# Min-max minimal hypersurfaces with higher multiplicity

Zhichao Wang  Department of Mathematics, University of British Columbia,
Vancouver, BC V6T 1Z2, Canada<EMAIL_ADDRESS>and  Xin Zhou  Department
of Mathematics, 531 Malott Hall, Cornell University, Ithaca, NY 14853, and
Department of Mathematics, University of California Santa Barbara, Santa
Barbara, CA 93106, USA<EMAIL_ADDRESS>
###### Abstract.

We exhibit the first set of examples of non-bumpy metrics on the
$(n+1)$-sphere ($2\leq n\leq 6$) in which the varifold associated with the
two-parameter min-max construction must be a multiplicity-two minimal
$n$-sphere. This is proved by a new area-and-separation estimate for certain
minimal hypersurfaces with Morse index two inspired by an early work of
Colding-Minicozzi. We also construct non-bumpy projective spaces in which the
first min-max hypersurfaces are one-sided, and non-bumpy balls in which the
free boundary min-max hypersurfaces are improper.

##  1\. Introduction

In the past decade, we have witnessed many important advancement in the
development for minimal hypersurfaces, including the solution of Yau
conjecture on minimal surfaces by Marques-Neves [MN17], and Song [Song18], and
the establishment of a Morse theory for the area functional by Marques-Neves
[MN16][MN18]. One key challenge in these works is the a priori existence of
integer multiplicity of the varifolds produced by the Almgren-Pitts min-max
theory [Alm62][Alm65][Pi] (see also [Colding-DeLellis03, DeTa]). Now we have
very well understanding of the multiplicity when the ambient manifold has a
bumpy metric ([Whi91]) thanks to the solution of the Multiplicity One
Conjecture [Zhou19]; (see also [CM20]). For non-bumpy metrics, there are known
trivial examples where some min-max varifolds have higher multiplicities,
while some others have multiplicity one. For instance, on a thin and long flat
torus, the min-max varifold can be two identical copies of the cross section,
but one can move one copy parallelly away to obtain a multiplicity one
varifold of the same mass. This made it tempting to conjecture that for any
metric there always exists a min-max varifold of multiplicity one. However, in
this paper, we will disprove this conjecture by constructing the first set of
nontrivial and non-bumpy examples, where the varifold associated with the two-
parameter min-max construction must have multiplicity two.

######  Theorem 1.1.

The $(n+1)$-sphere $S^{n+1}$ of dimension $3\leq(n+1)\leq 7$ admits metrics
$g$ with non-negative Ricci curvature so that the second volume spectrum
$\omega_{2}(S^{n+1},g)$ can only be achieved by a degenerate minimal
$n$-sphere with multiplicity 2.

Here $\\{\omega_{k}\\}$ is the volume spectrum of $M$ defined by Gromov
[Gro88], Guth [Guth09], and Marques-Neves [LMN16] as a sequence of non-
decreasing positive numbers

$0<\omega_{1}(M,g)\leq\omega_{2}(M,g)\leq\cdots\leq\omega_{k}(M,g)\to\infty,$

depending only on $M$ and $g$; see also [Lio16][GL17][Sab17][Wang21] for
further studies.

Our proof relies on a new area comparison argument. Roughly speaking, we show
that if a connected multiplicity-one minimal hypersurface $\Sigma$ is
sufficiently close to a multiplicity-two degenerate stable hypersurface
$S^{n}_{0}$, then the area of $\Sigma$ is strictly greater than that of
$S_{0}^{n}$. Similar area comparison arguments have played essential roles in
the study of Morse index and multiplicity of min-max minimal hypersurfaces
(see [MN12, Zhou15, Zhou17, KMN16]), but in a reserve way. In particular, the
catenoid estimates by Ketover-Marques-Neves [KMN16] imply that if $S^{n}_{0}$
is unstable, then by pushing away the two copies of $S^{n}_{0}$ and then
adding a catenoid type neck therein, one can strictly decrease the area. More
precisely, the area-decrease by pushing away the two copies of $S^{n}_{0}$
dominates the area-increase by adding the catenoid neck, when $S^{n}_{0}$ is
unstable. This idea has been further extended by Haslhofer-Ketover [HK19] (see
also [Dea]) to construct the second minimal 2-sphere in $S^{3}$ with a bumpy
metric associated with the second width. In our situation, we will show the
reverse phenomenon when $S^{n}_{0}$ is degenerate stable. A crucial ingredient
is to bound the area difference between $\Sigma$ and $2S^{n}_{0}$ away from
the neck region. To do so, we need a new distance-separation estimate between
$\Sigma$ and $S^{n}_{0}$ in terms of the size of the neck region; see Theorem
4.6 and 4.9. This part is inspired by an early work of Colding-Minicozzi
[CM02]; see the part on idea of proof for more details.

The Multiplicity One Theorem also implies that for bumpy metrics, the min-max
minimal hypersurfaces can only be two-sided. However, our method can also be
used to construct non-bumpy metrics on the projective spaces in which the
first width must be achieved by one-sided hypersurfaces.

######  Corollary 1.2.

The $(n+1)$-projective space $\mathbb{RP}^{n+1}$ with $3\leq(n+1)\leq 7$
admits metrics $g$ so that the first volume spectrum
$\omega_{1}(\mathbb{RP}^{n+1},g)$ can only be achieved by a one-sided
$\mathbb{RP}^{n}$ with multiplicity 2.

In [LZ16], Li-Zhou developed a free boundary min-max theory for compact
manifolds with boundary $(M,\partial M)$. The minimal hypersurface $\Sigma$
constructed therein may not be properly embedded, i.e. it may have interior
touching with the boundary $\operatorname{int}(\Sigma)\cap\partial
M\neq\emptyset$. The touching phenomenon had caused major challenges in the
application of this theory, e.g. [GLWZ19, Wang_21_2, SWZ]. It has been
conjectured that the touching could happen even for Euclidean domains [LZ16,
Conjecture 1.5]. Here we exhibit examples where the touching phenomenon does
happen.

######  Corollary 1.3.

$B^{n+1}$ admits a metric with minimal boundary and non-negative Ricci
curvature so that its first volume spectrum can only be achieved by its
boundary with multiplicity one.

###### Proof.

Let $(S^{n+1},g)$ be as in Theorem 1.1. Then the round $S_{0}^{n}$ divides the
$(n+1)$-sphere into two connected components, denoted by $M_{+}$ and $M_{-}$;
see Section 3 for explicit description. Note that $(M_{+},g)$ is an
$(n+1)$-ball that has positive Ricci curvature away from its boundary. By
[SWZ], $\omega_{1}(M_{+},g)$ is realized by a free boundary minimal
hypersurface $\Sigma$ with multiplicity one, whose index is bounded by one
from above. Then by reflection along $S_{0}^{n}$, one can obtain a smoothly
embedded minimal hypersurface $\widetilde{\Sigma}\subset(S^{n+1},g)$ with
index less than or equal to 2. Clearly,
$\omega_{1}(M_{+},g)\leq\mathrm{Area}(S_{0}^{n})$. It follows that

$\mathrm{Area}(\widetilde{\Sigma})\leq 2\mathrm{Area}(S_{0}^{n}).$

Then by Theorem 1.1, $\widetilde{\Sigma}$ can only be the $S_{0}^{n}$ with
multiplicity two, which implies that $\Sigma$ is the boundary of $M_{+}$. ∎

### Idea of the proof

We construct a sequence of Riemannian $(n+1)$-spheres $M_{k}$ (isometrically
embedded in $\mathbb{R}^{n+2}$) that converges locally smoothly to
$S_{0}^{n}\times\mathbb{R}$ (see Section 3 for details), where $S_{0}^{n}$ is
a round $n$-sphere in $\mathbb{R}^{n+1}$ and embedded in $M_{k}$ as the unique
degenerate stable minimal hypersurface for each $k$. Moreover, for each $k$,
the level sets (denoted by $\\{S_{t}\\}$) of the distance function to
$S_{0}^{n}$ have area lower bound $\Omega_{n}(1-|t|^{n+1})$, where
$\Omega_{n}$ is the volume of the unit $n$-sphere.

Suppose on the contrary that each $\omega_{2}(M_{k})$ is realized by a minimal
hypersurface $\Sigma_{k}$ that is not $S_{0}^{n}$. Then by our construction,
$\Sigma_{k}$ has to be connected, unstable, of index less than or equal to 2
(by [MN16]) and multiplicity one ([Zhou19]). Then the compactness [Sharp17]
gives that $\Sigma_{k}$ converges locally smoothly to an embedded minimal
hypersurface with integer multiplicity in $S_{0}^{n}\times\mathbb{R}$. By the
lower bounds for $\omega_{2}(M)$, the limit of $\Sigma_{k}$ is exactly
$S_{0}^{n}$ with multiplicity 2. Moreover, by the classification of embedded
minimal hypersurfaces with two ends ([Sch83_1]), after suitable scaling, the
blowup around each singular point is a standard catenoid. Let $r_{k}$ denote
the “radius” of link of the small catenoid in $\Sigma_{k}$, i.e. the distance
of the small catenoid to the center point. First, the Hausdorff distance
between $\Sigma_{k}$ and $S_{0}^{n}$ is bounded from above by $10r_{k}|\log
r_{k}|$ (to be discussed next). After replacing the annuli in neck regions of
$\Sigma_{k}$ by two minimizing $n$-disks, by using the one-sided minimizing
property of $\\{S_{t}\\}$, the area of the new hypersurface is at least
$2|S^{n}_{0}|-\mathcal{O}(r^{n+1}_{k}|\log r_{k}|^{n+1})$; see (3.4). On the
other hand, the neck regions of $\Sigma_{k}$ contribute at least
$c(n)r_{k}^{n}$ amount of area more than that of the $n$-disks; see (3.2).
Therefore, $|\Sigma_{k}|-2|S^{n}_{0}|\geq
c(n)r_{k}^{n}-\mathcal{O}(r^{n+1}_{k}|\log r_{k}|^{n+1})>0$, contradicting
$\omega_{2}(M_{k})\leq 2\omega_{1}(M_{1})\leq 2\mathrm{Area}(S_{0}^{n})$.

To bound the Hausdorff distance $d_{H}(\Sigma_{k},S_{0}^{n})$, the key
challenge arises from intermediate annuli regions
$A(y_{k},R_{k},\epsilon;M_{k})$, where $y_{k}$ is a singular point,
$R_{k}/r_{k}\nearrow\infty$, $R_{k}\searrow 0$. In fact, $\Sigma_{k}$ is close
in smooth topology to the catenoid of radius $r_{k}$ inside
$B(y_{k},R_{k};M_{k})$ and to $2S^{n}_{0}$ outside $B(y_{k},\epsilon;M_{k})$
by smooth convergence. However, $\Sigma_{k}$ may not be a 2-sheeted graph over
$S^{n}_{0}$ inside $A(y_{k},R_{k},\epsilon;M_{k})$, and we are forced to write
one component $\Sigma_{k}^{2}$ (of $\Sigma_{k}$) as a graph over the other
$\Sigma_{k}^{1}$. First, $d_{H}(\Sigma_{k}^{1},\Sigma_{k}^{2})$ when
restricted to $\partial B(y_{k},R_{k};M_{k})$ is at most
$3r_{k}\log\frac{R_{k}}{r_{k}}$; see (4.19). We need to prove that this bound
keeps roughly at the order $r_{k}\log\frac{s}{r_{k}}$ on $\partial
B(y_{k},s;M_{k})$ when $s\nearrow\epsilon$. Our proof relies on several new
monotonicity formulas inspired by Colding-Minicozzi [CM02]. In particular, we
study carefully the evolution of the averages of the height function $w_{k}$
(between $\Sigma_{k}^{1},\Sigma_{k}^{2}$):
$s\mapsto\mathop{\ooalign{$\int$\cr$-$}}_{\Sigma_{k}^{1}\cap\partial
B(y_{k},s;M_{k})}w_{k}$; see Proposition 4.4 and compare with [CM02, Lemma
2.1]. The main new challenge as compared with [CM02] is that $w_{k}$ is not a
graph function over a fixed hypersurface, and this causes many higher-order
terms in our monotonicity formulas, particularly as the height function
$w_{k}$ only satisfies a highly nonlinear PD-inequality (4.6). Note that some
extra care is needed when the two singular points are not too far from each
other; see Proposition 4.5.

### Outline

In Section 2, we will classify the limit cones of the singular set arising
from the compactness of minimal hypersurfaces. In Section 3, we describe the
concrete constructions and prove our main results by assuming the key
Hausdorff distance upper bound estimates, which will be proved in Section 4.
Finally in Appendix A, we derive a general inequality for minimal graphs over
another minimal hypersurface. Then we list some basic results of catenoids in
Appendix B.

### Acknowledgments

We would like to thank Professor Gang Tian for his constant support and
encouragement. X.Z. was supported by NSF grant DMS-1945178, and an Alfred P.
Sloan Research Fellowship.

##  2\. Blowing-up analysis

Let $\\{M_{k}\\}$ be a sequence of $(n+1)$-spheres embedded in
$\mathbb{R}^{n+2}=\mathbb{R}^{n+1}\times\mathbb{R}$. Denote by $S_{0}^{n}$ the
unit sphere in $\mathbb{R}^{n+1}$. Let $\Sigma_{k}$ be a closed embedded
connected minimal hypersurface in $M_{k}$. We use $B_{r}(p)$ and
$B(p,r;M_{k})$ to denote the geodesic ball in $\mathbb{R}^{n+1}$ and $M_{k}$,
respectively. In this section, $M_{k}$ and $\Sigma_{k}$ always satisfy the
following requirements.

  1. (1)

For each $k$, $S_{0}^{n}$ is a stable minimal hypersurface in $M_{k}$ with
constant Jacobi fields;

  2. (2)

$M_{k}$ locally smoothly converges to $S_{0}^{n}\times\mathbb{R}$ in
$\mathbb{R}^{n+2}$ as $k\to\infty$;

  3. (3)

$\Sigma_{k}$ converges to twice of $S_{0}^{n}$ in the sense of varifolds;

  4. (4)

$\Sigma_{k}$ has Morse index less than or equal to 2.

######  Example 1.

Let $M_{k}$ be the $(n+1)$-sphere given by

$x_{1}^{2}+x_{2}^{2}+\cdots+x_{n+1}^{2}+\frac{x_{n+2}^{2n}}{k^{2n}}=1.$

Suppose that $M_{k}$ contains an embedded minimal hypersurface $\Sigma_{k}$
with index less than or equal to 2 and

$\mathrm{Area}(S^{n}_{0})+\delta\leq\mathrm{Area}(\Sigma_{k})\leq
3\mathrm{Area}(S^{n}_{0})-\delta$

for some $\delta>0$. Then $\\{\Sigma_{k}\\}$ and $\\{M_{k}\\}$ satisfy our
requirements in this section.

By compactness [Sharp17], $\Sigma_{k}$ locally smoothly converges to
$S_{0}^{n}$ with multiplicity two away from a set $\mathcal{W}$ consisting of
one or two points. Let $p\in\mathcal{W}$. Then we can take a positive constant
$\epsilon<10^{-1000}$ small enough such that the convergence for $\Sigma_{k}$
is locally smooth in $B_{2\epsilon}(p)\setminus\\{p\\}$, and

(2.1)
$\mathrm{Area}(B(p,2\epsilon;M_{k})\cap\Sigma_{k})\leq\frac{5}{2}\cdot\frac{\Omega_{n-1}}{n}(2\epsilon)^{n},$

where $\Omega_{m}$ is the volume of unit $m$-spheres. Choose
$p_{k}\in\Sigma_{k}$ so that

$|A^{\Sigma_{k}}(p_{k})|=\max_{\Sigma_{k}\cap
B(p,\epsilon;M_{k})}|A^{\Sigma_{k}}(x)|.$

Clearly, $|A^{\Sigma_{k}}(p_{k})|\to\infty$ as $k\to\infty$. In the following,
we classify the limit cones of $\\{\Sigma_{k}\\}$ at $p$, which is either a
hyperplane with multiplicity or a catenoid.

######  Proposition 2.1 (Classification of limit cones).

Let $\\{c_{k}\\}$ be a sequence of positive numbers with $c_{k}\to\infty$.

  1. (1)

If $\lim_{k\to\infty}|A^{\Sigma_{k}}(p_{k})|/c_{k}=\ell>0$, then
$c_{k}(\Sigma_{k}-p_{k})$ locally smoothly converges to a catenoid contained
in $\mathbb{R}^{n+1}$;

  2. (2)

If $\lim_{k\to\infty}|A^{\Sigma_{k}}(p_{k})|/c_{k}=0$, then
$c_{k}(\Sigma_{k}-p_{k})$ locally smoothly converges to a hyperplane contained
in $\mathbb{R}^{n+1}$;

  3. (3)

If $\lim_{k\to\infty}|A^{\Sigma_{k}}(p_{k})|/c_{k}=\infty$, then
$c_{k}(\Sigma_{k}-p_{k})$ converges to a multiplicity-two hyperplane contained
in $\mathbb{R}^{n+1}$. Moreover, the convergence is locally smooth away from
at most two points including 0.

###### Proof.

Note that $c_{k}(\Sigma_{k}-p_{k})$ is always a minimal hypersurface in
$c_{k}(M_{k}-p_{k})$ with $\operatorname{index}\leq 2$. By the monotonicity
formula and (2.1),

(2.2) $\lim_{k\to\infty}\mathrm{Area}(B_{r}(0)\cap
c_{k}(\Sigma_{k}-p_{k}))\leq\frac{5}{2}\cdot\frac{\Omega_{n-1}}{n}r^{n}.$

Clearly, $c_{k}(M_{k}-p_{k})$ converges locally smoothly to
$\mathbb{R}^{n+1}$. Thus $c_{k}(\Sigma_{k}-p_{k})$ converges to an embedded
minimal hypersurface with integer multiplicity in $\mathbb{R}^{n+1}$ with
$\operatorname{index}\leq 2$. By the work of Tysk [Tys], the limit minimal
hypersurface, which has finite Morse index and polynomial volume growth (2.2),
must have a unique tangent cone at infinity; therefore using the work of
Schoen [Sch83_1]*Theorem 3 (see also [HHW]*Theorem 1.3), the limit minimal
hypersurface is either a catenoid or a hyperplane.

If $\lim_{k\to\infty}|A^{\Sigma_{k}}(p_{k})|/c_{k}=\ell<\infty$, then for any
compact set $\Omega\subset\mathbb{R}^{n+2}$,
$c_{k}(\Sigma_{k}-p_{k})\cap\Omega$ has uniformly bounded second fundamental
form, and this implies that the convergence is locally smooth. When $\ell>0$,
the limit hypersurface has at least one point that has non-zero curvature.
Thus the limit is a catenoid. When $\ell=0$, the limit hypersurface is flat
and hence is a hyperplane.

If $\lim_{k\to\infty}|A^{\Sigma_{k}}(p_{k})|/c_{k}=\infty$, then the
convergence is non-smooth. By Allard’s Regularity Theorem [Si]*Theorem 24.2,
the multiplicity of the convergence is larger than or equal to two. Thus the
limit is a stable minimal hypersurface in $\mathbb{R}^{n+1}$ with polynomial
(degree $n$) area growth, which can only be a hyperplane by Schoen-Simon [SS]
and Schoen-Simon-Yau [SSY]. Since the index of $\Sigma_{k}$ is bounded from
above by 2, the convergence is locally smooth away from at most two points.
Together with (2.2), we also have that the multiplicity of the convergence is
exactly 2. Hence Proposition 2.1 is proved. ∎

######  Remark 2.2.

Let $x_{k}\in\Sigma_{k}$ and $c_{k}\to\infty$. Then $c_{k}(\Sigma_{k}-x_{k})$
converges to a multiplicity-one catenoid or hyperplane with multiplicity one
or two in the sense of varifold. Suppose that the convergence is not locally
smooth. Then the limit is a multiplicity-two hyperplane.

Denote by $r_{k,1}=\sqrt{n(n-1)}|A^{\Sigma_{k}}(p_{k})|^{-1}$. Take
$y_{k,1}\in\Sigma_{k}$ so that $r_{k,1}^{-1}(\Sigma_{k}-y_{k,1})$ converges to
a standard catenoid $\mathcal{C}$ in $\mathbb{R}^{n+1}$; see Appendix B for
several properties of the geometry of $\mathcal{C}$. In the remaining of this
section, we are going to find a sequence of “bad balls”, in which $\Sigma_{k}$
is not flat enough even after rescaling.

So long as

$\max_{x\in
B(y_{k,1},\epsilon;M_{k})\cap\Sigma_{k}}|A^{\Sigma_{k}}(x)||x-y_{k,1}|<2\sqrt{n(n-1)},$

we let $y_{k,2}:=y_{k,1}$. Otherwise, take $y^{\prime}_{k,2}\in\Sigma_{k}$ so
that

$\max_{x\in
B(y_{k},\epsilon;M_{k})\cap\Sigma_{k}}|A^{\Sigma_{k}}(x)|x-y_{k}|=|A^{\Sigma_{k}}(y_{k,2}^{\prime})||y_{k,2}^{\prime}-y_{k,1}|\geq
2\sqrt{n(n-1)}.$

Then by Proposition 2.1,
$|A^{\Sigma_{k}}(y_{k,2}^{\prime})|(\Sigma_{k}-y_{k,2}^{\prime})$ converges
locally smoothly to a catenoid in $\mathbb{R}^{n+1}$. Then we can take
$y_{k,2}$ around $y_{k,2}^{\prime}$ and $r_{k,2}>0$ so that
$r_{k,2}^{-1}(\Sigma_{k}-y_{k,2})$ converges locally smoothly to a standard
catenoid. Let

$r_{k}=\max\\{r_{k,1},r_{k,2}\\}.$

Denote by

(2.3) $b_{k}:=|y_{k,1}-y_{k,2}|.$

By the locally smooth convergence of $\Sigma_{k}$ in
$B_{\epsilon}(p)\setminus\\{p\\}$, we know that

$b_{k}\to 0,\text{ as }k\to\infty.$

Note that in a standard catenoid (see Appendix B), we have
$|x|\cdot|A(x)|\leq\sqrt{n(n-1)}$. Therefore, we also have that

$b_{k}/r_{k}\to\infty,\text{ if }y_{k,1}\neq y_{k,2}.$

For simplicity, denote by $A(p,r,s;M_{k})=B(p,s;M_{k})\setminus B(p,r;M_{k})$,
$\mathcal{W}_{k}=\\{y_{k,1},y_{k,2}\\}$ and

(2.4) $d_{k}(x)=\operatorname{dist}_{M_{k}}(x,\mathcal{W}_{k}).$

We now claim that for all sufficiently large $k$,

(2.5) $\max_{x\in
B(p,\epsilon;M_{k})\cap\Sigma_{k}}|A^{\Sigma_{k}}(x)|d_{k}(x)<2\sqrt{n(n-1)}.$

Suppose not, then one can find $y_{k,3}$ and another small cantenoid. Observe
that each small catenoid will contribute index 1. This contradicts that
$\Sigma_{k}$ has index less than or equal to 2.

From now on, let $\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$ be the two connected
components of $\Sigma_{k}\setminus\cup_{j}B(y_{k,j},2r_{k,j};M_{k})$. Without
loss of generality, outside $B(p,\epsilon;M_{k})$, we choose the unit normal
vector field of $\Sigma_{k}^{1}$ pointing towards $\Sigma_{k}^{2}$.

######  Proposition 2.3.

Given $\delta>0$, there exists $K>0$ such that for all $k\geq K$, the
following statements hold true.

  1. (1)

The second fundamental forms satisfy for
$x\in\Sigma_{k}\setminus\cup_{j=1}^{2}B(y_{k,j},Kr_{k,j};M_{k})$,

(2.6) $d_{k}\cdot|A^{\Sigma_{k}}(x)|+d^{2}_{k}\cdot|\nabla
A^{\Sigma_{k}}(x)|<\delta.$

Moreover, $\Sigma_{k}^{1}$ is a minimal graph over $\Sigma_{k}^{2}$. Denote by
$w_{k}$ the graph function.

  2. (2)

For every
$x\in\Sigma_{k}^{1}\setminus\cup_{j=1}^{2}B(y_{k,j},Kr_{k,j};M_{k})$,

(2.7) $d_{k}^{2}\frac{|\nabla^{2}w_{k}|}{w_{k}}(x)+d_{k}\frac{|\nabla
w_{k}|}{w_{k}}+\frac{w_{k}}{d_{k}}<\delta.$

  3. (3)

For any $s>Kr_{k}$ and $x,y\in\Sigma_{k}^{1}\cap A(y_{k},s,4s;M_{k})\setminus
B(z_{k},s/4;M_{k})$,

$w_{k}(x)\leq(1+\delta)w_{k}(y).$

###### Proof.

Note that by Remark 2.2, the limit of $d^{-1}_{k}(x_{k})(\Sigma_{k}-x_{k})$
can only be a multiplicity-two hyperplane if $d_{k}(x_{k})/r_{k,j}\to\infty$.
Then the first item follows from standard blowup and contradiction arguments.
Observe the third one follows directly from the second one. Hence it suffices
to prove the second item.

Suppose on the contrary that there exists $\delta>0$ and $\alpha_{k}\to\infty$
such that for a subsequence of $k\geq\alpha_{k}$, there exists
$x_{k}\in\Sigma_{k}\setminus\cup_{j=1}^{2}B(y_{k,j},\alpha_{k}r_{k,j};M_{k})$
violating (2.7). So long as $\lim_{k\to\infty}d_{k}(x_{k})\neq 0$, after
renormalizations, the sequence, still denoted by $\\{w_{k}\\}$, will converge
locally smoothly to a Jacobi field of $S_{0}^{n}$ away from at most two points
(see [Si87]), which is a constant function. This will give the desired
contradiction.

It remains to consider the situation $d_{k}\to 0$ and
$d_{k}/r_{k,j}\geq\alpha_{k}\to\infty$. By Remark 2.2,
$d_{k}^{-1}(\Sigma_{k}-x_{k})$ converges to a hyperplane with multiplicity
two. Then the normalizations of $\\{w_{k}\\}$ will converge locally smoothly
to a positive Jacobi field of the hyperplane, which is also a constant
function. This will also give a contradiction. ∎

link of the catenoid around $y_{k,1}$
---
link of the catenoid around $y_{k,2}$
---
Figure I. Structure of $\Sigma_{k}$.

######  Remark 2.4.

For $\\{\Sigma_{k}\\}$ satisfying the requirements at the beginning of this
section, we summarize the properties of $\Sigma_{k}$ (see Figure I) for large
$k$:

  1. (1)

there exists $\\{y_{k,j}\\}$ containing one or two points such that for each
$j$, $\Sigma_{k}$ has a small catenoid with radius $r_{k,j}$ around $y_{k,j}$;

  2. (2)

$\Sigma_{k}$ has exactly two connected components (denoted as $\Sigma_{k}^{1}$
and $\Sigma_{k}^{2}$) after removing such two (possibly one) small catenoids;

  3. (3)

$\Sigma_{k}^{2}$ is a graph over $\Sigma_{k}^{1}$ and the graph function
satisfies Proposition 2.3.

##  3\. Proof of the main theorem

For any $a\geq 1$ and $3\leq(n+1)\leq 7$, let $M_{a}$ be the embedded
$(n+1)$-sphere in $\mathbb{R}^{n+2}$ given by

$x_{1}^{2}+x_{2}^{2}\cdots+x_{n+1}^{2}+\frac{x_{n+2}^{2n}}{a^{2n}}=1.$

For simplicity, let $S_{t}:=\\{x_{n+2}=t\\}\cap M_{a}$, where we omit the $a$
without ambiguity. These $(n+1)$-spheres satisfy the following properties:

  1. (A)

it has non-negative Ricci curvature;

  2. (B)

$S^{n}_{0}=S_{0}$ is the unique stable minimal hypersurface for each $a\geq
1$; it follows that each minimal hypersurface is two-sided;

  3. (C)

$M_{a}\to S_{0}^{n}\times\mathbb{R}$ locally smoothly as $a\to\infty$;

  4. (D)

$\\{S_{t}\\}_{t=-a}^{a}$ forms a foliation of $M_{a}$ and $\\{x_{n+2}=t\\}$ is
an embedded $n$-sphere with mean curvature vector pointing away from $S_{0}$
for $0<|t|<a$. In particular, any two embedded minimal hypersurfaces in
$M_{a}$ intersect each other;

  5. (E)

the area of $S_{t}\subset M_{a}$ satisfies that for all $t$,

(3.1)
$\mathrm{Area}(S_{t})=\Omega_{n}(1-\frac{t^{2n}}{a^{2n}})\geq\Omega_{n}(1-|t|^{2n}).$

  6. (F)

for each $t$ with $0<|t|<a$,

$|t|\leq\operatorname{dist}_{M_{a}}(S_{0},S_{t})\leq 2|t|.$

Before stating our main results, we introduce the key height estimates, which
will be proved in the next section (Theorem 4.6 and 4.9).

######  Theorem 3.1.

Let $\Sigma_{k}\subset M^{n+1}_{k}$ be a sequence of embedded minimal
hypersurfaces with $\operatorname{index}\leq 2$ and

$\mathrm{Area}(\Sigma_{k})\leq 3\mathrm{Area}(S_{0}^{n})-\delta$

for some $\delta>0$. Then for all sufficiently large $k$,

$\max_{\Sigma_{k}}\operatorname{dist}_{M_{k}}(x,S^{n}_{0})\leq 8r_{k}|\log
r_{k}|.$

######  Remark 3.2.

Note that the estimates are sharp for dimension $n+1=3$ (Theorem 4.6). In
higher dimensions $4\leq n+1\leq 7$, the height estimates are much better
(Theorem 4.9).

In the following, we prove a rigidity theorem for minimal hypersurfaces with
low index and area. This will be useful to identify the min-max solutions that
realize the second width.

######  Theorem 3.3.

Suppose that $\Sigma_{k}$ is a closed embedded minimal hypersurface in $M_{k}$
with $\operatorname{index}\leq 2$ and

$\mathrm{Area}(\Sigma_{k})\leq 2\mathrm{Area}(S_{0}^{n}).$

Then for sufficiently large $k$, $\Sigma_{k}$ is exactly $S^{n}_{0}$.

###### Proof.

Suppose not, then $\Sigma_{k}$ converges to a closed minimal hypersurface
(with multiplicity) in $S^{n}_{0}\times\mathbb{R}$. By the monotonicity
formula, the limit is $S^{n}_{0}\times\\{0\\}$ with multiplicity less than or
equal to 2. We claim the multiplicity has to be two. So long as the the
convergence has multiplicity one, the normalization of the difference between
$S^{n}_{0}$ and $\Sigma_{k}$ converges to a non-trivial Jacobi field on
$S^{n}_{0}$ (see [Si87]). Note that such a Jacobi field is a constant. Thus
$\Sigma_{k}$ lies on one side of $S^{n}_{0}$, which contradicts (D) at the
beginning of this section. Therefore, $\Sigma_{k}$ converges to $\Sigma$ with
multiplicity two.

By [Sharp17], the convergence is smooth away from a set $\mathcal{W}$
containing at most two points since the index of $\Sigma_{k}$ is less than or
equal to 2. Then $\\{\Sigma_{k}\\}$ and $\\{M_{k}\\}$ satisfy the conditions
in Section 2 (see Example 1). Let $p\in\mathcal{W}$. By the argument therein,
we can find $y_{k}(\in M_{k})\to p$ and $r_{k}>0$ such that
$r_{k}^{-1}(\Sigma_{k}-y_{k})$ locally smoothly converges to a standard
catenoid $\mathcal{C}\subset\mathbb{R}^{n+1}$, i.e. it has the center at 0 and

$\max_{x\in\mathcal{C}}|A(x)|=\sqrt{n(n-1)}.$

By Remark 2.4, here we have three possibilities:

  1. (1)

$\mathcal{W}=\\{p,q\\}$ consists of two different points. In this case, there
exists $z_{k}\to q$ and $\widetilde{r}_{k}$ such that
$\widetilde{r}_{k}^{-1}(\Sigma_{k}-z_{k})$ locally smoothly converges to a
standard catenoid. Without loss of generality, we assume that
$r_{k}\geq\widetilde{r}_{k}$ for all large $k$.

  2. (2)

$\mathcal{W}=\\{p\\}$ consists of only one point and $b_{k}$ described in
Proposition 2.3 is equal to 0. It follows that $\Sigma_{k}\setminus
B(y_{k},2r_{k};M_{k})$ has exactly two connected components. In this case, we
let $\widetilde{r}_{k}=0$.

  3. (3)

$\mathcal{W}=\\{p\\}$ consists of only one point and $b_{k}$ described in
Proposition 2.3 is not 0, i.e. there exists $z_{k}\to p$,
$|z_{k}-y_{k}|/r_{k}\to\infty$ and $\widetilde{r}_{k}>0$ such that
$\widetilde{r}_{k}^{-1}(\Sigma_{k}-z_{k})$ locally smoothly converges to a
standard catenoid. Without loss of generality, we assume that
$r_{k}\geq\widetilde{r}_{k}$ for large $k$.

Above all, outside two small balls with radii $r_{k}$ and $\widetilde{r}_{k}$,
$\Sigma_{k}$ has exactly two connected components; see Remark 2.4. Moreover,
we can take $R_{k}\to 0$ with $R_{k}/r_{k}\to\infty$,
$r_{k}^{-1}(\Sigma_{k}-y_{k})\cap B_{R_{k}/r_{k}}(0)$ is arbitrarily smoothly
close to a standard catenoid $\mathcal{C}\cap B_{R_{r}/r_{k}}(0)$; if $z_{k}$
exists, we can also take $\widetilde{R}_{k}\to 0$ with
$\widetilde{R}_{k}/\widetilde{r}_{k}\to\infty$ so that
$\widetilde{r}_{k}^{-1}(\Sigma_{k}-z_{k})\cap B_{R_{k}/r_{k}}(0)$ is
arbitrarily close to a standard catenoid $\mathcal{C}\cap B_{R_{k}/r_{k}}(0)$.

Denote by $\gamma_{k}^{1}$ and $\gamma_{k}^{2}$ the two components of
$\Sigma_{k}\cap\partial B(y_{k},R_{k};M_{k})$. Then each $\gamma_{k}^{j}$
($j=1$ or 2) bounds an area minimizing $n$-disk $\mathcal{D}_{k}^{j}$ in
$M_{k}$. By Proposition (B.3),

(3.2)
$\mathrm{Area}(B(y_{k},R_{k};M_{k})\cap\Sigma_{k})-\sum_{j}\mathrm{Area}(\mathcal{D}_{k}^{j})\geq\frac{\mathcal{A}_{n}}{2}r_{k}^{n}>0.$

Here $\mathcal{A}_{n}$ ($n\geq 3$) is defined by (B.2) and $\mathcal{A}_{2}$
can be any fixed real numbers because of (B.3). Similarly, in Case (1) or (3),
there exist $\widetilde{\gamma}_{k}^{j}\subset\Sigma_{k}$ ($j=1,2$)
surrounding $z_{k}$ such that each $\widetilde{\gamma}_{k}^{j}$ bounds an area
minimizing $n$-disk $\widetilde{\mathcal{D}}_{k}^{j}$ in $M_{k}$ with

(3.3)
$\mathrm{Area}(B(z_{k},\widetilde{R}_{k};M_{k})\cap\Sigma_{k})-\sum_{j}\mathrm{Area}(\widetilde{\mathcal{D}}_{k}^{j})\geq\frac{\mathcal{A}_{n}}{2}\widetilde{r}_{k}^{n}>0.$

Then we cut off $\Sigma_{k}\cap B(y_{k},R_{k};M_{k})$ and $\Sigma_{k}\cap
B(z_{k},\widetilde{R}_{k};M_{k})$, which are close to a standard catenoid
after scaling. After that, we add the $n$-disks $\\{\mathcal{D}_{k}^{j}\\}$
and $\\{\widetilde{\mathcal{D}}_{k}^{j}\\}$ to the new hypersurface; see
Figure II. This yields a closed hypersurface with two connected components
$\Gamma_{1}$ and $\Gamma_{2}$ which are both homologous to $S^{n}_{0}$ in
$M_{k}$.

$\Sigma_{k}^{2}$
---
$\Sigma_{k}^{1}$
---
$\mathcal{D}_{k}^{2}$
---
$\mathcal{D}_{k}^{1}$
---
$B(y_{k},R_{k};M_{k})$
---
$B(y_{k},\epsilon;M_{k})$
---
$S_{0}^{n}$
---
Figure II. Replacing the catenoids by two $n$-disks.

Note that Theorem 3.1 can be applied to obtain

$\max_{x\in\Sigma_{k}}\operatorname{dist}_{M_{k}}(x,S_{0}^{n})\leq 8r_{k}|\log
r_{k}|;$

together with a standard minimal foliation argument (applied to
$B(y_{k},\epsilon;M_{k})$), this implies that

$\max_{x\in\Gamma_{j}}\operatorname{dist}_{M_{k}}(x,S_{0}^{n})\leq
10r_{k}|\log r_{k}|.$

Let $\mathfrak{d}_{k}:=10r_{k}|\log r_{k}|$. Then by the one-sided minimizing
property of $S_{\mathfrak{d}_{k}}$ and $S_{-\mathfrak{d}_{k}}$,

$\mathrm{Area}(\Gamma_{j})+\mathrm{Area}(S^{n}_{0})>\mathrm{Area}(S_{\mathfrak{d}_{k}})+\mathrm{Area}(S_{-\mathfrak{d}_{k}}).$

By (3.1),

$\mathrm{Area}(S_{\mathfrak{d}_{k}})\geq\mathrm{Area}(S_{0}^{n})-\Omega_{n}|\mathfrak{d}_{k}|^{2n}\
\ \ \text{ and }\ \ \
\mathrm{Area}(S_{-\mathfrak{d}_{k}})\geq\mathrm{Area}(S_{0}^{n})-\Omega_{n}|\mathfrak{d}_{k}|^{2n};$

these imply that

(3.4)
$\mathrm{Area}(\Gamma_{1})+\mathrm{Area}(\Gamma_{2})>2\mathrm{Area}(S_{0}^{n})-4\Omega_{n}\mathfrak{d}_{k}^{2n}.$

By the construction of $\Gamma_{j}$,

$\displaystyle\mathrm{Area}(\Sigma_{k})$
$\displaystyle=\sum_{j=1}^{2}\Big{(}\mathrm{Area}(\Gamma_{j})-\mathrm{Area}(\mathcal{D}_{k}^{j})-\mathrm{Area}(\widetilde{\mathcal{D}}_{k}^{j})\Big{)}+\mathrm{Area}(B(y_{k},R_{k};M_{k})\cap\Sigma_{k})$
$\displaystyle\ \
+\mathrm{Area}(B(z_{k},\widetilde{R}_{k};M_{k})\cap\Sigma_{k})$
$\displaystyle\geq\sum_{j=1}^{2}\Big{(}\mathrm{Area}(\Gamma_{j})-\mathrm{Area}(\mathcal{D}_{k}^{j})\Big{)}+\mathrm{Area}(B(y_{k},R_{k};M_{k})\cap\Sigma_{k})$
$\displaystyle>2\mathrm{Area}(S_{0}^{n})-10^{2n}r_{k}^{2n-1}\cdot(4\Omega_{n}r_{k}|\log
r_{k}|^{2n})+\frac{\mathcal{A}_{n}}{2}r_{k}^{n}$
$\displaystyle>2\mathrm{Area}(S^{n}_{0})+\frac{\mathcal{A}_{n}}{4}r_{k}^{n}.$

Here the first inequality is from (3.3); we used (3.2) and (3.4) in the second
one; the last one follows from the fact that $r_{k}|\log r_{k}|^{2n}\to 0$ as
$k\to\infty$. This gives a contradiction and we finish the proof of Theorem
3.3. ∎

Now we are going to prove our main results.

######  Theorem 3.4.

For sufficiently large $k$, the second width $\omega_{2}(M_{k})$ can only be
realized by $S_{0}^{n}$ with multiplicity two.

###### Proof.

By the work of Marques-Neves [MN16], together with the Frankel’s property (D),
$\omega_{2}(M_{k})$ is realized by the area of a connected, closed, embedded,
minimal hypersurface $\Sigma_{k}$ with integer multiplicities and
$\operatorname{index}(\Sigma_{k})\leq 2$. Observe that

(3.5)
$\lim_{k\to\infty}\omega_{2}(M_{k})=\omega_{2}(S_{0}^{n}\times\mathbb{R})=2\mathrm{Area}(S_{0}^{n}).$

We first prove that $\Sigma_{k}$ is $S_{0}^{n}$. Suppose not, then
$\Sigma_{k}$ is unstable by Property (B); therefore using the direct corollary
of the Multiplicity One Theorem in [Zhou19], $\Sigma_{k}$ has multiplicity
one, i.e. $\omega_{2}(M_{k})=\mathrm{Area}(\Sigma_{k})$. On the other hand, by
the construction of optimal 1-sweepouts (c.f. [Zhou15]),
$\omega_{1}(M_{k})=\mathrm{Area}(S_{0}^{n})$. Then it follows that (c.f.
[MN17]*Proof of Theorem 5.1).

(3.6) $\omega_{2}(M_{k})\leq 2\omega_{1}(M_{k})=2\mathrm{Area}(S_{0}^{n}).$

By Theorem 3.3, $\Sigma_{k}$ has to be identical to $S_{0}^{n}$, but this
contradicts (3.5). Thus we conclude that $\Sigma_{k}$ has to be $S_{0}^{n}$.
Then the multiplicity follows from (3.5).

Therefore, $\omega_{2}(M_{k})$ can only be realized by $S_{0}^{n}$ with
multiplicity two. ∎

Let $M_{k}$ be the Riemannian $(n+1)$-sphere in Theorem 3.4. Denote by $N_{k}$
and $\mathbb{RP}^{n}_{0}$ the quotient spaces $M_{k}/\\{x\sim-x\\}$ and
$S_{0}^{n}/\\{x\sim-x\\}$, respectively. Then we have the following
properties:

  1. (1)

for each $k$, $N_{k}$ satisfies the Frankel’s property, i.e. any two embedded
minimal hypersurfaces intersect;

  2. (2)

$\mathbb{RP}^{n}_{0}$ is a one-sided embedded minimal hypersurface in $N_{k}$;

  3. (3)

as $k\to\infty$, $N_{k}$ locally smoothly converges to
$S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\}$;

  4. (4)

$\omega_{1}(S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\})=\mathrm{Area}(S_{0}^{n})=2\mathrm{Area}(\mathbb{RP}^{n}_{0})$.

Here the first three items follow from the Properties (A–D) of $M_{k}$ at the
beginning of this section. For the last one, since
$S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\}$ has a minimal foliation, it follows
that

$\omega_{1}(S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\})\leq\mathrm{Area}(S_{0}^{n})=2\mathrm{Area}(\mathbb{RP}^{n}_{0}).$

On the other hand, as the limit contains $S_{0}^{n}\times(0,\infty)$, it
follows that

$\omega_{1}(S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\})\geq\omega_{1}(S_{0}^{n}\times\mathbb{R})=\mathrm{Area}(S_{0}^{n}).$

Hence the last property is proved.

The following result directly implies Corollary 1.2.

######  Corollary 3.5.

For sufficiently large $k$, the first width $\omega_{1}(N_{k})$ can only be
realized by $\mathbb{RP}_{0}^{n}$ with multiplicity two.

###### Proof.

Notice that $N_{k}$ also satisfies the Frankel’s property. Then by Marques-
Neves [MN16], $\omega_{1}(N_{k})$ is realized by the area of a connected,
closed, embedded, minimal hypersurface $\Gamma_{k}$ with integer multiplicity
$m_{k}$ and $\operatorname{index}(\Gamma_{k})\leq 1$. Observe that

(3.7)
$\lim_{k\to\infty}\omega_{1}(N_{k})=\omega_{1}(S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\})=2\mathrm{Area}(\mathbb{RP}_{0}^{n}).$

It follows that $\mathrm{Area}(\Gamma_{k})$ is uniformly bounded. Then by
compactness [Sharp17], $\Gamma_{k}$ converges to an embedded minimal
hypersurface $\Gamma\subset S_{0}^{n}\times\mathbb{R}/\\{x\sim-x\\}$. Notice
that such a space is foliated by minimal hypersurfaces. Thus $\Gamma$ is
$S_{0}^{n}\times\\{t\\}$ or $\mathbb{RP}^{n}_{0}$. Recall that $\Gamma_{k}$
intersects $\mathbb{RP}^{n}_{0}$. Hence we conclude that the limit is
$\mathbb{RP}^{n}_{0}$. Together with (3.7), we then have that $m_{k}\leq 2$.

Case I: $m_{k}=1$.

Then $\Gamma_{k}$ locally smoothly converges to $2\mathbb{RP}^{n}_{0}$ away
from at most one point. Then by the same argument as in Theorem 3.3,

$\mathrm{Area}(\Gamma_{k})>2\mathrm{Area}(\mathbb{RP}^{n}_{0}),$

which contradicts $\omega_{1}(N_{k})\leq 2\mathrm{Area}(\mathbb{RP}^{n}_{0})$.

Case II: $m_{k}=2$.

Then $\Gamma_{k}$ converges to $\mathbb{RP}^{n}_{0}$ with multiplicity one. By
Allard’s regularity, the convergence is smooth. Let $\widetilde{\Gamma}_{k}\in
M_{k}$ be the double cover of $\Gamma_{k}$. Then $\widetilde{\Gamma}_{k}$
smoothly converges to $S_{0}^{n}$. This implies that $\widetilde{\Gamma}_{k}$
is identical to $S_{0}^{n}$. Hence $\Gamma_{k}$ is identical to
$\mathbb{RP}^{n}_{0}$. This completes the proof of Corollary 3.5. ∎

##  4\. Upper bounds for the Hausdorff distance

We use the notation in Section 2. Recall that $\Sigma_{k}\subset M_{k}^{n+1}$
is a closed embedded minimal hypersurface such that the Morse index is bounded
above by two and the area is uniformly bounded independent of
$k\in\mathbb{N}$. As $k\to\infty$, $M_{k}$ converges locally smoothly to the
product space $S_{0}^{n}\times\mathbb{R}$, and $\Sigma_{k}$ converges to a
minimal $n$-sphere in the limit space of $M_{k}$, namely
$S_{0}^{n}\times\mathbb{R}$. The convergence of $\Sigma_{k}$ is locally smooth
away from at most two points according to the Morse index and area bounds. At
a point near which the convergence is not smooth, the limit cones are either
planes or catenoids by Proposition 2.1. For simplicity, we use $A$ for
$A^{\Sigma_{k}}$ and sometimes omit the subscription $k$ when there is no
ambiguity. Denote by $\nabla$ and $\widetilde{\nabla}$ the Levi-Civita
connections of $\Sigma_{k}$ and $M_{k}$, respectively.

In this section, we prove the key height estimates, which says that the
Hausdorff distance between $\Sigma_{k}$ and $S^{n}_{0}$ is bounded by a
quantity associated with the catenoids arising from blowups. This result is
essentially used in the previous section (see Theorem 3.1).

Recall that by Remark 2.4, $\Sigma_{k}$ has two connected components (denoted
by $\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$) by removing one or two small
catenoids. Denote by $y_{k},z_{k}$ the centers and $r_{k},\widetilde{r}_{k}$
the radii of links of such catenoids (we used $y_{k,j}$ and $r_{k,j}$ in
Section 2 for general cases). Without loss of generality, we assume that
$r_{k}\geq\widetilde{r}_{k}$.

Then by Proposition 2.3, $\Sigma_{k}^{2}$ is a minimal graph over
$\Sigma_{k}^{1}$. Let $\mathbf{n}$ be the unit normal vector field of
$\Sigma_{k}^{1}$ , and let $\rho$ and $\widetilde{\rho}$ be the distance
functions to $y_{k}$ and $z_{k}$ in $M_{k}$. Denote by

(4.1)
$\bm{\eta}=\nabla\rho/|\nabla\rho|,\quad\widetilde{\bm{\eta}}=\nabla\widetilde{\rho}/|\nabla\widetilde{\rho}|;\quad\phi=|\nabla\rho|,\quad\widetilde{\phi}=|\nabla\widetilde{\rho}|.$

Recall that $b_{k}$, defined in (2.3), is the distance between the two blowup
points $y_{k},z_{k}$. For any

(4.2) $\epsilon\geq s\geq 2b_{k}>0\ \ \text{ or }\ \ \ b_{k}/2\geq s\geq
4r_{k},$

we set

(4.3) $\gamma_{s}=\Sigma_{k}^{1}\cap\partial B(y_{k},s;M_{k}).$

In the remaining part of this section, we assume that
$R_{k},\widetilde{R}_{k}\to 0$ are two sequence of real numbers satisfying
$R_{k}/r_{k}\to\infty$ and $\widetilde{R}_{k}/\widetilde{r}_{k}\to\infty$.
Moreover, we can also assume $r_{k}^{-1}(\Sigma_{k}\cap
B(y_{k},R_{k};M_{k})-y_{k})$ (resp. $\widetilde{r}_{k}^{-1}(\Sigma_{k}\cap
B(z_{k},\widetilde{R}_{k};M_{k})-y_{k})$) is arbitrarily close to the catenoid
$\mathcal{C}\cap B_{R_{k}/r_{k}}(0)$ (resp. $\mathcal{C}\cap
B_{\widetilde{R}_{k}/\widetilde{r}_{k}}(0)$) in the smooth topology. By
Proposition 2.1, for sufficiently large $k$, $s^{-1}(\gamma_{s}-y_{k})$ is
very close to the unit $(n-1)$-sphere in the smooth topology. In particular,
we have

(4.4)
$(1-\epsilon^{2})\Omega_{n-1}s^{n-1}<|\gamma_{s}|<(1+\epsilon^{2})\Omega_{n-1}s^{n-1},$

where $\Omega_{m}$ is the volume of unit $m$-spheres. Recall that $w_{k}$ is
the graph function of $\Sigma_{k}^{2}$ over $\Sigma_{k}^{1}$.

######  Lemma 4.1.

Given any $\delta>0$, then for all sufficiently large $k$, we have

(4.5) $\Delta w_{k}\leq w_{k}+C(n)\delta\cdot\frac{w_{k}^{3}}{d^{4}_{k}(x)}$

for $x\in\Sigma_{k}^{1}\setminus\big{[}B(y_{k},R_{k};M_{k})\cup
B(z_{k},\widetilde{R}_{k};M_{k})\big{]}$. In particular,

  1. (1)

for all large $k$ and $x\in\Sigma_{k}^{1}\cap
A(y_{k},R_{k},\epsilon;M_{k})\setminus B(z_{k},b_{k}/2;M_{k})$,

(4.6) $\Delta w_{k}\leq
w_{k}+\frac{1}{8}\cdot\frac{w_{k}^{3}}{|\rho(x)|^{4}};$

  2. (2)

for all large $k$ and $x\in\Sigma_{k}^{1}\cap A(z_{k},R_{k},b_{k}/2;M_{k})$,

$\Delta w_{k}\leq
w_{k}+\frac{1}{8}\cdot\frac{w_{k}^{3}}{|\widetilde{\rho}(x)|^{4}}.$

###### Proof.

By Proposition 2.3, $|A^{\Sigma_{k}}|$ has an upper bound over
$\Sigma_{k}^{1}\cap A(y_{k},s,2s;M_{k})\setminus B(z_{k},b_{k}/2;M_{k})$, so
the level sets of the distance to $\Sigma_{k}^{1}$ will form a desired
foliation as in Appendix A. By the Cauchy-Schwartz inequality, we have

(4.7) $\displaystyle\ 8|A||\nabla
w_{k}|^{2}\leq\frac{1}{2}|A|^{2}w_{k}+\frac{32|\nabla
w_{k}|^{4}}{w_{k}};\quad|\nabla^{2}w_{k}||A|w_{k}\leq\delta|A|^{2}w_{k}+\frac{1}{4\delta}|\nabla^{2}w_{k}|^{2}w_{k}.$

By plugging (2.6), (2.7) and (4.7) into (A.3) in Appendix A, we can show that

$\Delta w_{k}+|A|^{2}w_{k}\leq 3\delta
w_{k}+|A|^{2}w_{k}+C(n)\delta\cdot\Big{(}\frac{w_{k}^{3}}{d^{4}_{k}(x)}+w_{k}\Big{)}.$

Here in (A.3), we deal with the first and third terms by (4.7); for the second
term in (A.3), we used $|A|w_{k}\leq
d_{k}|A|\cdot\frac{w_{k}}{d_{k}}<\delta^{2}$; in the fourth term, we used
$w_{k}+|\nabla w_{k}|+|\nabla w_{k}|^{3}<3\delta$ by (2.7); the others can be
bounded similarly; for instance, we have

$|\nabla w_{k}|^{2}|\nabla^{2}w_{k}|<\Big{(}\frac{\delta
w_{k}}{d_{k}}\Big{)}^{2}\cdot\frac{w_{k}}{d_{k}^{2}}<\delta\cdot\frac{w_{k}^{3}}{d_{k}^{4}}.$

Hence the (4.5) is proved.

Then for $x\in\Sigma_{k}^{1}\cap A(y_{k},R_{k},\epsilon;M_{k})\setminus
B(z_{k},b_{k}/2;M_{k})$, it follows that

$d_{k}(x)\geq\frac{1}{4}\rho(x).$

Plugging this into (4.6), we then have

(4.8) $\Delta w_{k}\leq
w_{k}+\frac{1}{8}\cdot\frac{w_{k}^{3}}{|\rho(x)|^{4}}.$

The last case can be proved similarly. This completes the proof of Lemma 4.1.
∎

Inspired by Colding-Minicozzi [CM02]*(2.1), we define

(4.9)
$\mathcal{I}_{k}(s)=\frac{1}{\Omega_{n-1}s^{n-1}}\int_{\gamma_{s}}w_{k}|\nabla\rho|\,d\mathcal{H}^{n-1}(x),$
(4.10) $\tau_{k}(s)=\frac{1}{\Omega_{n-1}}\int_{\gamma_{s}}\langle\nabla
w_{k},\bm{\eta}\rangle\ \ \ \text{ and }\ \ \
F_{k}(s)=\frac{n}{\Omega_{n-1}s^{n}}\int_{\gamma_{s}}(\phi^{-1}-\phi),$

where $\phi=|\nabla\rho|$ is defined in (4.1). Compared with [CM02], we
introduce the weight $\phi=|\nabla\rho|$ in $\mathcal{I}_{k}$ as
$\Sigma_{k}^{1}$ is not flat. Note that by Proposition B.1,

(4.11) $\lim_{k\to\infty}\frac{\tau_{k}(R_{k})}{r_{k}^{n-1}}=2.$

###  4.1. Inequalities for the first derivatives

In this subsection, we take the derivative for the averages of the height
functions between two sheets. The coefficients of leading terms are the
functions $\tau_{k}$ defined above.

######  Proposition 4.2.

Using above notations, we then have

$\displaystyle\mathcal{I}_{k}^{\prime}(s)-\frac{\tau_{k}(s)}{s^{n-1}}$
$\displaystyle=\frac{1}{\Omega_{n-1}s^{n-1}}\int_{\gamma_{s}}w_{k}\Big{[}\phi^{-1}\mathrm{div}_{\Sigma_{k}}\widetilde{\nabla}\rho+\frac{1-n}{s}\phi\Big{]}\,d\mathcal{H}^{n-1}$
$\displaystyle\leq\frac{n}{\Omega_{n-1}s^{n}}\int_{\gamma_{s}}(\phi^{-1}-\phi)w_{k}+C_{0}s\mathcal{I}_{k}(s),$

where $C_{0}$ is a constant depending only on $n$ and can be changed from line
to line.

###### Proof.

Recall that $\phi=|\nabla\rho|$. A direct computation gives that

(4.12)
$\langle\nabla\phi,\bm{\eta}\rangle+\phi\mathrm{div}_{\gamma_{s}}\bm{\eta}=\langle\nabla\phi,\bm{\eta}\rangle+\phi\mathrm{div}_{\Sigma_{k}}\bm{\eta}=\mathrm{div}_{\Sigma_{k}}\phi\bm{\eta}=\mathrm{div}_{\Sigma_{k}}\nabla\rho=\mathrm{div}_{\Sigma_{k}}\widetilde{\nabla}\rho.$

Here we used that $\langle\nabla_{\bm{\eta}}\bm{\eta},\bm{\eta}\rangle=0$ in
the first equality; and the last one follows from the minimality of
$\Sigma_{k}$. Then by noting that $\frac{\partial}{\partial
s}=\frac{\nabla\rho}{|\nabla\rho|^{2}}=\frac{\bm{\eta}}{\phi}$, we have

$\displaystyle\mathcal{I}^{\prime}_{k}(s)$
$\displaystyle=\frac{1-n}{\Omega_{n-1}s^{n}}\int_{\gamma_{s}}w_{k}\phi\,d\mathcal{H}^{n-1}+\frac{1}{\Omega_{n-1}s^{n-1}}\int_{\gamma_{s}}\langle\nabla(w_{k}\phi),\frac{\bm{\eta}}{\phi}\rangle+w_{k}\phi\mathrm{div}_{\gamma_{s}}\big{(}\frac{\bm{\eta}}{\phi}\big{)}\,d\mathcal{H}^{n-1}$
$\displaystyle=\frac{1-n}{\Omega_{n-1}s^{n}}\int_{\gamma_{s}}w_{k}\phi\,d\mathcal{H}^{n-1}+\frac{1}{\Omega_{n-1}s^{n-1}}\int_{\gamma_{s}}\langle\nabla
w_{k},\bm{\eta}\rangle+w_{k}\big{[}\langle\nabla\phi,\frac{\bm{\eta}}{\phi}\rangle+\mathrm{div}_{\gamma_{s}}\bm{\eta}\big{]}\,d\mathcal{H}^{n-1}$
$\displaystyle=\frac{1-n}{\Omega_{n-1}s^{n}}\int_{\gamma_{s}}w_{k}\phi\,d\mathcal{H}^{n-1}+\frac{1}{\Omega_{n-1}s^{n-1}}\int_{\gamma_{s}}\langle\nabla
w_{k},\bm{\eta}\rangle+w_{k}\phi^{-1}\mathrm{div}_{\Sigma_{k}}\widetilde{\nabla}\rho\,d\mathcal{H}^{n-1},$

where the last equality is from (4.12). Then we are going to prove the
inequality. Recall that when restricting to $T(\partial B(p,s;M_{k}))$, we
have

(4.13)
$\widetilde{\nabla}^{2}\rho=\frac{1}{\rho}\delta_{ij}+\mathcal{O}(\rho).$

Therefore, we have that for $x\in\gamma_{s}$,

(4.14) $\displaystyle\mathrm{div}_{\Sigma_{k}}\widetilde{\nabla}\rho$
$\displaystyle=\mathrm{div}_{M_{k}}\widetilde{\nabla}\rho-\langle\widetilde{\nabla}_{\mathbf{n}}\widetilde{\nabla}\rho,\mathbf{n}\rangle$
$\displaystyle=\frac{n}{s}+\mathcal{O}(s)-\langle\widetilde{\nabla}_{E}\widetilde{\nabla}\rho,E\rangle\langle\mathbf{n},E\rangle^{2}$
$\displaystyle\leq\frac{1}{s}(n-\phi^{2})+C_{0}s.$

In the above calculation, we used $E$ for the following expression,

$E:=\frac{\mathbf{n}-\langle\mathbf{n},\widetilde{\nabla}\rho\rangle\widetilde{\nabla}\rho}{\sqrt{1-\langle\mathbf{n},\widetilde{\nabla}\rho\rangle^{2}}};$

this is the unit projection of $\mathbf{n}$ to $T(\partial B(p,s;M_{k}))$.
This finishes the proof of Proposition 4.2. ∎

In the next subsection, we will use the inequality in Proposition 4.2 to bound
$\mathcal{I}_{k}$, which requires the following results for $F_{k}(t)$.

######  Proposition 4.3.

Suppose that $[R,s]\subset[R_{k},b_{k}/2]$ or $[2b_{k},\epsilon]$. Then

$\displaystyle\frac{d}{ds}\Big{[}s^{-n}\Big{(}|\Sigma_{k}^{1}\cap
A(y_{k},R,s;M_{k})|+\frac{R}{n}\int_{\gamma_{R}}\phi\Big{)}\Big{]}$
$\displaystyle=s^{-n}\int_{\gamma_{s}}(\frac{1}{\phi}-\phi)+\frac{1}{s^{n+1}}\int_{A(y_{k},R,s;M_{k})\cap\Sigma_{k}^{1}}\mathcal{O}(\rho^{2}).$

For all sufficiently large $k$,

$\int_{R_{k}}^{b_{k}/2}F_{k}(t)dt+\int_{2b_{k}}^{\epsilon}F_{k}(t)dt<\epsilon.$

###### Proof.

By the divergence theorem,

(4.15)
$\int_{A(y_{k},R,s;M_{k})\cap\Sigma_{k}^{1}}\Delta\rho^{2}=\int_{\gamma_{s}}2s\phi-\int_{\gamma_{R}}2R\phi.$

Since $\Delta\rho^{2}=2n+\mathcal{O}(\rho^{2})$, we have

$|\Sigma_{k}^{1}\cap
A(y_{k},R,s;M_{k})|+\frac{R}{n}\int_{\gamma_{R}}\phi=\frac{s}{n}\int_{\gamma_{s}}\phi+\int_{A(y_{k},R,s;M_{k})\cap\Sigma_{k}^{1}}\mathcal{O}(\rho^{2}).$

Finally note that

$\frac{d}{ds}\Big{|}\Sigma_{k}^{1}\cap
A(y_{k},R,s;M_{k})\Big{|}=\int_{\gamma_{s}}\phi^{-1}.$

The first identity follows by plugging all the above identities. Then we have

$\displaystyle\int_{R_{k}}^{b_{k}/2}F_{k}(t)\,dt$
$\displaystyle\leq\frac{n}{\Omega_{n-1}(b_{k}/2)^{n}}\Big{(}|\Sigma_{k}^{1}\cap
A(y_{k},R_{k},b_{k}/2;M_{k})|+\frac{R_{k}}{n}\int_{\gamma_{R_{k}}}\phi\Big{)}$
$\displaystyle-\frac{1}{\Omega_{n-1}R_{k}^{n-1}}\int_{\gamma_{R_{k}}}\phi+C_{0}b_{k}^{2}.$

Recall that by (4.4), for all sufficiently large $k$, we have

$(1-\frac{\epsilon}{10})\Omega_{n-1}s^{n-1}\leq|\gamma_{s}|\leq(1+\frac{\epsilon}{10})\Omega_{n-1}s^{n-1};\
\ \ 1\leq\phi^{-1}\leq 1+\frac{\epsilon}{10}.$

Then by the co-area formula, we have

$\displaystyle\ \ \
\frac{n}{\Omega_{n-1}(b_{k}/2)^{n}}\Big{(}|\Sigma_{k}^{1}\cap
A(y_{k},R_{k},b_{k}/2;M_{k})|+\frac{R_{k}}{n}\int_{\gamma_{R_{k}}}\phi\Big{)}$
$\displaystyle\leq\frac{n}{\Omega_{n-1}(b_{k}/2)^{n}}\,\cdot\Omega_{n-1}\Big{[}\int_{R_{k}}^{b_{k}/2}(1+\frac{\epsilon}{10})s^{n-1}\,ds+\frac{R_{k}}{n}(1+\frac{\epsilon}{10})R_{k}^{n-1}\Big{]}=1+\frac{\epsilon}{10}.$

On the other hand,

$\frac{1}{\Omega_{n-1}R_{k}^{n-1}}\int_{\gamma_{R_{k}}}\phi\geq
1-\frac{\epsilon}{5}.$

Combining them together, we then have

$\int_{R_{k}}^{b_{k}/2}F_{k}(t)\,dt\leq\frac{3\epsilon}{10}+C_{0}\epsilon^{2}<\frac{\epsilon}{2}.$

Similarly,

$\int_{2b_{k}}^{\epsilon}F_{k}(t)\,dt<\frac{\epsilon}{2}.$

Hence Proposition 4.3 is proved. ∎

###  4.2. Hausdorff distance upper bounds in dimension three

In this part, we focus on the case $n=2$ and prove the upper bounds of the
Hausdorff distance between $\Sigma_{k}$ and $S_{0}^{2}$. Recall that
$\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$ are jointed by one or two small
catenoids with radii $r_{k}$ and $\widetilde{r}_{k}$; see Remark 2.4. Without
loss of generality, we assume that $r_{k}\geq\widetilde{r}_{k}$. Recall that
$\Sigma_{k}^{2}$ is a graph over $\Sigma_{k}^{1}$ with graph function $w_{k}$.

To prove the desired bound, we will derive several monotonicity formulas
associated with the average of $w_{k}$. Then the smooth convergence on
$\partial B(p,\epsilon;M_{k})$ will give the desired upper bound.

Now let

(4.16)
$\displaystyle\widetilde{\mathcal{I}}_{k}(s)=\mathcal{I}_{k}(s)-3r_{k}\log\frac{s}{r_{k}}-2sr_{k}\log\frac{s}{r_{k}}-10\Big{(}\int_{[R_{k},s]\setminus[b_{k}/2,2b_{k}]}F_{k}(t)\,dt\Big{)}\cdot
r_{k}\log\frac{s}{r_{k}};$ (4.17)
$\displaystyle\widetilde{\tau}_{k}(s)=\tau_{k}(s)+\frac{r_{k}^{2}}{2s}-sr_{k}\log\frac{s}{r_{k}}.$

Then by Proposition 4.3,

(4.18)
$\mathcal{I}_{k}(s)<\widetilde{\mathcal{I}}_{k}(s)+4r_{k}\log\frac{s}{r_{k}}.$

Recall that $R_{k}/r_{k}\to+\infty$ and $r_{k}^{-1}(\Sigma_{k}\cap
B(y_{k},R_{k};M_{k})-y_{k})$ is arbitrarily close to the catenoid
$\mathcal{C}\cap B_{R_{k}/r_{k}}(0)$ in the smooth topology. Then by Item
(1)in Appendix B,

(4.19) $\mathcal{I}_{k}(R_{k})\leq(1+\frac{1}{100})\cdot
2r_{k}\int_{1}^{R_{k}/r_{k}}\frac{ds}{\sqrt{s^{2}-1}}<(1+\frac{1}{100})\cdot
2r_{k}\log\frac{2R_{k}}{r_{k}}<3r_{k}\log\frac{R_{k}}{r_{k}}.$

Then by (4.11), (4.16) and (4.19),

(4.20) $\widetilde{\mathcal{I}}_{k}(R_{k})<0,\ \ \
\tau_{k}(R_{k})<(2+\frac{1}{10})r_{k}.$

Now we are ready to show that $\widetilde{\mathcal{I}}_{k}(s)$ is decreasing.

######  Proposition 4.4.

$\widetilde{\mathcal{I}}_{k}(s)$ and $\widetilde{\tau}_{k}(s)$ are decreasing
on $[R_{k},b_{k}/2]$ (resp. $[R_{k},\epsilon]$) if $b_{k}\neq 0$ (resp.
$b_{k}=0$). It follows that

$\displaystyle\mathcal{I}_{k}(s)<4r_{k}\log\frac{s}{r_{k}},\ \ \
\tau_{k}(s)<3r_{k}-\frac{r_{k}^{2}}{2s}+sr_{k}\log\frac{s}{r_{k}}\ \ \text{
for }\ s\in[R_{k},b_{k}/2].$

###### Proof.

We first assume that $b_{k}\neq 0$. To prove that
$\widetilde{\mathcal{I}}_{k}$ and $\widetilde{\tau}_{k}$ are decreasing on
$[R_{k},b_{k}/2]$, let

$s_{1}:=\sup\\{s\in(R_{k},b_{k}/2):\widetilde{\tau}^{\prime}_{k}(t)\leq 0,\
\widetilde{\mathcal{I}}_{k}^{\prime}(t)\leq 0\ \text{ for all }\
t\in[R_{k},s)\\}.$

We claim that $s_{1}=b_{k}/2$. Suppose on the contrary that $s_{1}<b_{k}/2$.
Then $\widetilde{\mathcal{I}}_{k}$ and $\widetilde{\tau}_{k}$ are decreasing
on $[R_{k},s_{1}]$, which together with (4.20) implies that

$\displaystyle\widetilde{\mathcal{I}}_{k}(s_{1})\leq\widetilde{\mathcal{I}}_{k}(R_{k})<0;\
\ \
\widetilde{\tau}_{k}(s_{1})\leq\widetilde{\tau}_{k}(R_{k})<\frac{5r_{k}}{2}.$

By (4.17) and (4.18), it follows that

$\displaystyle\mathcal{I}_{k}(s_{1})<\widetilde{\mathcal{I}}_{k}(s_{1})+4r_{k}\log\frac{s_{1}}{r_{k}}<4r_{k}\log\frac{s_{1}}{r_{k}};$
$\displaystyle\tau_{k}(s_{1})=\widetilde{\tau}_{k}(s_{1})-\frac{r_{k}^{2}}{2s_{1}}+s_{1}r_{k}\log\frac{s_{1}}{r_{k}}<3r_{k}-\frac{r_{k}^{2}}{2s_{1}}+s_{1}r_{k}\log\frac{s_{1}}{r_{k}}.$

Then by Proposition 4.2,

$\displaystyle\mathcal{I}_{k}^{\prime}(s_{1})$
$\displaystyle<\frac{\tau_{k}(s_{1})}{s_{1}}+5r_{k}\log\frac{s_{1}}{r_{k}}\cdot
F_{k}(s_{1})+C_{0}s_{1}r_{k}\log\frac{s_{1}}{r_{k}}$
$\displaystyle<\frac{3r_{k}}{s_{1}}+2r_{k}\log\frac{s_{1}}{r_{k}}+5r_{k}\log\frac{s_{1}}{r_{k}}\cdot
F_{k}(s_{1}).$

Here in the first inequality, we used that by Proposition 2.3 (3) and (4.4),
for $x\in\gamma_{s_{1}}$,

(4.21)
$w_{k}(x)\leq(1+1/100)^{2}\mathcal{I}_{k}(s_{1})<5r_{k}\log\frac{s_{1}}{r_{k}}.$

Then by (4.16), it follows that

(4.22)
$\widetilde{\mathcal{I}}_{k}^{\prime}(s_{1})<\mathcal{I}_{k}^{\prime}(s_{1})-\frac{3r_{k}}{s_{1}}-2r_{k}\log\frac{s_{1}}{r_{k}}-10F_{k}(s_{1})\cdot
r_{k}\log\frac{s_{1}}{r_{k}}<0.$

On the other hand, by the divergence theorem, the co-area formula, (4.6) and
(4.21),

(4.23)
$\displaystyle\tau_{k}^{\prime}(s_{1})=\frac{1}{2\pi}\int_{\gamma_{s_{1}}}\phi^{-1}\Delta
w_{k}$
$\displaystyle\leq\frac{1}{2\pi}\int_{\gamma_{s_{1}}}\phi^{-1}\Big{(}w_{k}+\frac{w_{k}^{3}}{8s_{1}^{4}}\Big{)}$
$\displaystyle<2s_{1}\cdot\Big{(}5r_{k}\log\frac{s_{1}}{r_{k}}+\frac{1}{8s_{1}^{4}}5^{3}r_{k}^{3}\big{(}\log\frac{s_{1}}{r_{k}}\big{)}^{3}\Big{)}<r_{k}\log\frac{s_{1}}{r_{k}}+\frac{r_{k}^{2}}{2s_{1}^{2}}.$

Here the first inequality follows from (4.6); the second one follows from
(4.21), (4.4) and $\phi^{-1}|_{\gamma_{s_{1}}}\leq 1+\epsilon$; in the last
one, we used that as $k\to\infty$,

$s_{1}\to 0;\ \ \frac{s_{1}}{r_{k}}\to\infty;\ \
\frac{r_{k}}{s_{1}}(\log\frac{s_{1}}{r_{k}})^{3}\to 0.$

Together with (4.17), it follows that

$\widetilde{\tau}_{k}^{\prime}(s_{1})<\tau_{k}^{\prime}(s_{1})-\frac{r_{k}^{2}}{2s_{1}^{2}}-r_{k}\log\frac{s_{1}}{r_{k}}<0.$

Combining with (4.22), this contradicts the choice of $s_{1}$.

If $b_{k}=0$, then the same argument above gives that
$\widetilde{\mathcal{I}}_{k}$ and $\widetilde{\tau}_{k}$ are decreasing on
$[R_{k},\epsilon]$. This finishes the proof of Proposition 4.4. ∎

Note that $\mathcal{I}_{k}(s)$ is not well-defined in a small neighborhood of
$b_{k}$. Here we use Proposition 2.3 and the divergence theorem to jump over
such a small interval.

######  Proposition 4.5.

Suppose that $b_{k}\neq 0$. Then
$\widetilde{\mathcal{I}}_{k}-3r_{k}\log\frac{s}{r_{k}}$ and
$\widetilde{\tau}_{k}$ are decreasing on $[2b_{k},\epsilon]$. It follows that

$\mathcal{I}_{k}(s)<7r_{k}\log\frac{s}{r_{k}},\ \ \ \
\tau_{k}(s)<6r_{k}+sr_{k}\log\frac{s}{r_{k}}\ \ \text{ for }\
s\in[2b_{k},\epsilon].$

In particular, we conclude that for sufficiently large $k$,

$\max_{x\in\Sigma_{k}^{1}\cap\partial
B(y_{k},\epsilon;M_{k})}w_{k}<\frac{15}{2}r_{k}\log\frac{\epsilon}{r_{k}}.$

###### Proof.

By applying Proposition 2.3 (3), and then Proposition 4.4, we have

(4.24) $\max\\{w_{k};x\in\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus
B(z_{k},b_{k}/4;M_{k})\\}\leq(1+\delta)\mathcal{I}_{k}(\frac{b_{k}}{2})<\frac{9}{2}r_{k}\log\frac{b_{k}}{2r_{k}}.$

Then by (4.4),

(4.25) $\mathcal{I}_{k}(2b_{k})<5r_{k}\log\frac{b_{k}}{2r_{k}}.$

Moreover, by the divergence theorem,

$\displaystyle\tau_{k}(2b_{k})-\tau_{k}(b_{k}/2)-\tau_{k}(b_{k}/4;z_{k})$
$\displaystyle=\frac{1}{2\pi}\int_{\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus B(z_{k},b_{k}/4;M_{k})}\Delta
w_{k}\,dx$ $\displaystyle\leq\frac{1}{2\pi}\int_{\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus
B(z_{k},b_{k}/4;M_{k})}w_{k}+\frac{w_{k}^{3}}{8\rho^{4}}\,dx$
$\displaystyle<10b_{k}^{2}r_{k}\log\frac{b_{k}}{r_{k}}+\frac{5^{3}r_{k}^{3}}{b_{k}^{2}}\Big{(}\log\frac{b_{k}}{r_{k}}\Big{)}^{3}$
$\displaystyle<\frac{1}{4}b_{k}r_{k}\log\frac{b_{k}}{2r_{k}}+\frac{r_{k}^{2}}{2b_{k}}.$

Here we used the notation

$\tau_{k}(s;z_{k})=\frac{1}{2\pi}\int_{\Sigma_{k}^{1}\cap\partial
B(z_{k},s;M_{k})}\langle\nabla w_{k},\bm{\eta}_{z}\rangle,$

and $\bm{\eta}_{z}$ is the unit normal to $\partial
B(z_{k},s;M_{k})\cap\Sigma_{k}^{1}$. The first inequality is from (4.6); we
used (4.24) in the second one, and the area upper bound of $\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus B(z_{k},b_{k}/4;M_{k})$ is from the
fact that $b_{k}^{-1}(\Sigma_{k}^{1}-y_{k})$ is very close to a hyperplane.
Recall that $\Sigma_{k}$ is very close to a catenoid with radius
$\widetilde{r}_{k}\leq r_{k}$ around $z_{k}$. Then by the same argument as in
Proposition 4.4, we also have

$\tau_{k}(b_{k}/4;z_{k})<3\widetilde{r}_{k}+\frac{1}{4}b_{k}\widetilde{r}_{k}\log\frac{b_{k}}{4\widetilde{r}_{k}}\leq
3r_{k}+\frac{1}{4}b_{k}r_{k}\log\frac{b_{k}}{4r_{k}}.$

Here the inequality is from $\widetilde{r}_{k}\leq r_{k}$, and the
monotonicity of $r\mapsto r\log\frac{b_{k}}{4r}$ when $b_{k}/4r_{k}>e^{-1}$.
It follows that

(4.26) $\displaystyle\tau_{k}(2b_{k})$
$\displaystyle<\tau_{k}(b_{k}/2)+\tau_{k}(b_{k}/4;z_{k})+\frac{1}{4}b_{k}r_{k}\log\frac{b_{k}}{2r_{k}}+\frac{r_{k}^{2}}{2b_{k}}$
$\displaystyle<3r_{k}-\frac{r_{k}^{2}}{b_{k}}+\frac{1}{2}b_{k}r_{k}\log\frac{b_{k}}{2r_{k}}+3r_{k}+\frac{1}{4}b_{k}r_{k}\log\frac{b_{k}}{4r_{k}}+\frac{1}{4}b_{k}r_{k}\log\frac{b_{k}}{2r_{k}}+\frac{r_{k}^{2}}{2b_{k}}$
$\displaystyle<6r_{k}-\frac{r_{k}^{2}}{4b_{k}}+b_{k}r_{k}\log\frac{b_{k}}{r_{k}}.$

In the next, we are going to prove that
$\widetilde{\mathcal{I}}_{k}(s)-3r_{k}\log\frac{s}{r_{k}}$ and
$\widetilde{\tau}_{k}$ are decreasing. Let

$s_{2}:=\sup\\{s\in[2b_{k},\epsilon]:\widetilde{\tau}^{\prime}_{k}(t)\leq 0,\
\widetilde{\mathcal{I}}_{k}^{\prime}(t)\leq\frac{3r_{k}}{t}\ \text{ for all }\
t\in[2b_{k},s]\\}.$

It suffices to prove that $s_{2}=\epsilon$. Suppose on the contrary that
$s_{2}<\epsilon$. Then $\widetilde{\mathcal{I}}_{k}-3r_{k}\log\frac{s}{r_{k}}$
and $\widetilde{\tau}_{k}$ are decreasing on $[2b_{k},s_{2}]$, and this
implies that

$\displaystyle\widetilde{\mathcal{I}}_{k}(s_{2})-3r_{k}\log\frac{s_{2}}{r_{k}}\leq\widetilde{\mathcal{I}}_{k}(2b_{k})-3r_{k}\log\frac{2b_{k}}{r_{k}}<\mathcal{I}(2b_{k})-6r_{k}\log\frac{2b_{k}}{r_{k}}<0,\,\text{by
\eqref{eq:Ik on 2bk}};$
$\displaystyle\widetilde{\tau}_{k}(s_{2})\leq\widetilde{\tau}_{k}(2b_{k})=\tau_{k}(2b_{k})+\frac{r_{k}^{2}}{4b_{k}}-2b_{k}r_{k}\log\frac{2b_{k}}{r_{k}}<6r_{k},\,\text{by
\eqref{eq:tau 2bk}}.$

Together with (4.18) and (4.17), we then have

(4.27)
$\displaystyle\mathcal{I}_{k}(s_{2})<\widetilde{\mathcal{I}}_{k}(s_{2})+4r_{k}\log\frac{s_{2}}{r_{k}}<7r_{k}\log\frac{s_{2}}{r_{k}};$
$\displaystyle\tau_{k}(s_{2})=\widetilde{\tau}_{k}(s_{2})-\frac{r_{k}^{2}}{2s_{2}}+s_{2}r_{k}\log\frac{s_{2}}{r_{k}}<6r_{k}-\frac{r_{k}^{2}}{2s_{2}}+s_{2}r_{k}\log\frac{s_{2}}{r_{k}}.$

Recall that by Proposition 4.2,

$\displaystyle\mathcal{I}_{k}^{\prime}(s_{2})$
$\displaystyle\leq\frac{\tau_{k}(s_{2})}{s_{2}}+9r_{k}\log\frac{s_{2}}{r_{k}}\cdot
F_{k}(s_{2})+C_{0}s_{2}r_{k}\log\frac{s_{2}}{r_{k}},$
$\displaystyle<\frac{6r_{k}}{s_{2}}+2r_{k}\log\frac{s_{2}}{r_{k}}+9r_{k}\log\frac{s_{2}}{r_{k}}\cdot
F_{k}(s_{2}),$

where in the first inequality, we used that for all $x\in\gamma_{s_{2}}$,

(4.28)
$w_{k}(x)\leq(1+1/10)\mathcal{I}_{k}(s_{2})<9r_{k}\log\frac{s_{2}}{r_{k}}.$

By (4.16), it follows that

(4.29)
$\widetilde{\mathcal{I}}_{k}^{\prime}(s_{2})<\mathcal{I}_{k}^{\prime}(s_{2})-\frac{3r_{k}}{s_{2}}-2r_{k}\log\frac{s_{2}}{r_{k}}-10F_{k}(s_{2})\cdot
r_{k}\log\frac{s_{2}}{r_{k}}<\frac{3r_{k}}{s_{2}}.$

On the other hand, by the divergence theorem and the co-area formula,

$\displaystyle\tau_{k}^{\prime}(s_{2})$
$\displaystyle=\frac{1}{2\pi}\int_{\gamma_{s_{2}}}\phi^{-1}\Delta
w_{k}\leq\frac{1}{2\pi}\int_{\gamma_{s_{2}}}\phi^{-1}\Big{(}w_{k}+\frac{w_{k}^{3}}{8s_{2}^{4}}\Big{)}\,dx$
$\displaystyle\leq 4s_{2}\cdot
9r_{k}\log\frac{s_{2}}{r_{k}}+9^{3}\frac{r_{k}^{3}}{s_{2}^{3}}(\log\frac{s_{2}}{r_{k}})^{3}<r_{k}\log\frac{s_{2}}{r_{k}}+\frac{r_{k}^{2}}{2s_{2}^{2}}.$

Here the first inequality is from (4.6); the second one follows from (4.27),
(4.28), (4.4) and $\phi^{-1}|_{\gamma_{s_{1}}}\leq 1+\epsilon$; in the last
one, we used that as $k\to\infty$,

$s_{2}\to 0;\ \ \frac{s_{2}}{r_{k}}\to\infty;\ \
\frac{r_{k}}{s_{2}}(\log\frac{s_{2}}{r_{k}})^{3}\to 0.$

Then (4.17) can be applied to get

$\widetilde{\tau}_{k}^{\prime}(s_{2})<\tau_{k}^{\prime}(s_{2})-\frac{r_{k}^{2}}{2s_{2}^{2}}-r_{k}\log\frac{s_{2}}{r_{k}}<0.$

Combining with (4.29), it contradicts the choice of $s_{2}$.

Then the bound of $w_{k}$ on $\Sigma_{k}^{1}\cap\partial
B(y_{k},\epsilon;M_{k})$ follows from Proposition 2.3(3). Hence Proposition
4.5 is proved. ∎

Recall that $\Sigma_{k}$ smoothly converges to $2S_{0}^{2}$ outside at most
two balls. In particular, $\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$ are graphs
over $S_{0}^{2}$ outside such two balls; see Remark 2.4. Observe that the
normalization of graph functions of $\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$ can
both give bounded Jacobi fields, which would be smooth across the
singularities. Then $w_{k}$ are equivalent to the difference of the two graph
functions. By the maximum principle, such two graph functions should have
opposite signs. Then the upper bound for $w_{k}$ implies the Hausdorff
distance between $\Sigma_{k}$ and $S_{0}^{2}$.

######  Theorem 4.6.

For all sufficiently large $k$,

$\max_{\Sigma_{k}}\operatorname{dist}_{M_{k}}(x,S^{2}_{0})\leq 8r_{k}|\log
r_{k}|.$

###### Proof.

Recall that $\Sigma_{k}$ locally smoothly converges locally smoothly to
$2\cdot S^{2}_{0}$ away from $\mathcal{W}$ which contains at most two points.
Then for any compact set $\Omega\subset S_{0}^{2}\setminus\mathcal{W}$,
$\Sigma_{k}$ can be decomposed into two minimal graph functions in the
neighborhood of $\Omega$ for sufficiently large $k$. Denote by $u_{k}^{1}$ and
$u_{k}^{2}$ the graph functions. Note that $u_{k}^{1}$ and $u_{k}^{2}$ are
defined on any compact domain in $S_{0}^{2}\setminus\mathcal{W}$ by taking
large $k$. Denote by

$\lambda_{k}(x,s)=\max_{\partial B_{s}(x)\cap
S_{0}^{2}}\\{-u_{k}^{1},u_{k}^{2}\\}\ \ \text{ and }\ \
\Lambda_{k,s}=\max_{x\in\mathcal{W}}\lambda_{k}(x,s).$

Then by a standard argument (see [Si87]), $u_{k}^{1}/\Lambda_{k,\epsilon}$ and
$u^{2}_{k}/\Lambda_{k,\epsilon}$ locally smoothly converges to $c_{1}$ and
$c_{2}$, which are Jacobi fields of $S_{0}^{2}\setminus\mathcal{W}$. By the
minimal foliation argument, $c_{1}$ and $c_{2}$ are bounded, which implies
that they can be extended smoothly across $\mathcal{W}$. Therefore, $c_{1}$
and $c_{2}$ are constants. Moreover, by the definition of
$\Lambda_{k,\epsilon}$, $c_{1}=-1$ or $c_{2}=1$. Without loss of generality,
we assume that $c_{2}=1$. Note that $\Sigma_{k}$ does not lie in one side of
$S_{0}^{2}$, which implies that $c_{1}\leq 0$. Then by the smooth convergence
of $\Sigma_{k}$ in $B_{2\epsilon}(\mathcal{W})\setminus
B_{\epsilon/2}(\mathcal{W})$, for sufficiently large $k$,

$\lim_{k\to\infty}\max_{\Sigma_{k}^{1}\cap
B_{\epsilon}(\mathcal{W})}\frac{w_{k}}{\Lambda_{k,\epsilon}}=\lim_{k\to\infty}\max_{\partial
B_{\epsilon}(\mathcal{W})\cap\Sigma}\frac{u_{k}^{2}-u_{k}^{1}}{\Lambda_{k,\epsilon}}=1-c_{1}\geq
1.$

By Propositions 4.4 and 4.5, for sufficiently large $k$,

$\max_{x\in\Sigma_{k}^{1}\cap\partial
B_{\epsilon}(\mathcal{W})}w_{k}<(\frac{15}{2}+\frac{1}{100})r_{k}\log\frac{\epsilon}{r_{k}}.$

Thus we conclude that for all large $k$,

$\Lambda_{k,\epsilon}<(\frac{15}{2}+\frac{1}{50})r_{k}|\log r_{k}|.$

By the minimal foliation argument inside $B_{\epsilon}(p)$ and Harnack
inequalities outside such a ball,

$\max_{x\in\Sigma_{k}}\operatorname{dist}_{M}(x,S_{0}^{2})<8r_{k}|\log
r_{k}|.$

This completes the proof of Lemma 4.6. ∎

###  4.3. Upper bounds for Hausdorff distance in high dimensions

In this part, we restrict $3\leq n\leq 6$ and prove the upper bounds of the
Hausdorff distance between $\Sigma_{k}$ and $S_{0}^{n}$. Recall that
$\Sigma_{k}^{1}$ and $\Sigma_{k}^{2}$ are jointed by one or two small
catenoids with radii $r_{k}$ and $\widetilde{r}_{k}$; see Remark 2.4. Without
loss of generality, we assume that $r_{k}\geq\widetilde{r}_{k}$. Recall that
$\Sigma_{k}^{2}$ is a graph over $\Sigma_{k}^{1}$ with graph function
$w_{k}>0$.

To prove the desired bound, we start at one catenoid and derive new
monotonicity formulas. Then the smooth convergence on $\partial
B(p,\epsilon;M_{k})$ will give the desired upper bound. Now let

(4.30)
$\displaystyle\widetilde{\mathcal{I}}_{k}(s)=\mathcal{I}_{k}(s)+4s^{-\frac{1}{2}}r_{k}^{\frac{3}{2}}-sr_{k}-10r_{k}\int_{[R_{k},s]\setminus[b_{k}/2,2b_{k}]}F_{k}(t)\,dt;$
(4.31)
$\displaystyle\widetilde{\tau}_{k}(s)=\tau_{k}(s)-9s^{n}r_{k}-r_{k}^{\frac{3}{2}}s^{n-\frac{5}{2}}.$

Then by (4.3),

(4.32)
$\mathcal{I}_{k}(s)\leq\widetilde{\mathcal{I}}_{k}(s)+\frac{r_{k}}{20},$

Recall that $R_{k}/r_{k}\to 0$ and $r_{k}^{-1}(\Sigma_{k}\cap
B(y_{k},R_{k};M_{k})-y_{k})$ is arbitrarily close to the catenoid
$\mathcal{C}\cap B_{R_{k}/r_{k}}(0)$ in the smooth topology. Then by Item (1)
in Appendix B,

(4.33) $\mathcal{I}_{k}(R_{k})\leq(1+\frac{1}{100})\cdot
2r_{k}\int_{1}^{\infty}\frac{ds}{\sqrt{s^{2(n-1)}-1}}<\frac{27}{10}r_{k}.$

And by (4.11), (4.30) and (4.33),

(4.34) $\widetilde{\mathcal{I}}_{k}(R_{k})<\frac{14}{5}r_{k},\ \ \
\tau_{k}(R_{k})<(2+\frac{1}{10})r_{k}^{n-1}.$

Now we are ready to show that $\widetilde{\mathcal{I}}_{k}(s)$ is decreasing
in two disjoint intervals.

######  Proposition 4.7.

$\widetilde{\mathcal{I}}_{k}(s)$ and $\widetilde{\tau}_{k}(s)$ are decreasing
on $[R_{k},b_{k}/2)$ (resp. $[R_{k},\epsilon]$) if $b_{k}\neq 0$ (resp.
$b_{k}=0$). It follows that

$\displaystyle\mathcal{I}_{k}(s)<3r_{k},\ \ \
\tau_{k}(s)<3r_{k}^{n-1}+9s^{n}r_{k}+r_{k}^{\frac{3}{2}}s^{n-\frac{5}{2}}\ \
\text{ for }\ s\in[R_{k},b_{k}/2].$

###### Proof of Proposition 4.7.

Suppose $b_{k}\neq 0$. To prove that $\widetilde{\mathcal{I}}_{k}$ and
$\widetilde{\tau}_{k}$ are decreasing on $[R_{k},b_{k}/2]$, we let

$s_{1}:=\sup\\{s\in(R_{k},b_{k}/2):\widetilde{\tau}^{\prime}_{k}(t)\leq 0,\
\widetilde{\mathcal{I}}_{k}^{\prime}(t)\leq 0\ \text{ for all }\
t\in[R_{k},s)\\}.$

We claim that $s_{1}=b_{k}/2$. Suppose on the contrary that $s_{1}<b_{k}/2$.
Then it follows that $\widetilde{\mathcal{I}}_{k}(s)$ and
$\widetilde{\tau}_{k}(s)$ are decreasing on $[R_{k},s_{1}]$. Hence by (4.34),

$\displaystyle\widetilde{\mathcal{I}}_{k}(s_{1})\leq\widetilde{\mathcal{I}}_{k}(R_{k})<\frac{14r_{k}}{5};\
\ \
\widetilde{\tau}_{k}(s_{1})\leq\widetilde{\tau}_{k}(R_{k})<\frac{5r_{k}^{n-1}}{2}.$

Together with (4.32) and (4.31), we have that

(4.35)
$\displaystyle\mathcal{I}_{k}(s_{1})\leq\widetilde{\mathcal{I}}_{k}(s_{1})+\frac{r_{k}}{20}<\frac{29r_{k}}{10};$
$\displaystyle\tau_{k}(s_{1})=\widetilde{\tau}_{k}(s_{1})+9s_{1}^{n}r_{k}+s_{1}^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}}<\frac{5}{2}r_{k}^{n-1}+9s_{1}^{n}r_{k}+s_{1}^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}}.$

Note that the first inequality together with (4.4) and Proposition 2.3 (3)
implies that for $x\in\gamma_{s_{1}}$,

(4.36) $w_{k}(x)\leq(1+1/100)\mathcal{I}_{k}(s_{1})<3r_{k}.$

Then by Proposition 4.2,

$\displaystyle\mathcal{I}_{k}^{\prime}(s_{1})$
$\displaystyle\leq\frac{\tau_{k}(s_{1})}{s_{1}^{n-1}}+\frac{n}{\Omega_{n-1}s_{1}^{n}}\int_{\gamma_{s_{1}}}(\phi^{-1}-\phi)w_{k}+C_{0}s_{1}\mathcal{I}_{k}(s_{1})$
$\displaystyle\leq\frac{5}{2}\Big{(}\frac{r_{k}}{s_{1}}\Big{)}^{n-1}+9s_{1}r_{k}+s_{1}^{-\frac{3}{2}}r_{k}^{\frac{3}{2}}+3r_{k}F_{k}(s_{1})+C_{0}s_{1}\cdot
3r_{k}$
$\displaystyle<2\Big{(}\frac{r_{k}}{s_{1}}\Big{)}^{\frac{3}{2}}+r_{k}+3r_{k}F_{k}(s_{1}),$

where the second inequality is from (4.36); the last one follows from $n-1\geq
2$, $s_{1}\to 0$ and $r_{k}/s_{k}\to 0$ as $k\to\infty$. Together with (4.30),
we then have

(4.37)
$\widetilde{\mathcal{I}}_{k}^{\prime}(s_{1})=\mathcal{I}^{\prime}_{k}(s_{1})-2\Big{(}\frac{r_{k}}{s_{1}}\Big{)}^{\frac{3}{2}}-r_{k}-10r_{k}F_{k}(s_{1})<0.$

On the other hand, by the divergence theorem and the co-area formula,

$\displaystyle\tau_{k}^{\prime}(s_{1})=\frac{1}{\Omega_{n-1}}\int_{\gamma_{s_{1}}}\phi^{-1}\Delta
w_{k}$
$\displaystyle\leq\frac{1}{\Omega_{n-1}}\int_{\gamma_{s_{1}}}\phi^{-1}\Big{(}w_{k}+\frac{w_{k}^{3}}{8s_{1}^{4}}\Big{)}$
$\displaystyle<(1+\frac{1}{100})^{2}\cdot\frac{29}{10}s_{1}^{n-1}r_{k}+\frac{3^{3}}{8}\cdot
4s_{1}^{n-5}r_{k}^{3}$ $\displaystyle<3s_{1}^{n-1}r_{k}+r_{k}^{2}s_{1}^{n-4}.$

Here the first inequality follows from (4.6); in the second one, we used
(4.4), (4.35), (4.36) and $\phi^{-1}\Big{|}_{\gamma_{s_{1}}}<1+\frac{1}{100}$;
the last one is from $r_{k}/s_{1}\to 0$. Together with (4.31),

$\widetilde{\tau}_{k}^{\prime}(s_{1})=\tau_{k}^{\prime}(s_{1})-9ns_{1}^{n-1}r_{k}-{(n-\frac{5}{2})}r_{k}^{\frac{3}{2}}s_{1}^{n-\frac{7}{2}}<0.$

Combining with (4.37), this contradicts the choice of $s_{1}$.

If $b_{k}=0$, then the same argument above gives that
$\widetilde{\mathcal{I}}_{k}$ and $\widetilde{\tau}_{k}$ are decreasing on
$[R_{k},\epsilon]$. Hence Proposition 4.7 is proved. ∎

Note that $\mathcal{I}(s)$ is not well-defined in a small neighborhood of
$b_{k}$. Here we use Proposition 2.3 and the divergence theorem to jump over
such a small interval.

######  Proposition 4.8.

Suppose that $b_{k}\neq 0$. Then $\widetilde{\mathcal{I}}_{k}$ and
$\widetilde{\tau}_{k}$ are decreasing on $[2b_{k},\epsilon]$. It follows that

$\displaystyle\mathcal{I}_{k}(s)<4r_{k},\ \ \ \
\tau_{k}(s)<6r_{k}^{n-1}+9s^{n}r_{k}+r_{k}^{\frac{3}{2}}s^{n-\frac{5}{2}}\ \
\text{ for }\ s\in[2b_{k},\epsilon].$

In particular, we conclude that for sufficiently large $k$,

$\max_{x\in\Sigma_{k}^{1}\cap\partial
B(y_{k},\epsilon;M_{k})}w_{k}<\frac{9}{2}r_{k}.$

###### Proof.

By applying Proposition 2.3 (3), and then Proposition 4.7, together with
(4.4), we have

(4.38) $\max\\{w_{k};x\in\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus
B(z_{k},b_{k}/4;M_{k})\\}\leq(1+\delta)\mathcal{I}_{k}\big{(}\frac{b_{k}}{2}\big{)}\leq\frac{10}{3}r_{k},$

as well as

(4.39) $\mathcal{I}_{k}(2b_{k})<\frac{11r_{k}}{3}.$

Moreover, by the divergence theorem and (4.6),

$\displaystyle\ \ \
\tau_{k}(2b_{k})-\tau_{k}(b_{k}/2)-\tau_{k}(b_{k}/4;z_{k})=\frac{1}{\Omega_{n-1}}\int_{\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus B(z_{k},b_{k}/4;M_{k})}\Delta
w_{k}\,dx$ $\displaystyle\leq\frac{1}{\Omega_{n-1}}\int_{\Sigma_{k}^{1}\cap
A(y_{k},b_{k}/2,2b_{k};M_{k})\setminus
B(z_{k},b_{k}/4;M_{k})}\Big{(}w_{k}+\frac{w_{k}^{3}}{8\rho^{4}}\Big{)}\,dx$
$\displaystyle<\frac{10}{3}r_{k}\cdot(1+\frac{1}{100})(2b_{k})^{n}+(1+\frac{1}{100})(\frac{10}{3})^{3}\cdot\frac{1}{8}r_{k}^{3}(2b_{k})^{n-4}<4r_{k}(2b_{k})^{n}+r_{k}^{2}(2b_{k})^{n-3}.$

Here

$\tau_{k}(s;z_{k})=\frac{1}{\Omega_{n-1}}\int_{\Sigma_{k}^{1}\cap\partial
B(z_{k},s,M_{k})}\langle\nabla w_{k},\bm{\eta}_{z}\rangle,$

and $\bm{\eta}_{z}$ is the unit normal to $\partial
B(z_{k},s;M_{k})\cap\Sigma_{k}^{1}$. The first inequality is from (4.6); we
used (4.38) in the second one, and the area upper bound is from the fact that
$b_{k}^{-1}(\Sigma_{k}^{1}-y_{k})$ is very close to a hyperplane. Recall that
$\Sigma_{k}$ is very close to a catenoid with radius $\widetilde{r}_{k}\leq
r_{k}$ around $z_{k}$. Then by the same argument as in Proposition 4.7, we
also have

$\tau_{k}(b_{k}/4;z_{k})<3\widetilde{r}_{k}^{n-1}+9\Big{(}\frac{b_{k}}{4}\Big{)}^{n}\widetilde{r}_{k}+\Big{(}\frac{b_{k}}{4}\Big{)}^{n-\frac{5}{2}}\widetilde{r}_{k}^{\frac{3}{2}}\leq
3r_{k}^{n-1}+9\Big{(}\frac{b_{k}}{4}\Big{)}^{n}r_{k}+\Big{(}\frac{b_{k}}{4}\Big{)}^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}}.$

It follows that

$\displaystyle\tau_{k}(2b_{k})$
$\displaystyle<\tau_{k}(b_{k}/2)+\tau_{k}(b_{k}/4;z_{k})+4r_{k}(2b_{k})^{n}+r_{k}^{2}(2b_{k})^{n-3}$
$\displaystyle<6r_{k}^{n-1}+9\Big{(}\frac{b_{k}}{2}\Big{)}^{n}r_{k}+\Big{(}\frac{b_{k}}{2}\Big{)}^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}}+9\Big{(}\frac{b_{k}}{4}\Big{)}^{n}r_{k}+\Big{(}\frac{b_{k}}{4}\Big{)}^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}}+4r_{k}(2b_{k})^{n}+r_{k}^{2}(2b_{k})^{n-3}$
$\displaystyle<6r_{k}^{n-1}+9(2b_{k})^{n}r_{k}+(2b_{k})^{n-\frac{5}{2}}r_{k}^{\frac{3}{2}},$

which implies that

$\widetilde{\tau}_{k}(2b_{k})=\tau_{k}(2b_{k})-9(2b_{k})^{n}r_{k}-r_{k}^{\frac{3}{2}}(2b_{k})^{n-\frac{5}{2}}<6r_{k}^{n-1}.$

Then one can prove that $\widetilde{\mathcal{I}}_{k}(s)$ and
$\widetilde{\tau}_{k}(s)$ are decreasing on $[2b_{k},\epsilon]$ by the same
argument as in Proposition 4.7.

Applying Proposition 2.3(3) and (4.4) again,

$\max_{x\in\Sigma_{k}^{1}\cap\partial
B(y_{k},\epsilon;M_{k})}w_{k}\leq(1+1/10)\mathcal{I}_{k}(\epsilon)\leq\frac{9}{2}r_{k}.$

This completes the proof of Proposition 4.7. ∎

Then using the same argument as Theorem 4.6, one can prove the Hausdorff
distance upper bounds between $\Sigma_{k}$ and $S_{0}^{n}$. The only
modification is to replace Propositions 4.4 and 4.5 by Propositions 4.7 and
4.8.

######  Theorem 4.9.

For all sufficiently large $k$,

$\max_{\Sigma_{k}}\operatorname{dist}_{M_{k}}(x,S^{n}_{0})<5r_{k}.$

##  Appendix A Minimal graphs

Let $\mathcal{N}\subset M$ be a two-sided minimal hypersurface in
$(M^{n+1},g)$ with a chosen unit normal vector field $\mathbf{n}$. Let $d$ be
the oriented distance function to $\mathcal{N}$, and $\mathcal{N}_{s}$ be the
level set of $d$. Then for some small $\mathfrak{d}>0$,
$\\{\mathcal{N}_{t}\\}_{t\in(-\mathfrak{d},\mathfrak{d})}$ forms a foliation
of a neighborhood of $\mathcal{N}$. Denote by $\widetilde{\nabla}$ and
$\nabla$ the Levi-Civita connections of $M$ and $\mathcal{N}_{s}$
respectively. Then $\widetilde{\nabla}d$ is the unit normal vector field on
$\mathcal{N}_{s}$. In this section, we always assume that $M$ and
$\mathcal{N}$ satisfy the following conditions:

(A.1) $|A|_{\mathcal{N}}<\epsilon/{\mathfrak{d}},\quad|R|\leq
1,\quad|\widetilde{\nabla}R|<C(n).$

Here $R$ is the Riemannian curvature tensor of $M$; $\epsilon$ is a small
constant depending only on $n$, e.g. $\epsilon=10^{-1000n}$; $C(n)>1$ is a
constant that can be changed from line to line.

Let $\Sigma$ be a minimal graph over $\mathcal{N}$ with

$\max_{x\in\Sigma}\operatorname{dist}_{M}(x,\mathcal{N})<\mathfrak{d}.$

Denote by $u$ the graph function. Then such a function can be extended to a
neighborhood of $\mathcal{N}$ by taking

$u(p,s)=u(p)-s.$

When restricted to $\mathcal{N}_{s}$, $u$ is the graph function of $\Sigma$
over $\mathcal{N}_{s}$. Moreover, $u=0$ when restricted to $\Sigma$, and hence
$\widetilde{\nabla}u|_{\Sigma}$ is the normal vector field on $\Sigma$ and is
non-zero everywhere.

For $p\in\Sigma\cap\mathcal{N}_{s}$, let $\\{e_{i}\\}$ be an orthonormal basis
of $T_{p}\mathcal{N}_{s}$. Let $A$ and $H$ denote respectively the second
fundamental form and mean curvature of $\mathcal{N}_{s}$ with respect to
$\widetilde{\nabla}d$. A direct computation gives that

$\displaystyle\frac{\partial}{\partial
s}H=-\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}d)-|A|^{2};$
$\displaystyle\widetilde{\nabla}_{\widetilde{\nabla}d}\widetilde{\nabla}u=\widetilde{\nabla}_{\widetilde{\nabla}d}\nabla
u=-A(\nabla u);\ \ \ \frac{\partial}{\partial
s}|\widetilde{\nabla}u|^{2}=-2A(\nabla u,\nabla u);$
$\displaystyle\mathrm{div}_{M}A(\nabla
u)=\mathrm{div}_{\mathcal{N}_{s}}A(\nabla
u)=\langle\nabla^{2}u,A\rangle+\operatorname{Ric}(\nabla
u,\widetilde{\nabla}d)+\langle\nabla u,\nabla H\rangle;$
$\displaystyle\frac{\partial}{\partial
s}\mathrm{div}_{M}\widetilde{\nabla}u=-\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}u+\nabla
u)+|A|^{2}-2\langle\nabla^{2}u,A\rangle-\langle\nabla u,\nabla H\rangle;$
$\displaystyle\frac{\partial}{\partial s}|\nabla
H|^{2}=-2\langle\nabla(\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}d)+|A|^{2}),\nabla
H\rangle-2A(\nabla H,\nabla H);$ $\displaystyle\frac{\partial}{\partial
s}|A|^{2}=2R(\widetilde{\nabla}d,e_{i},\widetilde{\nabla}d,e_{j})A(e_{i},e_{j})-2A(e_{i},e_{j})A(e_{j},e_{k})A(e_{k},e_{i});$
$\displaystyle\frac{\partial}{\partial s}|\nabla
A|^{2}=2R_{sjsk,i}A_{jk,i}+4R_{silk}A_{jl}A_{ki,j}+4R_{silj}A_{kl}A_{jk,i}-2A_{jk,i}A_{jk,l}A_{li}-4A_{ik,j}A_{lk,j}A_{il};$
$\displaystyle\frac{\partial}{\partial
s}|\nabla^{2}u|^{2}=-2A_{kj,i}u_{k}u_{ij}+2R_{sikj}u_{k}u_{ij}-4u_{ij}u_{kj}A_{ik}.$

We pause to state a standard differential inequality, whose proof is left to
readers.

######  Lemma A.1.

Let $f:[0,\mathfrak{d}]\to\mathbb{R}$ be a non-negative differentiable
function. Suppose that

$f^{\prime}(t)\leq a+bf(t)$

for real numbers $a$ and $b$. Then for each $t\in[0,\mathfrak{d}]$,

$f(t)\leq e^{bt}f(0)+\frac{a}{b}(e^{bt}-1).$

Now Lemma A.1 can be applied to bound those terms on $\mathcal{N}_{s}$ by
their restriction on $\mathcal{N}$.

######  Lemma A.2.

$\displaystyle|\nabla u(x,t)|\leq 2|\nabla u(x)|;\ \ \ \ |A(x,t)|\leq
2|A(x)|+2t;$ $\displaystyle\Big{|}|A(x,t)|^{2}-|A(x)|^{2}\Big{|}\leq
C(n)\Big{(}|A(x)|t+t+|A(x)|^{3}t\Big{)};$ $\displaystyle|\nabla A(x,t)|\leq
C(n)\Big{(}|\nabla A(x)|+t(|A(x)|+1)\Big{)};$ $\displaystyle|\nabla
H(x,t)|\leq C(n)\Big{(}t+t(|A(x)|+t)|\nabla A(x)|+|A(x)|^{2}t^{2}\Big{)};$
$\displaystyle|\nabla^{2}u(x,t)|\leq C(n)\Big{(}|\nabla^{2}u(x)|+t(1+|\nabla
A(x)|)|\nabla u(x)|\Big{)}.$

###### Proof.

Note that $\frac{\partial}{\partial s}|A|^{2}\leq 2|A|+2|A|^{3}$. Then by
classical OD-inequalities, we have

$\arctan|A(x,t)|\leq\arctan|A(x)|+t.$

Since $|tA(x)|\leq\delta|A(x)|<\epsilon$,

(A.2) $|A|(x,t)\leq 2|A(x)|+2t.$

The others can be derived similarly as follows: note that

$\frac{\partial}{\partial s}|\nabla A|^{2}\leq C(n)\Big{(}|\nabla
A(x,t)|+|A(x,t)||\nabla A(x,t)|+|A(x,t)||\nabla A(x,t)|^{2}\Big{)}.$

Fix $x\in\mathcal{N}$. Now let $f(t)=\sqrt{1+|\nabla A(x,t)|^{2}}$. It follows
that

$f^{\prime}(t)\leq C(n)\Big{(}1+|A(x,t)|+|A(x,t)|f\Big{)}\leq
C(n)\Big{(}1+|A(x)|+(1+|A(x)|)f(t)\Big{)}$

Here the second inequality comes from (A.2). Then Lemma (A.1) can be applied
to obtain

$|\nabla A(x,t)|+1\leq e^{C(n)(1+|A(x)|)t}(1+|\nabla
A(x)|)+e^{C(n)(1+|A(x)|)t}-1,$

which yields

$|\nabla A(x,t)|\leq 2|\nabla A(x)|+C(n)(1+|A(x)|)t.$

Here we used the condition that $C(n)|A(x)|t\leq C(n)|A(x)|\mathfrak{d}\ll 1$
by (A.1). ∎

Now we are ready to derive our inequality for $u$. Indeed, by the minimality
of $\Sigma$,

$\displaystyle 0=\mathrm{div}_{\Sigma}\widetilde{\nabla}u$
$\displaystyle=\mathrm{div}_{M}\widetilde{\nabla}u-\widetilde{\nabla}^{2}u\Big{(}\frac{\widetilde{\nabla}u}{|\widetilde{\nabla}u|},\frac{\widetilde{\nabla}u}{|\widetilde{\nabla}u|}\Big{)}$
$\displaystyle=\mathrm{div}_{M}\widetilde{\nabla}u-(A+\nabla^{2}u)\Big{(}\frac{\nabla
u}{|\widetilde{\nabla}u|},\frac{\nabla u}{|\widetilde{\nabla}u|}\Big{)}.$

It follows that

$\displaystyle\frac{(A+\nabla^{2}u)(\nabla u,\nabla u)}{1+|\nabla u|^{2}}$
$\displaystyle=\mathrm{div}_{\mathcal{N}}\nabla
u+\int_{0}^{s}\frac{\partial}{\partial t}\mathrm{div}_{M}\widetilde{\nabla}u$
$\displaystyle=\Delta_{\mathcal{N}}u+\int_{0}^{s}-\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}u+\nabla
u)+|A|^{2}-2\langle\nabla^{2}u,A\rangle-\langle\nabla u,\nabla H\rangle\,dt.$

Since $\widetilde{\nabla}u=\nabla u-\widetilde{\nabla}d$, then

$\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}u+\nabla
u)=-\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}d)+2\operatorname{Ric}(\widetilde{\nabla}d,\nabla
u).$

It follows that

$\displaystyle(\Delta u$
$\displaystyle+|A|^{2}u)(x,0)+\int_{0}^{s}\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}d)\,dt$
$\displaystyle\leq(|A|+|\nabla^{2}u|)|\nabla u|^{2}(x,s)+\int_{0}^{s}2n|\nabla
u|+\Big{|}|A(x)|^{2}-|A(x,t)|^{2}\Big{|}+2|\nabla^{2}u||A|+|\nabla u||\nabla
H|\,dt.$

Combining all of them, we conclude that

(A.3) $\displaystyle\Delta_{\mathcal{N}}u+|A|_{\mathcal{N}}^{2}u+$
$\displaystyle\int_{0}^{u(x)}\operatorname{Ric}(\widetilde{\nabla}d,\widetilde{\nabla}d)\,dt$
$\displaystyle\leq 8|A|_{\mathcal{N}}|\nabla
u|^{2}+|A|_{\mathcal{N}}^{3}u^{2}+|\nabla^{2}u||A|_{\mathcal{N}}u+C(n)\Big{(}u+|\nabla
u|+|\nabla u|^{3}\Big{)}u+$ $\displaystyle+C(n)\Big{\\{}|\nabla
u(x)|^{2}|\nabla^{2}u|+|\nabla u|^{3}|\nabla A|u+|A|u^{2}(1+|\nabla
u|)+|\nabla^{2}u|u^{2}+$ $\displaystyle+|\nabla A||\nabla
u|u^{2}(|A|+u)+|\nabla u|u^{2}(1+|A||\nabla u|+|\nabla
A|u+|A|^{2}u)\Big{\\}}_{\mathcal{N}}.$

##  Appendix B Catenoids

In this section, we collect some basic results for the catenoids. In
$\mathbb{R}^{n+1}$, given $r>0$, there is an associated catenoid given by

$|x_{n+1}|=\int_{r}^{t}\frac{ds}{\sqrt{(s/r)^{2(n-1)}-1}},\ \
t=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}.$

Here $r$ is called the radius. When $r=1$, this catenoid is said to be
standard. Let $t,h,R$ be the solutions of

(B.1) $t^{2}+h^{2}=R^{2}\ \ \ \text{ and }\ \ \
|h|=\int_{1}^{t}\frac{ds}{\sqrt{s^{2(n-1)}-1}}.$

For a standard catenoid $\mathcal{C}$, we have that

  1. (1)

for $n=2$, $\log t<|h|<\log(2t)$ and for $n\geq 3$, $|h|<1.31103$;

  2. (2)

$|A(x)|\leq\sqrt{n(n-1)}$ and $|x||A|\to 0$ as $|x|\to\infty$;

  3. (3)

for each connected component $\gamma$ of $\partial B_{R}(0)\cap\mathcal{C}$,
it bounds an $n$-dimensional ball $D$ with area $\frac{\Omega_{n-1}}{n}t^{n}$,
where $\Omega_{n-1}$ is the area of unit $(n-1)$-sphere;

  4. (4)

for $4\leq(n+1)\leq 7$,

(B.2) $\mathcal{A}_{n}:=\lim_{R\to\infty}\mathrm{Area}(\mathcal{C}\cap
B_{R}(0))-2\cdot\frac{\Omega_{n-1}}{n}t^{n}>0;$

  5. (5)

for $n=2$,

(B.3) $\mathrm{Area}(\mathcal{C}\cap
B_{R}(0))-2\mathrm{Area}(D)=2\pi(|h|+t\sqrt{t^{2}-1})-2\pi t^{2}>2\pi(\log
R-1).$

Note that outside $B_{2}(0)$, $\mathcal{C}$ has two connected components
$\Sigma^{1}$ and $\Sigma^{2}$. Here we assume
$\Sigma^{1}\subset\\{x_{n+1}<0\\}$. Then $\Sigma^{2}$ is a graph over
$\Sigma^{1}$. Let $w$ denote the graph function. Denote by $\bm{\eta}$ the
unit outward normal to $\partial B_{R}(0)\cap\Sigma^{1}$ of
$B_{R}(0)\cap\Sigma^{1}$.

######  Proposition B.1.

$\lim_{R\to\infty}\frac{1}{\Omega_{n-1}}\int_{\Sigma_{k}^{1}\cap\partial
B_{R}(0)}\langle\nabla w,\bm{\eta}\rangle=2.$

###### Proof of Proposition B.1.

By computation,

$\partial_{h}=(-\sqrt{t^{2n-2}-1}Y,1),$

where $Y=(y_{1},y_{2},\cdots,y_{n})$ with $|Y|=1$. Then the normal line at
$(tY,h)$ intersects $\mathcal{C}$ at another point

$P=(t_{h}Y,z_{h}),$

where $z_{h}>0$ is given by

(B.4) $z_{h}=h+(t_{h}-t)\sqrt{t^{2n-2}-1}.$

$x_{n+1}$
---
$h$
---
$t$
---
$\partial_{h}$
---
$\bm{\eta}$
---
$(tY,h)$
---
$Y$
---
$(t_{h}Y,z_{h})$
---
Figure III. The normal line of a catenoid.

Observe that $z_{h}>0$ and $h<0$. Then by (B.4), $t_{h}>t$. By (B.1),
$z_{h}<2\log t_{h}<t_{h}$ and $|h|<2\log t<t$ for large $t$. Plugging them
into (B.4), it follows that

$t_{h}+t>z_{h}-h=(t_{h}-h)\sqrt{t^{2n-2}-1}\geq 3(t_{h}-t),$

which implies that $t_{h}<2t$. Plugging it back into (B.4) again,

$2\log(2t)+2\log t>z_{h}-h=(t_{h}-t)\sqrt{t^{2n-2}-1}>(t_{h}-t)(t-1).$

It follows that $t<t_{h}<t+1$ for large $t$. Observe that by (B.1),

$\partial_{h}t_{h}=\partial_{h}z_{h}\cdot\sqrt{t_{h}^{2n-2}-1};\quad\partial_{h}t=-\sqrt{t^{2n-2}-1}.$

Differentiating both sides in (B.4), we then have

$\partial_{h}t_{h}=1+(\partial_{h}t_{h}+\partial_{h}t)\sqrt{t^{2n-2}-1}+(t_{h}-t)(n-1)\frac{t^{2n-3}\cdot\partial_{h}t}{\sqrt{t^{2n-2}-1}}.$

Combining them together, it follows that

$\partial_{h}z_{h}=\frac{t^{2n-2}-(t_{h}-t)(n-1)t^{2n-3}}{1-\sqrt{(t_{h}^{2n-2}-1)(t^{2n-2}-1)}}\to-1\
\ \ \text{ as }\ \ \ h\to-\infty.$

Note that

$w^{2}=(t_{h}-t)^{2}+(z_{h}-h)^{2}.$

Then by computation,

$\displaystyle\partial_{h}w^{2}$
$\displaystyle=2(t_{h}-t)(\sqrt{t_{h}^{2n-2}-1}\partial_{h}z_{h}+\sqrt{t^{2n-2}-1})+2(z_{h}-h)(\partial_{h}z_{h}-1)$
$\displaystyle=2\cdot\Big{(}\frac{\sqrt{t_{h}^{2n-2}-1}}{\sqrt{t^{2n-2}-1}}+1\Big{)}\cdot(z_{h}-h)\partial_{h}z_{h},$

which implies that

$\lim_{h\to-\infty}\partial_{h}w=\lim_{h\to-\infty}-\frac{2(z_{h}-h)}{w}=\lim_{h\to-\infty}-2\cdot\frac{(t_{h}-t)\sqrt{t^{2n-2}-1}}{t^{n-1}(t_{h}-t)}=-2.$

On the other hand,

$\frac{1}{\Omega_{n-1}}\int_{\Sigma\cap\\{x_{n+1}=h\\}}\langle\nabla
w,\bm{\eta}\rangle=\frac{1}{\Omega_{n-1}}\int_{\Sigma\cap\\{z=h\\}}\frac{-\partial_{h}w}{t^{n-1}}=-\partial_{h}w.$

This finishes the proof of Proposition B.1. ∎

## References


Send correspondence to Ipek Oguz<EMAIL_ADDRESS>
# Deep Angiogram: Trivializing Retinal Vessel Segmentation

Dewei Hu  Vanderbilt University, Dept. of Electrical and Computer Engineering
Xing Yao  Vanderbilt University, Dept. of Computer Science  Jiacheng Wang
Vanderbilt University, Dept. of Computer Science  Yuankai K. Tao  Vanderbilt
University, Dept. of Biomedical Engineering, Nashville, TN, USA  Ipek Oguz
Vanderbilt University, Dept. of Computer Science

###### Abstract

Among the research efforts to segment the retinal vasculature from fundus
images, deep learning models consistently achieve superior performance.
However, this data-driven approach is very sensitive to domain shifts. For
fundus images, such data distribution changes can easily be caused by
variations in illumination conditions as well as the presence of disease-
related features such as hemorrhages and drusen. Since the source domain may
not include all possible types of pathological cases, a model that can
robustly recognize vessels on unseen domains is desirable but remains elusive,
despite many proposed segmentation networks of ever-increasing complexity. In
this work, we propose a contrastive variational auto-encoder that can filter
out irrelevant features and synthesize a latent image, named deep angiogram,
representing only the retinal vessels. Then segmentation can be readily
accomplished by thresholding the deep angiogram. The generalizability of the
synthetic network is improved by the contrastive loss that makes the model
less sensitive to variations of image contrast and noisy features. Compared to
baseline deep segmentation networks, our model achieves higher segmentation
performance via simple thresholding. Our experiments show that the model can
generate stable angiograms on different target domains, providing excellent
visualization of vessels and a non-invasive, safe alternative to fluorescein
angiography.

###### keywords:

deep learning, vessel enhancement, vessel segmentation, domain generalization

##  1 INTRODUCTION

Retinal fundus photography is a cheap, fast and non-invasive modality that
reveals essential anatomical features including optic disc, optic cup, macula,
fovea, vessels and lesions such as hemorrhages and exudates [1]. Therefore, it
is widely used for the diagnosis of diseases such as diabetic retinopathy [2],
glaucoma [3] and age-related macular degeneration [4]. While fundus
photography is broadly used as a low-cost screening tool, it does not provide
sufficient contrast to resolve clinically relevant vascular features and
exogenous indocyanine green angiography (ICG)/fluorescein angiography (FA)
remain the standard of care for visualization/quantifying retinal
vasculopathies. An algorithm that can provide accurate vessel segmentation
from these fundus images would have profound impact on future clinical
practice. In recent years, deep learning models [5] have achieved remarkable
success in this task. Nevertheless, the domain shift induced by variations in
image contrast and presence of unseen pathological features in testing data
can dramatically degrade the performance of deep models.

Recent research explored three main types of domain generalization methods
[6]: domain randomization, representation learning and general learning
strategy. Domain randomization augments the training data to extend the source
domain [7], improving the likelihood that an unseen target domain overlaps
with the training domain. Representation learning refers to the
disentanglement of features that are invariant to different domains [8]. A
typical general learning strategy is meta-learning: for example, Li et al.
simulate the domain shift by splitting the source domain into meta-train and
meta-test [9].

In this work, we leverage both domain randomization and representation
learning approaches to train a model that has superior generalizability across
different domains. We augment the source domain by the contrast limited
adaptive histogram equalization (CLAHE) [10] with clip limit
$\epsilon\in\mathcal{N}$. In addition to well-enhanced contrast for vessels,
the augmented images also have exaggerated irrelevant structures including
noise and lesions. Inspired by the idea of disentangling the shared features
in two images presented in our previous work [11, 12], we leverage a
variational auto-encoder (VAE) to extract the representation of vessels.
However, as we showed in [11], this latent image may have an arbitrary style
that contains unwanted features. We tackle this challenge by introducing a
contrastive loss such that vessels are the only features in the synthetic
image. We name the result a deep angiogram. Then, the segmentation task is
simply reduced to Otsu thresholding [13]. Without the irrelevant features, the
visibility of the vasculature is drastically improved in the deep angiogram
compared to other vessel enhancement approaches [14]. We evaluate the
generalizability of our model by the segmentation performance on the target
domains. For baseline models, we trained two segmentation networks on the
source domain that take the green channel fundus image and the principle
component analysis (PCA) image as the input respectively. The result indicates
that the proposed method generalizes better on target domains and achieves
higher segmentation performance than deep segmentation networks, by simple
thresholding.

##  2 METHODS

(a) model structure

(b) source and target domains

Figure 1: (a) The deep angiogram model structure. $x$ is the input fundus
image, $x^{\prime}=C_{\epsilon}(x)$ is the CLAHE-enhanced image with clip
limit $\epsilon$. $y$ is the ground truth. $E_{\theta}$ is a residual U-Net
that serves as the encoder of the VAE. $D_{\varphi}$ is the corresponding
decoder. $\mathcal{L}_{cont}$ and $\mathcal{L}_{seg}$ represent the
contrastive loss and segmentation loss. The dashed line on $D_{\varphi}$
indicates it will not be applied in testing. (b) The source target domains. a,
DRIVE, b, HRF, c, STARE, d, ARIA.

###  2.1 Causal Feature Extraction

Fig. 1(a) shows our VAE model composed by the encoder $E_{\theta}$ and the
decoder $D_{\varphi}$. The input image is $x$ and the supervision is provided
by the label $y$. As we have previously shown [11, 12], when the latent
manifold of the VAE has the same dimension with input $x$, the encoder is able
to enhance the shared features in $x$ and $y$. Intuitively, if an image is
regarded as a collection of representations, then $(x\cap y)\subseteq
E_{\theta}(x)$ should hold to guarantee that there is no essential information
missing in the output $\hat{y}$. In the context of causal learning, $x\cap y$
is the set of causal features for the final prediction. In this
implementation, the fundus image $x$ includes information of many anatomical
structures such as optic disc, vessels, macula and lesions, whereas the causal
features for the segmentation task contain just the vessels, so ideally the
latent image should be a vessel map without any irrelevant features, i.e.,
$(x\cap y)=E_{\theta}(x)$.

As suggested in Fig. 1, since we want to put most of the workload on the
encoder $E_{\theta}$, it is designed to have more learnable parameters than
the decoder $D_{\varphi}$. Both $E_{\theta}$ and $D_{\varphi}$ have residual
U-Net architecture. Note that the decoder $D_{\varphi}$ will not be applied in
the testing since its purpose is to simply provide supervision to $E_{\theta}$
during training. The segmentation loss for the decoder is set to be a
combination of cross-entropy and Dice loss:

$\mathcal{L}_{seg}=-\frac{1}{N}\sum_{n=1}^{N}y_{n}\log\hat{y}_{n}+\left(1-\frac{2\sum_{n=1}^{N}y_{n}\hat{y}_{n}}{\sum_{n=1}^{N}y_{n}^{2}+\hat{y}_{n}^{2}}\right)$
(1)

Fundus | Deep Angiogram | Label
---|---|---
ARIA |  |
STARE |  |
Figure 2: Test examples from the target domains, ARIA and STARE. Below each
image, close-up panels show two highlighted areas (red and yellow boxes) for
easier comparison. Deep angiograms provide excellent vessel clarity.

###  2.2 Domain Randomization

There are two major causes for distribution shift of fundus images. First,
within a well-curated dataset (e.g., DRIVE [15]), the image contrast is
usually consistent. A model trained on such a dataset may struggle with a
poor-contrast test image. Second, since a given dataset is unlikely to
exhaustively provide samples of all possible pathologies, unseen features such
as drusen and hemorrhages can be problematic during testing.

To improve the robustness of the model, we randomize the source domain data by
CLAHE [10] in addition to other commonly used augmentation methods (e.g.,
rotation). For an input image $x$, we apply CLAHE $C_{\epsilon}$ to all the
color channels with a random clip limit $\epsilon\in\mathcal{N}(5,1)$. In the
resultant image $x^{\prime}$, the contrast of vessels are strongly enhanced,
as well as the background noise. Then as in Fig. 1, we introduce a contrastive
loss $\mathcal{L}_{cont}$ for the latent image to guarantee that the model is
not distracted by this exaggerated noise and provides stable visualization for
input with various contrasts. The loss function is defined as the sum of the
$L_{2}$ loss and the structural similarity (SSIM) loss.

$\mathcal{L}_{cont}=\|E_{\theta}(x)-E_{\theta}(x^{\prime})\|_{2}+SSIM(E_{\theta}(x)-E_{\theta}(x^{\prime}))$
(2)

The SSIM loss is defined as
$SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})}$,
where $\mu$ and $\sigma$ represent the mean and standard deviation of the
image, and $c_{1}$ and $c_{2}$ are constants.

###  2.3 Experiments

Baseline Methods. Since the color image is more sensitive to domain shift, it
is common to convert the fundus image to grayscale as pre-processing,
typically by extracting the green channel or using principle component
analysis (PCA). We train a segmentation network that has the same architecture
as $E_{\theta}$ with either the green channel or the PCA as input. We compare
these two networks to Otsu thresholding of deep angiograms.

Datasets. We use four publicly available fundus datasets as shown in Fig.
1(b). The DRIVE dataset [15] consists of 20 labelled images of size $565\times
584$. The HRF dataset [16] contains 45 labelled images of size $3504\times
2336$. The STARE dataset [17] includes 20 labelled images of size $700\times
605$. The ARIA dataset [18] includes 138 labelled images of size $768\times
576$. DRIVE and HRF are set as source domain, whereas STARE and ARIA are used
for testing.

Implementation Details. All networks are trained and tested on an NVIDIA RTX
2080TI 11GB GPU. We use a batch size of 4 and train for 300 epochs. We use the
Adam optimizer with the initial learning rate of $5\times 10^{-4}$ for the
proposed VAE, $1\times 10^{-3}$ for the baseline segmentation networks. The
learning rate for both networks decay by 0.5 every 3 epochs.

##  3 RESULTS and Conclusion

|
---|---
Figure 3: Quantitative evaluation of the segmentation results on the two
target domains. From left to right: Dice coefficient, accuracy, sensitivity,
specificity. Blue, segmentation network trained on the green channel. Orange,
segmentation network trained on the PCA image. Green, segmentation obtained by
thresholding the deep angiogram.

Fig. 2 shows a test example from each of the target domains. We observe that
for different datasets, the manual annotations includes varying amounts of
detail: the label for the STARE dataset contains many more small vessels than
ARIA. In the ARIA example, the deep angiogram is able to enhance the thin
vessels with very poor contrast. This is also evident by the big vessels seen
at the bottom left quadrant of the image where the illumination is low.
Moreover, the angiogram filters out the circular artifacts seen within the red
box. In the STARE example, our model extracts most of the vasculature
including the faintly visible fine vessels. These tiny vessels have relatively
lower intensity in the deep angiogram, which suggests lower confidence.
Compared to the manual label, the deep angiogram can also delineate the vessel
diameter more precisely.

We quantitatively evaluate the vessel segmentation performance in Fig. 3. By
simple thresholding on deep angiogram, we obtain get better vessel maps than
the segmentation networks that use the green channel and PCA image as inputs.

The proposed method can effectively extract a specific type of feature from a
complex context. Specific to retinal vessels, our model can generate stable
deep angiograms that dramatically enhance small vessels with poor contrast for
color fundus images from unseen domains. Hence, deep angiogram is a low-cost
method that can be performed using standard fundus photography technologies,
including portable handheld systems. The ability to resolve vascular features
without the need for exogenous contrast injections significantly reduces the
clinical expertise/equipment/cost of retinal angiography. Integration of these
technologies with recent demonstrations of cellphone-based fundus photography
methods and remote diagnostic technologies can move retinal disease screening
out of the clinic and dramatically expand the impact of color fundus
photography.

##  4 ACKNOWLEDGEMENTS

This work is supported by the Vanderbilt University Discovery Grant Program.

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