{\beta_{q},\eta_{q},\ell_{q}}:=$
$\displaystyle\textstyle\sum_{f=F_{q}}T[\eta_{q},f]_{*}\big{(}\Delta^{n,q\,|\,\beta_{q},\ell_{q}}\big{)}\mbox{
and }$ $\displaystyle\mathbb{O}^{\eta_{q},\ell_{q}}_{q,\xi}:=$
$\displaystyle\textstyle\bigcup_{f\in
F_{q}}O^{\eta_{q},\ell_{q}}_{f,q,\xi}:=\bigcup_{f\in
F_{q}}T[\eta_{q},f]\big{(}O_{\xi}\cap\Omega_{q}\big{)}.$

$\bigcup^{n-k}_{q=0}\mathbb{O}^{\eta_{q},\ell_{q}}_{q,\xi}$ is an open
neighborhood of $Q^{n-k}_{1}\times\\{0\\},$ for any $\xi\in[r[n],1]$, and from
Prop. 2.2 and Prop. 2.4, we notice that the inductively chosen parameters

(64) $\ell_{q}\geq 1,\eta_{q}>0\mbox{ and }\beta_{q}>0,\mbox{ for }0<q\leq
n-k,$

are independent of $C^{n}=\mathbb{R}^{n-k}\times C^{k}\in{\cal{SC}}_{n}$. We
make one fixed choice of these parameters and keep it for the rest of this
paper.

Figure 5: An illustration of the support of a local bump shielding
$Q^{n-k}_{1}\times\\{0\\}$.

###### Definition 2.7

_(Local Bumps)_ We call the pseudo metric

(65)
$\textstyle\boldsymbol{\square}(Q^{n-k}_{1}):=\sum^{n-k}_{q=0}\boldsymbol{\square}^{\beta_{q},\eta_{q},\ell_{q}}\mbox{
an {(n-k)-local bump} on }C^{n}=\mathbb{R}^{n-k}\times C^{k}.$

We define the associated (n-k)-local bump metric

(66) $\textstyle(\Phi_{\square}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}:=\left(\big{(}\prod^{n-k}_{q=0}\prod_{f=F_{q}}(\mathbf{F}^{\phi,Q_{\ell}^{m}}_{\beta_{q}}\circ
T[\eta_{q},f]^{-1})\big{)}\cdot\Psi_{\circ}\right)^{4/(n-2)}\cdot g_{C^{n}}.$

and we call
$\mathbb{O}^{C}_{n-k}:=\bigcup^{n-k}_{q=0}\mathbb{O}^{\eta_{q},\ell_{q}}_{q,r[n]}$
the core of the local bump.

$\Phi_{\square}$ is the multiplicative representation of a local bump with
$\Phi_{\square}\in{\cal{C}}[C^{n},\Lambda[n]]$ and we have
$\operatorname{supp}|\Phi_{\square}-1|=\operatorname{supp}\boldsymbol{\square}(Q^{n-k}_{1})$

(67) $L_{C^{n}}(\Phi_{\square}\cdot\Psi_{\circ})\geq\lambda/10\cdot\langle
A\rangle^{2}\cdot\Phi_{\square}\cdot\Psi_{\circ}.$

This follows from 2.6 (59) and (61) since the
$\operatorname{supp}|\mathbf{F}^{\phi,Q_{\ell}^{m}}_{\beta_{q}}\circ
T[\eta_{q},f]^{-1}-1|$ for any two different ($q$)-bump elements are disjoint
and the relation (67) remains invariant under translations and scalings. We
use Prop. 2.4 to understand the effect of local bumps.

###### Proposition 2.8

_(Shielding)_ For $\mathbf{G}\in{\cal{C}}[C^{n},\Lambda[n]]$ with
_$\operatorname{supp}|\mathbf{G}-1|\cap\operatorname{supp}|\Phi_{\square}-1|=\emptyset$_
and the $\ell_{q},\eta_{q},\beta_{q}$ from (64), we choose for any ($q$)-face
$f\in F_{q}$ some

(68) $\mathcal{T}_{f}=\partial
U_{f}\in\mathcal{P}[q,\ell_{q},\mathbf{G}_{\beta_{q}}]\mbox{ for an open
}O_{r[n]}\cap\Omega_{q}\subset U_{f}\cap\Omega_{q}\subset O_{1}\cap\Omega_{q}$

where $\mathbf{G}_{\beta_{q}}:=\mathbf{G}\circ T[\eta_{q},f]$ on
$O_{1}\cap\Omega$. Combining these $U_{f}$ we get a local shield for
$Q^{n-k}_{1}$:

(69)
$\textstyle\mathbb{U}(Q^{n-k}_{1}):=\bigcup^{n-k}_{q=0}\mathbb{U}^{\beta_{q},\eta_{q},\ell_{q}}_{q}:=\bigcup^{n-k}_{q=0}\bigcup_{f\in
F_{q}}T[\eta_{q},f]\big{(}U_{f}\cap\Omega_{q}\big{)}\supset\mathbb{O}^{C}_{n-k}\supset
Q^{n-k}_{1}\times\\{0\\},$ (70) $\partial\mathbb{U}\mbox{ is a local {inner
area minimizer} relative to }\mathbb{U}\mbox{ and
}(\mathbf{G}\cdot\Phi_{\square}\cdot\Psi_{\circ})^{4/(n-2)}\cdot g_{C^{n}}.$

That is,
$\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial\mathbb{U})\leq\mu^{\Lambda,n-1}_{\mathcal{S}}(\partial\big{(}\mathbb{U}\setminus
A\big{)})$ for sufficiently small open sets $A\subset C$.

This means we take the standard (q)-bump element aligned to $Q_{\ell}^{q}$,
defined in 2.5, and use $\mathbf{G}\circ T[\eta_{q},f]$ for the standard
(q)-bump element to apply Prop.2.4. Then the resulting $\mathcal{T}_{f}$ and
$U_{f}$ are shifted by $T[\eta_{q},f]$ to their associated face $f$ of
$Q_{1}^{n-k}$.

Proof From Prop. 2.4 we see that $\mathbb{U}$ is an open neighborhood of
$Q^{n-k}_{1}\times\\{0\\}$. For (70) we note that the union of two locally
inner area minimizing $U$ and $V$ is again locally inner area minimizing: for
an open $A$ we first use this inner minimality of $U$, for $A\setminus V$, and
then that of $V$, for $A\cap V$, and write $(U\cup V)\setminus A=\big{(}(U\cup
V)\setminus(A\setminus V)\big{)}\setminus A\cap V.$
This applies to $\partial\mathbb{U}$ since, dropping $\Omega_{q}$ for $q=0$,
$\partial\mathbb{U}\cap\bigcup^{n-k}_{q=0}\bigcup_{f\in
F_{q}}T[\eta_{q},f]\big{(}U_{f}\cap\Omega_{q}\big{)}=\emptyset$. We only need
to consider $A\cap\bigcup^{n-k}_{q=0}\bigcup_{f\in
F_{q}}T[\eta_{q},f]\big{(}U_{f}\cap\Omega_{q}\big{)}=\emptyset$ and we argue
by induction over all $T[\eta_{q},f]\big{(}U_{f}\cap\Omega_{q}\big{)}$ in the
union $\mathbb{U}$. $\Box$

Groupings We also need bumps shielding arbitrarily large cubes
$Q^{n-k}=Q^{n-k}_{a_{1},...,a_{n-k}}=[-a_{1},a_{1}]\times\cdots\times[-a_{n-k},a_{n-k}]\times\\{0\\}\subset\mathbb{R}^{n-k}\times
C^{k}$, for some $a_{i}>0$. To control the interplay with other bumps we must
keep the size of both, the (support of) local bumps in the $C^{k}$-directions
and of the tunnel parameters $\beta_{q}$, unchanged. This means, we cannot
just scale the bump construction. Alternatively, when we try to adjust the
parameters in (65) to shield a larger cube $Q$, we find from 2.4 that,
starting from an initial $\beta_{0}$-tunnel for the $(0)$-faces and $(0)$-bump
elements, the subsequent $\beta_{i}$ have to be chosen the smaller the larger
$Q$ becomes.
The way out are _groupings_ of local bumps, each shielding a translated copy
of the unit cube $Q^{n-k}_{1}$. Concretely, we consider disjoint unions
$\bigcup_{v\in S}(Q^{n-k}_{1}+v)\subset
Q^{n-k}_{L,...,L}\subset\mathbb{R}^{n-k}$, for finite, otherwise arbitrary
subsets $S\subset\mathbb{Z}^{n-k}$ and some $L=L(S)>0$ large enough.
We repeat the inductive process we have used to define local bumps and place
($m$)-bump elements, $0\leq m\leq n-k$, along the ($m$)-faces of the cubes in
this union. We count all faces in this union with multiplicity one even when
they belong to more than one cube $Q^{n-k}_{1}+v$. Applying 2.7 and 2.8 to
this periodic version of the local bump construction yields:

###### Corollary 2.9

_(Periodic Groupings)_ We get periodic bumps and shields from the obvious
extensions of the definitions (65) and (69):

(71) $\textstyle\boldsymbol{\square}(\bigcup_{v\in S}(Q^{n-k}_{1}+v)),\mbox{
the bumped metric }(\Phi_{\square}\cdot\Psi_{\circ})^{4/(n-2)}\cdot
g_{C^{n}}\mbox{ and }\mathbb{U}(\bigcup_{v\in S}(Q^{n-k}_{1}+v)),$

and we define the core of the periodic bump by

(72) $\textstyle\mathbb{O}^{C}_{n-k}(\bigcup_{v\in
S}(Q^{n-k}_{1}+v)):=\bigcup_{v\in
S}\big{(}\bigcup^{n-k}_{q=0}\mathbb{O}^{\eta_{q},\ell_{q}}_{q,r[n]}+v\big{)}.$

For $\mathbf{G}\in{\cal{C}}[C^{n},\Lambda[n]]$ with
_$\operatorname{supp}|\mathbf{G}-1|\cap\operatorname{supp}|\Phi_{\square}-1|=\emptyset$_
, and the $r$-distance neighborhood $U_{r}$ in $C^{n}$ of the family $S$ of
cubes, we have:

  1. (i)

$\mathbb{U}(\bigcup_{v\in
S}(Q^{n-k}_{1}+v))\supset\mathbb{O}^{C}_{n-k}(\bigcup_{v\in
S}(Q^{n-k}_{1}+v))\supset\bigcup_{v\in S}(Q^{n-k}_{1}+v)$,

  2. (ii)

$\partial\mathbb{U}$ is locally inner area minimizing relative to
$\mathbb{U}$, i.e. $\mathbb{U}$ shields $\mathbb{O}^{C}_{n-k}(\bigcup_{v\in
S}(Q^{n-k}_{1}+v))$,

  3. (iii)

$L_{C^{n}}(\Phi_{\square}\cdot\Psi_{\circ})\geq\lambda/10\cdot\langle
A\rangle^{2}\cdot\Phi_{\square}\cdot\Psi_{\circ}$.

These results extend to $\bigcup_{v\in S}(Q^{n-k}_{1}+v)\subset
Q^{n-k}_{L,...,L}\subset\mathbb{R}^{n-k}$ after scaling by $2^{a}$,
$a\in\mathbb{Z}^{\geq 1}$, where we interpret $2^{a}\cdot Q^{n-k}_{1}$ as a
union of $2^{a\cdot(n-k)}$ appropriately translated copies of $Q^{n-k}_{1}$.
Rescaling the resulting bump by $2^{-a}$ yields the $2^{a}$-rescaled periodic
bumps $\boldsymbol{\square}[a]$ and shields with

(73) $\textstyle\mathbb{U}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v))\supset\mathbb{O}^{C}_{n-k}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v))\supset\bigcup_{v\in S}(Q^{n-k}_{1}+v),$

with $\mathbb{U}[a]:=2^{-a}\cdot\mathbb{U}(2^{a}\cdot\bigcup_{v\in
S}(Q^{n-k}_{1}+v))$ and
$\mathbb{O}^{C}_{n-k}[a]:=2^{-a}\cdot\mathbb{O}^{C}_{n-k}(2^{a}\cdot\bigcup_{v\in
S}(Q^{n-k}_{1}+v))$.

Proof We note that, unless to faces of different cubes coincide, all bump
elements have again disjoint support. From this we can repeat the same bump
element-wise argument as in the proof of Prop. 2.8 for the same parameters
with the same conclusions. $\Box$

###### Remark 2.10

(Estimates for $\operatorname{supp}\boldsymbol{\square}$) We set
$r^{\circ}[n]:=\min\\{\beta_{0}\cdot\eta_{0},...,\beta_{n-7}\cdot\eta_{n-7}\\}\cdot
r[n]/4$ as a lower bound for all radii, independent of the dimension of the
factor $\mathbb{R}^{n-k}$, and note that

(74) $\textstyle U_{2^{-a}}(\bigcup_{v\in S}(Q^{n-k}_{1}+v))\setminus
U_{2^{-a}\cdot
r^{\circ}[n]}(\mathbb{R}^{n-k}\times\\{0\\})\supset\operatorname{supp}\boldsymbol{\square}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v)).$

Estimates in terms of $\delta_{\langle A\rangle}$ are compatible with
transfers between $H$ and tangent cones. From (2.3)(i) and $\delta_{\langle
A\rangle}(z)\leq dist(z,\Sigma_{C})$, we get for any $p\in\Sigma$

(75) $\displaystyle\textstyle\overline{B_{2^{-a}\cdot
r^{\circ}[n]}(p)}\cap\operatorname{supp}\boldsymbol{\square}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v))$ $\displaystyle\subset...$ $\displaystyle\textstyle\\{z\in
C\setminus\Sigma_{C}\,|\,\delta_{\langle A\rangle}(z)\leq 2^{-a}\cdot
r^{\circ}[n]\\}\cap\operatorname{supp}\boldsymbol{\square}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v))$ $\displaystyle=\emptyset.$

and from (74) complementary to (75):

(76) $\textstyle\\{z\in C\setminus\Sigma_{C}\,|\,\delta_{\langle
A\rangle}(z)\geq
2^{-a}\\}\cap\operatorname{supp}\boldsymbol{\square}[a](\bigcup_{v\in
S}(Q^{n-k}_{1}+v))=\emptyset.$

$\Box$

###### Example 2.11

(Shielding Balls) The rescaling of periodic bumps shrinks the distance of the
support to $\bigcup_{v\in S}(Q^{n-k}_{1}+v)$ but it also improves the control
over the shielded subset. For
$B^{n-k}_{1}(0)\subset\mathbb{R}^{n-k}\times\\{0\\}$ we interpret the
rescaling as a partition of the cubes into $2^{a\cdot(n-k)}$-subcubes of side
length $2^{-a}$, $a\in\mathbb{Z}^{\geq 1}$, and consider the _minimal set_
$S_{a,n-k}\subset 2^{-a}\cdot\mathbb{Z}^{n-k}$ of such cubes with
$B^{n-k}_{1}(0)\subset\bigcup_{v\in S_{a,n-k}}(Q^{n-k}_{2^{-a}}+v)$. We have
from $\mathrm{diam}(Q^{n-k}_{1})=\sqrt{n-k}$ :

(77)
$\textstyle\overline{B^{n-k}_{1+{2^{-a}}\cdot\sqrt{n-k}}(0)}\supset\bigcup_{v\in
S_{a,n-k}}(Q^{n-k}_{2^{-a}}+v)\supset\overline{B^{n-k}_{1}(0)}.$

From this and (74) we have for some $a_{n}\in\mathbb{Z}^{\geq 1}$ large enough
and any $m\in\mathbb{Z}^{\geq 1}$

  * •

$B^{n}_{2}(0)\supset B^{n}_{1+{2\cdot
2^{-a_{n}}}\cdot\sqrt{n-k}}(0)\supset\operatorname{supp}\boldsymbol{\square}[\kappa_{n}\cdot(a_{n}+m)](\bigcup_{v\in
S_{a_{n},n-k}}(Q^{n-k}_{2^{-a}}+v))$,

  * •

$\operatorname{supp}\boldsymbol{\square}[\kappa_{n}\cdot a_{n}](\bigcup_{v\in
S_{a_{n},n-k}}(Q^{n-k}_{2^{-a}}+v))\cap\operatorname{supp}\boldsymbol{\square}[\kappa_{n}\cdot(a_{n}+m)](\bigcup_{v\in
S_{a_{n},n-k}}(Q^{n-k}_{2^{-a}}+v))=\emptyset,$

where $\kappa_{n}:=[\,\mbox{smallest integer}\geq|\ln(r^{\circ}[n])|\,]$, for
the radius $r^{\circ}[n]>0$ in 2.10. $\Box$

##  3 From Local to Global Deformations

We turn to a general hypersurface $H\in{\cal{G}}_{n}$ and gradually scale $H$
around some $p\in\Sigma_{H}$, by large $\tau\gg 1$. Eventually there is some
not necessarily unique area minimizing cone $C$ that Hausdorff approximates
$\tau\cdot H$ on the unit ball. On compact parts of $C\setminus\Sigma_{C}$
this can be improved to a smooth norm convergence, cf. [L2, Appendix A]. This
smooth approximation comes with a uniquely determined smooth map, which we
call an $\mathbf{ID}$-map, locally parameterizing $\tau\cdot H$ as a section
of the normal bundle over $C$, we can use to transfer local bump
configurations on $C$ to $H\setminus\Sigma_{H}$. The challenge to build global
bumps along $\Sigma_{H}$, from local transfers, is to compensate the
differences between the singular sets of $H$ and such cones.

  1. (1)

Basic Idea We would like to have ball covers of $\Sigma_{H}$ with (i) common
ball radii and controlled intersection numbers $c(n)$, depending only on $n$
and so that (ii) each of the balls admits a good cone approximation to place
cone bumps on $H\setminus\Sigma_{H}$ and to estimate their effect. Yet, we
cannot accomplish both goals (i) and (ii) at the same time. For any
$p\in\Sigma_{H}$ and $\varepsilon>0$ there is a critical radius, the
$\varepsilon$_-accuracy radius_ $\mathcal{R}_{\varepsilon}(p)$. Balls of
radius $r\in(0,\mathcal{R}_{\varepsilon}(p))$, suitably rescaled, admit fine
approximations by cones of a (yet to define) accuracy $\varepsilon$. One
hurdle is that $\mathcal{R}_{\varepsilon}(p)$ does not depend continuously on
$p\in\Sigma$.

  2. (2)

Self-Similar Covers To resolve this discontinuity issue, we define a hierarchy
on $\Sigma$, the _accuracy decomposition_ $\Sigma=\dot{\bigcup}_{i\geq
0}\Sigma_{i}$, with
$\Sigma_{0}=\\{p\in\Sigma\,|\,\mathcal{R}_{\varepsilon}(p)>1/\varepsilon\\}$
and
$\Sigma_{i}=\\{p\in\Sigma\,|\,s^{i-1}/\varepsilon\geq\mathcal{R}_{\varepsilon}(p)>s^{i}/\varepsilon\\}$,
for $i\geq 1$ and some $s\in(0,1/100)$, in Ch. 3.1. For each $\Sigma_{i}$ we
can find covers by balls of radius $s^{i}$ with the two conditions (i) and
(ii) in (1). Scaling by $s$ shifts the family index by $1$ and makes the cover
self-similar under scaling by $s$ in the sense that properties we derive for
one $\Sigma_{i}$ carry over to the other families. For small $s>0$, the bumps
placed on ball families for different $i$ do not interfere.

  3. (3)

Auto-Alignment For the placement of bumps on families of equally sized balls
we observe that singular and highly curved directions in any two neighboring
balls align like compass needles in a common field. This controls not only the
relative position of the singular set in these balls but also the positioning
of the balls themselves.

###  3.1 $\mathbf{ID}$-Maps and Ball Covers

Around any $q\in\Sigma_{H}$ in some $H\in{\cal{G}}_{n}$, we have tangent cones
of $H$: $H_{i}=\tau_{i}\cdot H$, for some sequence
$\tau_{i}\rightarrow\infty$, $i\rightarrow\infty$, flat norm subconverges to
some (generally non-unique) area minimizing tangent cone
$C^{n}\subset\mathbb{R}^{n+1}$. Allard theory shows that this convergence can
be upgraded.

$\mathbf{ID}$-maps For $B_{R}(q)\cap C\setminus\Sigma_{C}$, $R>0$, large $i$
and $B_{R}(q_{i})\cap H_{i}$ for suitable $q_{i}\in H\setminus\Sigma$, there
is a local $C^{5}$-section $\Gamma_{i}:B_{R}(q)\cap C\rightarrow
B_{R}(q_{i})\cap H_{i}\subset\nu$ of the normal bundle $\nu\mbox{ of
}B_{R}(q)\cap C$ up to minor adjustments near $\partial B_{R}(q)$. For
$i\rightarrow\infty$, $\Gamma_{i}$ converges in $C^{k}$-norm to the zero
section, which we identify with $B_{R}(q)\cap C$. We call
$\mathbf{ID}:=\Gamma_{i}$ the asymptotic identification map, briefly the
$\mathbf{ID}$-map. Since they are assigned canonically, we generally omit
writing indices. We also recall the definition of the $\mathcal{S}$-pencil
$\mathbb{P}(p,\omega)$ pointing to $p\in\Sigma$, for some $\omega>0$:
$\mathbb{P}(p,\omega):=\\{x\in H\setminus\Sigma_{H}\,|\,\delta_{\langle
A\rangle_{H}}(x)>\omega\cdot d_{H}(x,p)$. We consider the $\mathbf{ID}$-maps
on bounded subsets of $\mathbb{P}(p,\omega)$, the truncated
$\mathcal{S}$-pencils $\mathbb{TP}$.

(78) $\mathbb{TP}_{H}(p,\omega,R,r):=B_{R}(p)\setminus
B_{r}(p)\cap\mathbb{P}(p,\omega)\subset H.$

For increasingly large $\tau>0$,
$\tau\cdot\mathbb{TP}_{H}(p,\omega,R/\tau,r/\tau)$ is gradually better
$C^{5}$-approximated by the truncated $\mathcal{S}$-pencil in _some_ tangent
cone, i.e., the twisting of $\mathbb{P}(p,\omega)$ slows down as
$\tau\to\infty$, it _asymptotically freezes_. More precisely, we have [L4,
Prop. 3.10]:

######  Proposition 3.1 ($\mathcal{S}$-Freezing)

Let $H\in{\cal{G}}_{n}$ and $p\in\Sigma_{H}$. We pick an $\varepsilon>0$, some
$R>1>r>0$ and an $\omega\in(0,1)$. Then there is a smallest
$\tau^{*}=\tau^{*}(\varepsilon,\omega,R,r,p)\geq 1$ so that for any
$\tau>\tau^{*}$ there is some tangent cone $C^{\tau}_{p}$ so that for the
$\mathbf{ID}$-map to $\tau\cdot H$:

  * •

$|\mathbf{ID}-id_{C^{\tau}_{p}}|_{C^{5}(\mathbb{TP}_{C^{\tau}_{p}}(0,\omega,R,r))}\leq\varepsilon$,
where we view the zero section of $\nu$ as the identity $id$,

  * •

$\Psi_{H}$ induces the uniquely# determined solution
$\Psi_{\circ}>0(C^{\tau}_{p})$ on $C^{\tau}_{p}$ from Prop. 1.2 and we have,
normalizing $\Psi_{\circ}$ appropriately,
$|\Psi_{H}\circ\mathbf{ID}/\Psi_{\circ}-1|_{C^{2,\alpha}(\mathbb{TP}_{C^{\tau}_{p}}(0,\omega,R,r))}\leq\varepsilon$.

We call $\varepsilon$ the accuracy of the $\mathbf{ID}$-map and
$\mathcal{R}_{\varepsilon}(H,p):=1/\tau^{*}(\varepsilon,\varepsilon,1/\varepsilon,\varepsilon,p)$
the $\boldsymbol{\varepsilon}$-accuracy radius of $p\in\Sigma_{H}$. That is,
any ball $B_{\rho}(p)\subset H$, $0<\rho\leq\mathcal{R}_{\varepsilon}$,
rescaled by $(\varepsilon\cdot\rho)^{-1}$ admits an $\varepsilon$-accurate
$\mathbf{ID}$-map over
$\mathbb{TP}_{C}(0,\varepsilon,1/\varepsilon,\varepsilon)\subset
B_{1/\varepsilon}(0)\setminus B_{\varepsilon}(0)\cap C$, for some tangent cone
$C$.

###### Remark 3.2

For Euclidean area minimizers, the flat norm $d^{\flat}$-convergence implies
Hausdorff $d_{\cal{H}}$-convergence from volume growth estimates [Gi, Prop.
5.14]. We already used this argument in Step 2 of 2.4 for area minimizers in
$(C,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ based on Corollary
1.4 (13).
The spaces $\mathcal{C}_{n}$ and ${\cal{H}}^{\mathbb{R}}_{n}$ are compact
under compact convergence in the flat norm topology. $\mathcal{K}_{n}$ is
compact in flat norm topology. $\mathcal{SC}_{n}\subset\mathcal{C}_{n}$ and
${\cal{SH}}^{\mathbb{R}}_{n}\subset{\cal{H}}^{\mathbb{R}}_{n}$ are closed and,
hence, compact. Consequently $\overline{\cal T}_{H}\subset\mathcal{SC}_{n}$,
also we have $\overline{\cal T}_{p}={\cal T}_{p}$. $\Box$

We also recall a variant for singular cones, [L4, Cor. 3.11], where we use the
Hausdorff distance on ${\cal{SC}}_{n}$ measured in terms of the distance of
the unit spheres $S_{C}=\partial B_{1}(0)\cap C$.

######  Corollary 3.3 (Cone Transitions)

Let $\alpha\in(0,1)$, $\varepsilon$, $\omega>0$ and $R>1>r>0$. Then there is a
$\zeta(\varepsilon,\omega,R,r)>0$ so that for any $C$ and
$C^{\prime}\in{\cal{SC}}_{n}$ with
$d_{\cal{H}}(S_{C},S_{C^{\prime}})\leq\zeta$:

  * •

$|\mathbf{ID}-id_{C}|_{C^{5}(\mathbb{TP}_{C}(0,\omega,R,r))}\leq\varepsilon$,

  * •

$\Psi_{\circ}(C^{\prime})$ on $C^{\prime}$ induces $\Psi_{\circ}(C)$ on $C$
with
$|\Psi_{\circ}(C)/\Psi_{\circ}(C^{\prime})\circ\mathbf{ID}-1|_{C^{2,\alpha}(\mathbb{TP}_{C}(0,\omega,R,r))}\leq\varepsilon.$

Self-Similar Ball Covers We write $\Sigma$ as a disjoint union of subsets
$\Sigma_{i}$, $i\in\mathbb{Z}^{\geq 1}$, sorted according to their
$\varepsilon$-accuracy radii. Then we cover these subsets by balls with radii
$s^{i}$, for some small $s>0$. This makes the scenario self-similar under
scaling by powers of $1/s$.

###### Definition 3.4

_(Accuracy Decomposition)_ For $H\in{\cal{G}}_{n}$, some $\varepsilon>0$ and
$s\in(0,s_{0})$ we define the $\boldsymbol{\varepsilon}$-accuracy
decomposition of $\Sigma=\dot{\bigcup}_{i\geq 0}\Sigma_{i}:$

(79)
$\Sigma_{0}=\\{p\in\Sigma\,|\,\mathcal{R}_{\varepsilon}(p)>1/\varepsilon\\}\mbox{
and
}\Sigma_{i}=\\{p\in\Sigma\,|\,s^{i-1}/\varepsilon\geq\mathcal{R}_{\varepsilon}(p)>s^{i}/\varepsilon\\}.$

In general, the $\Sigma_{i}$ will be uncountable, but each of them contains a
countable and dense subset $\Sigma^{*}_{i}\subset\Sigma_{i}$ and we set
$\Sigma^{*}:=\bigcup_{i\geq 0}\Sigma^{*}_{i}\subset\Sigma$. We write the
elements of $\Sigma^{*}_{i}$ accordingly as $a_{i,1},a_{i,2},...$. We assign a
ball $\overline{B_{s^{i}}(p)}$ to any of the points $p\in\Sigma^{*}_{i}$ and
set

(80)
$\textstyle{\cal{B}}[i]:=\\{\overline{B_{s^{i}}(p)}\,|\,p\in\Sigma^{*}_{i}\\}\mbox{
and }{\cal{B}}:=\bigcup_{i\geq 0}{\cal{B}}[i].$

We note that ${\cal{B}}[i]$ is a cover of $\Sigma_{i}$ and, hence, ${\cal{B}}$
is a cover of $\Sigma$ since for any $x\in\Sigma_{i}$ there is some
$p_{x}\in\Sigma^{*}_{i}$ with $d_{H}(x,p_{x})<s^{i}/2$ and thus
$x\in\overline{B_{s^{i}}(p_{x})}$.

###### Proposition 3.5

_(Self-Similar Covers)_ For $H\in{\cal{G}}_{n}$, with
$\Sigma_{H}\neq\emptyset$, some accuracy $\varepsilon\in(0,10^{-5})$,
$R\in[10^{2},\varepsilon^{-1}/10^{2}]$ and any self-similarity factor
$s\in(0,s_{0})$, the following holds:

  1. (i)

For each $i\geq 0$, there is a locally finite family
${\cal{A}}[i]\subset{\cal{B}}[i]$, from (80), of closed balls

${\cal{A}}[i]=\\{\overline{B_{s^{i}}(p)}\subset H\,|\,p\in A[i]\\},\mbox{ for
some discrete set }A[i]\subset\Sigma^{*}_{i}.$

  2. (ii)

For any $k\geq 0$ we have: $\bigcup_{i\leq k}{\cal{A}}[i]$ covers
$\bigcup_{i\leq k}\Sigma_{i}$.

  3. (iii)

$q\notin\overline{B_{s^{i}}(p)}$, $p\notin\overline{B_{s^{k}}(q)}$ for any two
different $\overline{B_{s^{i}}(p)}\in{\cal{A}}[i]$,
$\overline{B_{s^{k}}(q)}\in{\cal{A}}[k]$.

For $i\geq 0$, there is a subset $A^{\bullet}[i]\subset A[i]$ and a family
${\cal{A}}^{\bullet}[i]=\\{\overline{B_{R\cdot s^{i}}(p)}\subset H\,|\,p\in
A^{\bullet}[i]\\}$ with

(81) $\textstyle\bigcup_{x\in A[i]}\overline{B_{s^{i}}(x)}\subset\bigcup_{z\in
A^{\bullet}[i]}\overline{B_{R\cdot s^{i}}(z)}\mbox{ and
}z^{*}\notin\overline{B_{(R-1)\cdot s^{i}}(z)},\mbox{ for }z\neq
z^{*},z,z^{*}\in A[i]$

and so that, for some constant $c(H)$, ${\cal{A}}^{\bullet}[i]$ splits into
disjoint families ${\cal{A}}^{\bullet}[i,1],...,{\cal{A}}^{\bullet}[i,c]$ with

(82) $B_{10\cdot R\cdot s^{i}}(p)\cap B_{10\cdot R\cdot
s^{i}}(q)=\emptyset,\mbox{ for $\overline{B_{R\cdot
s^{i}}(p)},\overline{B_{R\cdot s^{i}}(q)}$ in the same family
}{\cal{A}}^{\bullet}[i,j].$

We call the ${\cal{A}}^{\bullet}[i]$ the layers of the ball cover
${\cal{A}}^{\bullet}:=\bigcup_{i\geq 1}{\cal{A}}^{\bullet}[i]$ of $\Sigma$
around the points of $A^{\bullet}:=\bigcup_{i\geq
1}A^{\bullet}[i]=\bigcup_{i\geq 1,c\geq j\geq 1}A^{\bullet}[i,j]$. The
${\cal{A}}^{\bullet}[i,j]$ are the sublayers of the ${\cal{A}}^{\bullet}[i]$.

###### Remark 3.6

1\. To explain the meaning of 3.5 we anticipate that in the assembly of global
bumps in Ch.3.3 we transfer a periodic bump to $\overline{B_{R\cdot
s^{i}}(p)}\in{\cal{A}}^{\bullet}[i]$ that shields all balls
$\overline{B_{s^{i}}(x)}\subset\overline{B_{R\cdot s^{i}}(p)}$ with
$\overline{B_{s^{i}}(x)}\in{\cal{A}}[i]$. The balls in
${\cal{A}}^{\bullet}[i]$ may intersect but we can ensure that the support of
the bumps in the at most $c(n,R)$ different sublayers
${\cal{A}}^{\bullet}[k,i]$ are disjoint. For balls in
${\cal{A}}^{\bullet}[i]$, ${\cal{A}}^{\bullet}[j]$, with $i\neq j$, the bumps
are disjoint when $s>0$ is sufficiently small.
2\. The covering number $c(H)$ depends only on $n$ if $\varepsilon\cdot R\leq
r_{H,\Phi}$, for $r_{H,\Phi}>0$ in [L1, Th. 3.4], or when
$H\in{\cal{H}}^{\mathbb{R}}_{n}$, since, cf. (85) below, in both cases the
Ahlfors constants depend only on $n$. From (82) we see that for any
$t\in(0,10)$ and any point $z\in H$:

(83) $\char 35\relax\\{p\in A^{\bullet}[i]\,|\,z\in B_{t\cdot R\cdot
s^{i}}(p)\\}\leq c.$

3\. After scaling the balls in ${\cal{A}}^{\bullet}[i]$ by $(\varepsilon\cdot
s^{i}/\varepsilon)^{-1}=s^{-i}$, $B_{s^{i}}(p)$ transforms to $B_{1}(p)$ and
$B_{R\cdot s^{i}}(p)$ becomes $B_{R}(p)$ with a tangent cone approximating
with accuracy $\varepsilon$ over
$\mathbb{TP}_{C}(0,\varepsilon,1,\varepsilon)$ respectively over
$\mathbb{TP}_{C}(0,\varepsilon,R,\varepsilon)$, since
$R\leq\varepsilon^{-1}/10$. We observe that
$\mathcal{R}_{\varepsilon}(p)<\mathrm{diam}(H,d_{\mathcal{S}})$, for any $p\in
H$. Thus, the Ahlfors regularity estimates (11) apply to the balls in these
covers. $\Box$

Proof We inductively define two different selection maps $\kappa$ on
$\Sigma^{*}$. In a first step we select $A\subset\Sigma^{*}$ for the small
balls $B_{s^{i}}$ and then $A^{\bullet}\subset A$ for the large balls
$B_{R\cdot s^{i}}$.

Step 1 (Small Cover ${\cal{A}}$) We define
$\kappa_{i}:\Sigma^{*}_{i}\rightarrow\mathbb{Z}_{2}.$ When $\kappa_{i}(p)=0$,
then we delete $\overline{B_{s^{i}}(p)}$, otherwise, we keep it. To start the
(double) induction over $i$, we define $\kappa_{i}$ on the _first non-empty_
$\Sigma^{*}_{i}$:

Start on $\Sigma^{*}_{i}$: For $a_{i,1}\in\Sigma^{*}_{i}$, we set
$\kappa_{i}(a_{i,1}):=1$.

Step on $\Sigma^{*}_{i}$: We assume $\kappa_{i}$ has been defined for
$a_{i,j}$, $j\leq m$. Then we set

$\kappa_{i}(a_{i,m+1}):=0,\,\mbox{ if }\,a_{i,m+1}\in\textstyle\bigcup_{j\leq
m}\\{\overline{B_{s^{i}}(a_{i,j})}\,|\,a_{i,j}\in\Sigma^{*}_{i},\kappa_{i}(a_{i,j})=1\\},\mbox{
otherwise }\kappa_{i}(a_{i,m+1}):=1.$

Start on $\Sigma^{*}_{i+1}$: Assuming $\kappa_{a}$ on $\Sigma^{*}_{a}$ has
been defined for any $a\leq i$ we continue with the definition of
$\kappa_{i+1}$ on $\Sigma^{*}_{i+1}$. We start with
$a_{i+1,1}\in\Sigma^{*}_{i+1}$ and set $\kappa_{i+1}(a_{i+1,1}):=0$ when

$a_{i+1,1}\in\textstyle\bigcup_{l\leq
i}\\{\overline{B_{s^{l}}(a_{l,j})}\,|\,a_{l,j}\in\Sigma^{*}_{l},\kappa_{l}(a_{l,j})=1\\}\,\mbox{
and }\kappa_{i+1}(a_{i+1,1}):=1\mbox{ otherwise.}$

Step on $\Sigma^{*}_{i+1}$: We assume $\kappa_{i+1}$ has been defined for
$a_{i+1,j}$, $j\leq m$. When

$\textstyle\textstyle a_{i+1,m+1}\in$ $\textstyle\bigcup_{j\leq
m}\\{\overline{B_{s^{i+1}}(a_{i+1,j})}\,|\,a_{i+1,j}\in\Sigma^{*}_{i+1},\kappa_{i+1}(a_{i+1,j})=1\\}\,\cup$
$\textstyle\bigcup_{l\leq
i}\\{\overline{B_{s^{l}}(a_{l,j})}\,|\,a_{l,j}\in\Sigma^{*}_{l},\kappa_{l}(a_{l,j})=1\\},\mbox{
we set }\kappa_{i+1}(a_{i+1,m+1}):=0$

and otherwise, $\kappa_{i+1}(a_{i+1,m+1}):=1$. Now we define for $i\geq 0$ the
following sets of balls and of center points of these balls

  * •

$\textstyle{\cal{A}}[i]:=\\{\overline{B_{s^{i}}(p)}\,|\,p\in\Sigma^{*}_{i},\kappa_{i}(p)=1\\},\,{\cal{A}}:=\bigcup_{i\geq
0}{\cal{A}}[i],$

  * •

$\textstyle
A[i]:=\\{p\in\Sigma^{*}_{i}\,|\,\kappa_{i}(p)=1\\},\,A:=\bigcup_{i\geq
0}A[i].$

We observe that
$\overline{B_{s^{i}/3}(x)}\cap\overline{B_{s^{j}/3}(y)}=\emptyset$, for $x\neq
y,x\in A[i],y\in A[j]$. From this the Ahlfors $n$-regularity, [L1, Remark
3.7], of $(H,d_{H},\mu_{H})$ on $H\in{\cal{G}}_{n}$ shows that in a given ball
$B\subset H$ there are at most _finitely_ many balls belonging to
${\cal{A}}[i]$, for any given $i\geq 0$. We infer that

(84) $\textstyle\bigcup_{i\leq k}{\cal{A}}[i]\mbox{ covers }\bigcup_{i\leq
k}\Sigma_{i}\mbox{ for any }k\geq 0.$

Otherwise, there were a $q\in\Sigma_{l}\setminus\bigcup_{p\in\bigcup_{i\leq
k}A[i]}\overline{B_{s^{i}}(p)}$, for some $l\leq k$. The local finiteness of
each ${\cal{A}}[i]$ shows that $\bigcup_{p\in\bigcup_{i\leq
k}A[i]}\overline{B_{s^{i}}(p)}$ is a closed subset of $H$ and, thus, $q$
belongs to the _open_ complement. Since $\Sigma^{*}_{l}\subset\Sigma_{l}$ is
dense, there is some
$q^{\prime}=a_{l,m}\in\Sigma^{*}_{l}\setminus\bigcup_{p\in\bigcup_{i\leq
k}A[i]}\overline{B_{s^{i}}(p)}$. For $\kappa[l]_{1}(a_{l,m})=0$,
$a_{l,m}\in\bigcup_{p\in
A[l]}\overline{B_{s^{i}}(p)}\subset\bigcup_{p\in\bigcup_{i\leq
k}A[i]}\overline{B_{s^{i}}(p)}$. Alternatively, for
$\kappa[l]_{1}(a_{l,m})=1$, we have
$a_{l,m}\in\overline{B_{s^{i}}(a_{l,m})}\subset\bigcup_{p\in\bigcup_{i\leq
k}A[i]}\overline{B_{s^{i}}(p)}$. That is, both cases lead to contradictions.

Step 2 (Large Cover ${\cal{A}}^{\bullet}$) Now we select subfamilies
${\cal{A}}^{\bullet}[i]$ of $\\{\overline{B_{R\cdot s^{i}}(p)}\subset
H\,|\,p\in A[i]\\}$ with (81) and (82). We define another selection map
$\kappa_{i}^{\bullet}:A[i]\rightarrow\mathbb{Z}^{\geq 0}$, writing the
elements of $A[i]$ as $p_{i,1},p_{i,2},..$. When $\kappa_{i}^{\bullet}(p)=0$,
then we delete $\overline{B_{R\cdot s^{i}}(p)}$, otherwise, we keep it.

Start: For $p_{i,1}\in A[i]$, we set $\kappa_{i}^{\bullet}(p_{i,1}):=1$.

Step: We assume $\kappa_{i}^{\bullet}$ has been defined for $p_{i,j}$, $j\leq
m$. We set

$\kappa_{i}^{\bullet}(p_{i,j+1}):=0,\mbox{ when
}\overline{B_{s^{i}}(p_{i,j+1})}\subset\textstyle\bigcup_{k\leq
j}\\{\overline{B_{R\cdot
s^{i}}(p_{i,k})}\,|\,\kappa_{i}^{\bullet}(p_{i,k})\geq 1\\},\mbox{ and
otherwise }$ $\kappa_{i}^{\bullet}(p_{i,j+1}):=\min\big{(}\\{k\leq
j\,|\,\kappa_{i}^{\bullet}(p_{i,k})\geq 1\mbox{ with
}d(p_{i,j+1},p_{i,k})>20\cdot R\cdot s^{i}\\}\cup\\{j+1\\}\big{)}.$

For $j\geq 1$ we define the following sets of balls and of center points of
these balls

  * •

$\textstyle{\cal{A}}^{\bullet}[i,j]:=\\{\overline{B_{R\cdot
s^{i}}(p)}\,|\,p\in
A[i],\kappa_{i}^{\bullet}(p)=j\\},\,{\cal{A}}^{\bullet}:=\bigcup_{i\geq
0}{\cal{A}}^{\bullet}[i]:=\bigcup_{i\geq 0,j\geq 1}{\cal{A}}^{\bullet}[i,j],$

  * •

$\textstyle A^{\bullet}[i,j]:=\\{p\in
A[i]\,|\,\kappa_{i}^{\bullet}(p)=j\\},\,A^{\bullet}:=\bigcup_{i\geq
0}A^{\bullet}[i]:=\bigcup_{i\geq 0,j\geq 1}A^{\bullet}[i,j]$.

From the induction step we notice that $\textstyle\bigcup_{x\in
A[i]}\overline{B_{s^{i}}(x)}\subset\bigcup_{z\in
A^{\bullet}[i]}\overline{B_{R\cdot s^{i}}(z)}$ and
$z^{*}\notin\overline{B_{(R-1)\cdot s^{i}}(z)}$, for $z\neq z^{*},z,z^{*}\in
A^{\bullet}[i]$. From Ahlfors $n$-regularity (11) we have for any $z\in H$:

(85) $\mu_{H}(B_{R/4\cdot s^{i}}(z))\geq A\cdot(R/4\cdot s^{i})^{n}\mbox{ and
}\mu_{H}(B_{20\cdot R\cdot s^{i}}(z))\leq B\cdot(20\cdot R\cdot s^{i})^{n},$

Now we set $c:=[\mbox{\emph{the smallest integer}}\geq 100^{n}\cdot B/A]$ and
we claim ${\cal{A}}^{\bullet}[i,j]=\emptyset\mbox{ for }j>c.$

Otherwise, we had some $\overline{B_{R\cdot
s^{i}}(p)}\in{\cal{A}}^{\bullet}[i,c+1]$. Then there are at least $c$
different $p_{m}\in A^{\bullet}[i,j_{m}]$ with $B_{10\cdot R\cdot
s^{i}}(p)\cap B_{10\cdot R\cdot s^{i}}(p_{m})\neq\emptyset$. Since the
$B_{R/4\cdot s^{i}}(x_{m})$ are pairwise disjoint we get $\mu_{H}(B_{20\cdot
R\cdot s^{i}}(p))\geq 100^{n}\cdot B/A\cdot A\cdot(R/4\cdot s^{i})^{n}$. But
this contradicts the upper estimate for $\mu_{H}(B_{20\cdot R\cdot
s^{i}}(p))$, and, hence, we have ${\cal{A}}^{\bullet}[i,c+1]=\emptyset$.
$\Box$

###  3.2 Transfer of Local Bumps

We cannot transfer shields on cones $C$ via $\mathbf{ID}$-maps to a general
singular $H\in{\cal{G}}$ since they intersect $\Sigma_{C}$. Instead we
transfer periodic bumps from cones and use them to reproduce shields on $H$.
(But we use the shields on $C$ to derive estimates for shields on $H$ in 3.9
below.) We consider $\bigcup_{v\in S}(Q^{n-k}_{1}+v)\subset
Q^{n-k}_{L,...,L}\subset\mathbb{R}^{n-k}$, for some $L>0$, and the periodic
bump $\boldsymbol{\square}(\bigcup_{v\in S}(Q^{n-k}_{1}+v))$ on
$\mathbb{R}^{n-k}\times C^{k}\in{\cal{SC}}_{n}$ for $\omega_{0}(n,L)$,
$R_{0}(n,L)$, $r_{0}(n,L)$ with $\omega\in(0,\omega_{0})$, $R>R_{0}$,
$r\in(0,r_{0})$ and some
$\beta_{\bullet}(\omega,r,R)\in(0,\beta_{n-k}\cdot\eta_{n-k})$:

(86) $\textstyle\operatorname{supp}\boldsymbol{\square}(\bigcup_{v\in
S}(Q^{n-k}_{1}+v))\subset\delta^{-1}_{\langle
A\rangle^{*}}(\mathbb{R}^{\geq\beta_{\bullet}})\cap
B_{R}(0)\subset\mathbb{TP}_{\mathbb{R}^{n-k}\times C^{k}}(0,\omega,R,r)$

###### Corollary 3.7

_(Periodic Bumps on $H$)_ For $H\in{\cal{G}}$, some $p\in\Sigma_{H}$ and a
tangent cone $(\mathbb{R}^{n-k}\times C^{k},0)$ in $p$ so that the
$\mathbf{ID}$-map over $\mathbb{TP}_{\mathbb{R}^{n-k}\times
C^{k}}(0,\omega,R,r)\subset C^{n}$, for $\omega\in(0,\omega_{0})$, $R>R_{0}$,
$r\in(0,r_{0})$, with $\Psi_{\circ}$ appropriately normalized, satisfies, for
some $\varepsilon>0$:

(87)
$|\mathbf{ID}-id_{C}|_{C^{5}(\mathbb{TP}_{C}(0,\omega,R,r))}\leq\varepsilon\mbox{
and
}|\Psi_{H}\circ\mathbf{ID}/\Psi_{\circ}-1|_{C^{2,\alpha}(\mathbb{TP}_{C}(0,\omega,R,r))}\leq\varepsilon.$

For some small $\varepsilon_{0}(n)\in(0,1/10)$,
$\varepsilon\in[0,\varepsilon_{0}]$, (86) shows that
$\Phi^{H}_{\square}:=\Phi_{\square}\circ\mathbf{ID}^{-1}$ is well-defined and
extends to $H\setminus\Sigma_{H}$ with

(88) $L_{H^{n}}(\Phi^{H}_{\square}\cdot\Psi_{H})\geq\lambda/16\cdot\langle
A\rangle^{2}\cdot\Phi^{H}_{\square}\cdot\Psi_{H}\mbox{ on
}H\setminus\Sigma_{H}.$

We call $\big{(}\Phi^{H}_{\square}\cdot\Psi_{H}\big{)}^{4/(n-2)}\cdot g_{H}$ a
periodic bump metric and the pseudo metric
$\big{(}(\Phi^{H}_{\square}-1)\cdot\Psi_{H}\big{)}^{4/(n-2)}\cdot g_{H}$ a
periodic bump.

###### Remark 3.8

We have $\Phi^{H}_{\square}\in{\cal{C}}(H^{n},\Lambda[n])$,
$\operatorname{supp}|\Phi^{H}_{\square}-1|=\operatorname{supp}(\boldsymbol{\square}\circ\mathbf{ID}^{-1})$.
When we add
$\boldsymbol{\square}_{H}:=\boldsymbol{\square}\circ\mathbf{ID}^{-1}$ to
$\Psi_{H}^{4/(n-2)}\cdot g_{H}$ this may change the conformal class since
$\mathbf{ID}$ need not to be conformal. Using the multiplicative local bump
$\Phi^{H}_{\square}$ resolves this issue. The two resulting deformations
coincide in the limit of $\varepsilon\rightarrow 0$ and we continue speaking
of added local bumps but technically we use the multiplicative version. $\Box$

Tightness of Bumps on $H$ We use the Ahlfors $n$-regularity of
$(H,d^{\Lambda}_{\mathcal{S}},\mu^{\Lambda}_{\mathcal{S}})$ to extend the
tightness result 2.4 to the $\mathbf{ID}$-images of bump elements in
$H\in{\cal{G}}_{n}$ by some perturbation argument when $H$ is close to a
product tangent cone in the sense that (87) of Lemma 3.7 holds for
$\varepsilon\in[0,\varepsilon_{1}]$, for some
$\varepsilon_{1}[n,\ell]\in(0,\varepsilon_{0}[n,\ell])$. In general
$\Sigma_{H}$ has no local product structure. For a given product tangent cone
$\mathbb{R}^{n-k}\times C^{k}$ we can, however, use induced structures when
$H$ is (locally) closely approximated by $\mathbb{R}^{n-k}\times C^{k}$ in a
common ambient space $\mathbb{R}^{n+1}$.

(89) $\Omega_{\ell}:=Q_{\ell}^{n-k}\times\mathbb{R}^{k+1},\mbox{ matching
(\ref{ome}) from }Q_{\ell}^{n-k}\times C^{k}=\mathbb{R}^{n-k}\times C^{k}\cap