uniformly converge, and hence the local CAT(0) property is preserved. We note
that by a finer intrinsic investigation one can deduce that the induced metric
is actually $C^{1,1}$ with curvature defined almost everywhere, see [KP94]. We
will apply the following lemma to the level sets.

######  Lemma 3.20.

For every $\varepsilon,A>0$ and every CAT(0)-metric $d$ on $S$ of area at most
$A$, for every $p\in S$ and every complete geodesic $\psi$ through $p$ in
$(S,d)$ there exists a closed homotopically non-trivial Lipschitz curve $\chi$
through $p$ in $(S,d)$ consisting from three subsequent arcs $\chi_{1}$,
$\chi_{2}$ and $\chi_{3}$ such that the arcs $\chi_{1}$ and $\chi_{3}$ are
subarcs of $\psi$ and we have ${\rm len}(\chi_{1})\leq A/\varepsilon$, ${\rm
len}(\chi_{2})\leq\varepsilon,$ ${\rm len}(\chi_{3})\leq A/\varepsilon$.

We refer to [BH99] for a discussion of the notion of area in general
CAT(0)-space, but remark again that we will apply Theorem 3.20 only to metrics
coming from $C^{1,1}$-Riemannian structures. It is possible that some arcs are
empty, particularly $\psi$ might be a closed geodesic of length at most
$A/\varepsilon$, in which case it may be taken as $\chi$. It is possible that
the arcs $\chi_{1}$ and $\chi_{3}$ overlap as subarcs of $\psi$.

###### Proof.

Let $\tilde{P}$ be the full preimage of $p$ in $\tilde{S}$ and
$\tilde{p}\in\tilde{P}$ be some lift. The lift $\tilde{d}$ of $d$ is a
(globally) CAT(0)-metric. Let $\tilde{\psi}$ be the lift of $\psi$ through
$\tilde{p}$ and $R(\tilde{p},l,\varepsilon)$ be the set of points $r$ at
distance at most $\varepsilon/2$ from $\tilde{\psi}$ (on both sides) such that
the closest point to $r$ on $\tilde{\psi}$ is at distance at most $l/2$ from
$\tilde{p}$. There is a natural surjective map from
$R(\tilde{p},l,\varepsilon)$ to an $(\varepsilon\times l)$-Euclidean rectangle
$E$: we send the corresponding segment of $\tilde{\psi}$ to the segment $I$ of
$E$, connecting the midpoints of the sides of length $\varepsilon$, and then
extend it to a map from $R(\tilde{p},l,\varepsilon)$ to $E$ so that the
nearest-point projections to $\tilde{\psi}$ and to $I$ commute. The Fermi
Lemma, [AB06, Lemma 3.1], states that this map is 1-Lipschitz, hence ${\rm
area}(R(\tilde{p},l,\varepsilon))\geq\varepsilon l$.

By $l_{0}$ we denote the minimal number such that
$R(\tilde{p},l_{0},\varepsilon)$ intersects
$R(\tilde{p}^{\prime},l_{0},\varepsilon)$ for some
$\tilde{p}^{\prime}\in\tilde{P}\setminus\\{\tilde{p}\\}$. Due to the area
bound, $l_{0}\leq A/\varepsilon$.

Since the sets $R(\tilde{p},l_{0},\varepsilon)$ and
$R(\tilde{p}^{\prime},l_{0},\varepsilon)$ intersect, there exists a curve
$\tilde{\chi}$ connecting $\tilde{p}$ with $\tilde{p}^{\prime}$, consisting of
three subsequent arcs $\tilde{\chi}_{1}$, $\tilde{\chi}_{2}$ and
$\tilde{\chi}_{3}$ such that $\tilde{\chi}_{1}\subset\tilde{\psi}$, ${\rm
len}(\tilde{\chi}_{1})\leq l_{0}$; ${\rm
len}(\tilde{\chi}_{2})\leq\varepsilon$;
$\tilde{\chi}_{3}\subset\tilde{\psi}^{\prime}$, where $\tilde{\psi}^{\prime}$
is the lift of $\psi$ passing through $\tilde{p}^{\prime}$, ${\rm
len}(\tilde{\chi}_{3})\leq l_{0}$. The projection of $\tilde{\chi}$ to $S$ is
the desired curve $\chi$. ∎

For a spacetime $\Omega^{+}(\rho,\tau)$ the flow $\xi^{t}$ is minus the
gradient flow of the cosmological time.

######  Lemma 3.21 ([BBZ11, Proposition 6.1]).

Let $\chi:[a,b]\rightarrow\Omega^{+}(\rho,\tau)$ be a spacelike Lipschitz
curve contained in the past of the level set $L_{1}$ of the cosmological time,
and $\chi^{\prime}$ be its projection to $L_{1}$ along the flow $\xi^{t}$.
Then ${\rm len}(\chi)\leq{\rm len}(\chi^{\prime})$.

The following lemma uses a decomposition of $\Omega^{+}(\rho,\tau)$ into _thin
blocks_ and _solid blocks_. Roughly speaking, the intersection of the level
sets $L_{a}$ of the cosmological time with the thin blocks recovers the
geodesic lamination $\mathcal{L}$ which is obtained as the image of the Gauss
map of the spacelike part of $\partial\tilde{\Omega}^{+}(\rho,\tau)$. The
connected components of $L_{a}\setminus\mathcal{L}$ are the intersections of
$L_{a}$ with the solid blocks. As we only need to use the following lemma, we
don’t need to define further those blocks, and we refer to [BBZ11] for more
details.

######  Lemma 3.22 ([BBZ11, Proposition 6.2]).

Let $\chi:[a,b]\rightarrow\Omega^{+}(\rho,\tau)$ be a spacelike Lipschitz
curve and $\chi^{\prime}$ be its projection to $L_{1}$ along the flow
$\xi^{t}$. Assume that $\chi$ is contained in a single block of the canonical
decomposition. Then

$\exp(-{\rm
len}(\chi^{\prime}))\leq\frac{\textsc{ct}(\chi(a))}{\textsc{ct}(\chi(b))}\leq\exp({\rm
len}(\chi^{\prime}))~{},$ ${\rm len}(\chi)\leq{\rm
len}(\chi^{\prime})\exp({\rm len}(\chi^{\prime}))\textsc{ct}(\chi(a))~{}.$

The proof of the next Lemma is based on the ideas of Barbot–Beguin–Zeghib in
their proof of Theorem 3.5 in [BBZ11].

######  Lemma 3.23.

For every $\varepsilon>0$ and every compact set
$U\subset\mathcal{H}^{1}(\rho)$ there exists a constant $\alpha>0$ such that
for every convex Cauchy surface $\Sigma\subset\Omega^{+}(\rho,\tau)$, $\tau\in
U$, if the systole of $\Sigma$ is greater than $\varepsilon$, then the infimum
of the cosmological time $\textsc{ct}_{\rho,\tau}$ over $\Sigma$ is at least
$\alpha$.

###### Proof.

Due to Lemma 3.19, we may consider only those $\Sigma$, for which the supremum
of the cosmological time is bounded from above by some $\beta>0$. By applying
scaling we can assume that $\beta=1$. Support functions of $\tilde{L}_{1}$
converge uniformly when cocycles converge [BF20, Lemma 4.11]. This implies
convergence of the areas of $L_{1}$-surfaces [BF17, Lemma 2.12]. Hence, since
$U$ is compact, there exists $A>0$ depending only on $U$ such that for every
$\tau\in U$, we have ${\rm area}(L_{1}(\tau))\leq A$, where $L_{1}(\tau)$ is
the $L_{1}$-surface in $\Omega^{+}(\rho,\tau)$. Fix now some $\tau\in U$.

Pick $p\in\Sigma$ first in a thin block of $\Omega^{+}(\rho,\tau)$. The
intersection of this block with $L_{1}$ is a simple geodesic $\psi^{\prime}$.
Let $p^{\prime}$ be the projection of $p$ to $L_{1}$ along the flow $\xi^{t}$.
We apply Lemma 3.20 to $p^{\prime}$, $\psi^{\prime}$ and $\varepsilon/2$ and
get a curve $\chi^{\prime}\subset L_{1}$, consisting of three arcs
$\chi_{1}^{\prime},\chi_{2}^{\prime}$ and $\chi_{3}^{\prime}$ with ${\rm
len}(\chi_{1}^{\prime}),{\rm
len}(\chi_{3}^{\prime})\leq\frac{2A}{\varepsilon}$ and ${\rm
len}(\chi^{\prime}_{2})\leq\frac{\varepsilon}{2}.$

Let $\chi$ be the $\xi^{t}$-projection of $\chi^{\prime}$ to $\Sigma$. We
transfer the decomposition of $\chi^{\prime}$ into the arcs to the
corresponding decomposition of $\chi$. Due to Lemma 3.21, we have ${\rm
len}(\chi_{2})\leq{\rm len}(\chi^{\prime}_{2})\leq\frac{\varepsilon}{2}$.
Thus, as by assumption the systole of $\Sigma$ is greater than $\varepsilon$,
one of the arcs $\chi_{1}$ or $\chi_{3}$ has length
$\geq\frac{\varepsilon}{2}$. Applying Lemma 3.22 to this arc we obtain

$\textsc{ct}(p)\geq\frac{\varepsilon^{2}}{4A\exp(\frac{2A}{\varepsilon})}~{}.$

Now take $p$ in a solid block of $\Omega^{+}(\rho,\tau)\cap\Sigma$, and let
$p^{\prime}$ its $\xi^{t}$-projection onto $L_{1}$. Set $\delta:={\rm
diam}(S,h)$. There exists a point $q^{\prime}$ in a thin block of
$\Omega^{+}(\rho,\tau)$ and a Lipschitz arc $\chi$ of length $\leq\delta$
connecting $p^{\prime}$ and $q^{\prime}$ and belonging to the solid block of
$p^{\prime}$ except the endpoint at $q^{\prime}$. Let $q$ be the projection of
$q^{\prime}$ along $\xi^{t}$. Then Lemma 3.22 gives us a bound

$\textsc{ct}(p)\geq\textsc{ct}(q)\exp(-\delta)~{}.$

Putting it all together, it follows that for every $p\in\Sigma$ we have

$\textsc{ct}(p)\geq\exp(-\delta)\frac{\varepsilon^{2}}{4A\exp(\frac{2A}{\varepsilon})}~{}.$

∎

Finally, we can give the proof of Proposition 3.11.

###### Proof of Proposition 3.11.

Lemma 3.18 shows that $\tau_{i}$ converge to some
$\tau\in\mathcal{H}^{1}(\rho)$. Let $\textsc{ct}_{i}$ be the cosmological time
of $\Omega^{+}(\rho,\tau_{i})$. Suppose that for all $v\in V$ the cosmological
time $\textsc{ct}_{i}(f_{i}(v))$ is unbounded from above. From [Bon05b, Lemma
7.4], the area of the level sets of the cosmological time is an increasing
function of the cosmological time. As the area of convex Cauchy surfaces is
non-decreasing with respect to the inclusion [BF17, Lemma 3.24], the areas of
$\partial{\rm conv}(\mathbf{p}_{i})$ grow to infinity, which is a
contradiction. Hence there is $v\in V$ such that $\textsc{ct}_{i}(f_{i}(v))$
is uniformly bounded from above. Lemma 3.19 implies that the supremum of
$\textsc{ct}_{i}$ over $\partial{\rm conv}(\mathbf{p}_{i})$ is uniformly
bounded from above, particularly for all $v\in V$, $\textsc{ct}_{i}(f_{i}(v))$
is uniformly bounded from above. Lemma 3.23 shows that it is also uniformly
bounded from below, by continuity of the systole, see e.g. [Pro22b, Lemma
2.20] (in the reference hyperbolic cone-metrics are considered instead of flat
ones, but the proof of the statement remains exactly the same). It follows
that up to a subsequence, $\mathbf{p}_{i}$ converge to
$\mathbf{p}\in\mathcal{P}(\rho,V)$. We need to observe that the vertices do
not collapse. For every distinct $v,w\in V$ all segments between $f_{i}(v)$
and $f_{i}(w)$ are spacelike. By applying the reversed triangle inequality for
the Minkowski plane, we see that the intrinsic distance between $f_{i}(v)$ and
$f_{i}(w)$ in $\partial{\rm conv}(\mathbf{p})$ is bounded from above by the
infimum of lengths of these segments. Hence, the vertices do not collapse and
we obtain $\mathbf{p}\in\overline{\mathcal{P}}_{c}(\rho,V)$. Since
$d\in\mathcal{\mathcal{M}}_{-}(S,V)$, we have
$\mathbf{p}\in\mathcal{P}_{c}(\rho,V)$. ∎

##  4\. Proof of Theorem II’

###  4.1. The Teichmüller tangent vector field associated to a flat metric

Recall from Lemma 2.10 that $\mathcal{P}_{c}(S,V)$ is an analytic manifold of
dimension $12\mathbf{g}-12+3n$. Recall from (3) the fibration
$\mu:\mathcal{P}(S,V)\rightarrow T\mathcal{T}(S)$. We will also denote by
$\mu$ its restriction to $\mathcal{P}_{c}(S,V)$, which is also a fibration.
The composition with the natural projection
$p:T\mathcal{T}(S)\rightarrow\mathcal{T}(S)$ gives a fibration
$\pi:\mathcal{P}_{c}(S,V)\rightarrow\mathcal{T}(S)$. We have that
$\pi^{-1}(\rho)=\mathcal{P}_{c}(\rho,V)$ is a manifold of dimension
$6\mathbf{g}-6+3n$. By Theorem 1.4 the restriction of $\mathcal{I}$ to
$\pi^{-1}(\rho)$ is a $C^{1}$ diffeomorphism, and in turn, by Lemma 2.13, the
map $\mathcal{I}:\mathcal{P}_{c}(S,V)\to\mathcal{\mathcal{M}}_{-}(S,V)$ is a
$C^{1}$ submersion.

It also follows that $\mathcal{I}^{-1}(d)$ is a $C^{1}$ submanifold of
$\mathcal{P}_{c}(S,V)$ of dimension $6\mathbf{g}-6$. From Theorem 1.4, the
intersection $\mathcal{I}^{-1}(d)\cap\pi^{-1}(\rho)$ is a point, and is
transverse. In turn, $\pi|_{\mathcal{I}^{-1}(d)}$ is a $C^{1}$ diffeomorphism.

The restriction of $\mu:\mathcal{P}_{c}(S,V)\to T\mathcal{T}(S)$ to
$\mathcal{I}^{-1}(d)$ is a $C^{1}$ injective immersion. Moreover,
$\mathcal{I}^{-1}(d)$ is the image of the $C^{1}$ section
$(\pi|_{\mathcal{I}^{-1}(d)})^{-1}\circ p$, hence $\mu|_{\mathcal{I}^{-1}(d)}$
is proper, and in turn an embedding.

######  Notation 4.1.

We will denote by $\mathbf{p}_{d}(\rho)=\mathcal{I}^{-1}(d)\cap\pi^{-1}(\rho)$
and by $X_{d}(\rho)=\mu(\mathbf{p}_{d}(\rho))$ the resulting tangent vector of
Teichmüller space at the point $\rho$.

We will consider $\mathbf{p}_{d}$ as a map
$\mathcal{T}(S)\rightarrow\mathcal{P}_{c}(S,V)$.

######  Lemma 4.2.

The vector field $X_{d}$ over Teichmüller space is $C^{1}$.

###### Proof.

That is immediate from its definition, as $\mathbf{p}_{d}$ is also $C^{1}$. ∎

We will also need to check a stronger regularity property for
$\mathbf{p}_{d}$.

######  Notation 4.3.

Let $\mathscr{C}$ be a cellulation of $(S,V)$, by
$\mathcal{P}_{c}(S,\mathscr{C})$ denote the space of marked spacetimes
$\mathbf{p}$ such that the face cellulation of ${\rm conv}(\mathbf{p})$ is
exactly $\mathscr{C}$. By
$\overline{\mathcal{P}}_{c}(S,\mathcal{C})\subset\mathcal{P}_{c}(S,V)$ denote
the subspace of marked spacetimes $\mathbf{p}$ such that $\mathcal{C}$
subdivides the face cellulation.

We note that $\mathcal{P}_{c}(S,\mathscr{C})$,
$\overline{\mathcal{P}}_{c}(S,\mathscr{C})$ are subsets of an analytic variety
$\mathcal{A}(\mathscr{C})\subset\mathcal{P}_{c}(S,V)$ determined by the
coplanarity conditions for the quadruples of points belonging to the same
face. These subsets are open and closed respectively with respect to the usual
topology. (Here by an analytic variety we mean the intersection of finitely
many analytic submanifolds. Note that we cannot guarantee that the
intersection is transverse, and hence that $\mathcal{A}(\mathscr{C})$ is an
analytic submanifold.)

######  Lemma 4.4.

Let $\rho_{t}$, $t\in(-\varepsilon,\varepsilon)$, be an analytic curve in
$\mathcal{T}(S)$. Then there exists a cellulation $\mathscr{C}$ of $(S,V)$,
which is a subdivision of the face cellulation $\mathscr{C}_{0}$ of the
boundary of ${\rm conv}(\mathbf{p}_{d}(\rho_{0}))$, and $\varepsilon_{0}$,
$0<\varepsilon_{0}<\varepsilon$, such that for all $t\in(0,\varepsilon_{0}]$
we have $\mathbf{p}_{d}(\rho_{t})\in\mathcal{P}_{c}(S,\mathscr{C})$. Moreover,
$\mathbf{p}_{d}(\rho_{t})$ is analytic over $[0,\varepsilon_{0}]$.

###### Proof.

Denote $\mathbf{p}_{d}(\rho_{t})$ by $\mathbf{p}_{t}$. Let
$\mathscr{T}_{1},\ldots,\mathscr{T}_{r}$ be all the triangulations refining
the boundary cellulation $\mathscr{C}_{0}$ of ${\rm conv}(\mathbf{p}_{0})$ and
let $U$ be a neighborhood of $\mathbf{p}_{0}$ in $\mathcal{P}_{c}(S,V)$. For
every $i=1,\ldots,r$, an element $\mathbf{p}=(\rho,\tau,f)\in U$ and a
triangulation $\mathscr{T}_{i}$ determine a polyhedral surface $\Sigma_{i}$ in
$\Omega^{+}(\rho,\tau)$, which, provided that $U$ is small enough, is a
spacelike Cauchy surface. We also assume that $U$ is small enough so that the
combinatorics of any such surface $\Sigma_{i}$ is a subdivision of
$\mathscr{C}_{0}$. We denote by $\mathcal{P}_{i}(S,\mathscr{C})$ the set of
such triples $(\rho,\tau,f)\in U$ that the resulting $\Sigma_{i}$ has
combinatorics exactly $\mathscr{C}$ (which may be empty). By
$\mathcal{I}_{i}:U\rightarrow\mathcal{\mathcal{M}}_{-}(S,V)$ we denote the map
sending $\mathbf{p}\in U$ to the induced metric on $\Sigma_{i}$ (the induced
metrics are flat with negative singular curvatures provided that $U$ is
sufficiently small). The maps $\mathcal{I}_{i}$ are analytic. It follows from
Claim 2 in the proof of Lemma 2.13 that the differentials of all
$\mathcal{I}_{i}$ at $\mathbf{p}_{0}$ coincide with the differential of
$\mathcal{I}$. In turn, Lemma 3.10 implies that $\mathcal{I}_{i}^{-1}(d)$ are
$(6g-6)$-dimensional analytic submanifolds around $\mathbf{p}_{0}$, transverse
to $\pi$. By $\mathbf{p}_{i,t}$ we denote
$\pi^{-1}(\rho_{t})\cap\mathcal{I}_{i}^{-1}(d)$, which is then an analytic
curve, provided that $\varepsilon$ is sufficiently small. By possibly
decreasing $\varepsilon$ we assume that it is small enough so that
$\mathbf{p}_{i,t}$ and $\mathbf{p}_{t}$ belong to $U$ for all
$t\in(-\varepsilon,\varepsilon)$ and for all $i$.

We first claim that for every $i$ there exists a cellulation $\mathscr{C}_{i}$
of $(S,V)$ (which is a subdivision of $\mathscr{C}_{0}$) and
$\varepsilon_{i}$, $0<\varepsilon_{i}<\varepsilon$, such that for all
$t\in(0,\varepsilon_{i}]$ we have
$\mathbf{p}_{i,t}\in\mathcal{P}_{i}(S,\mathscr{C}_{i})$. (We remark that a
cellulation $\mathscr{C}_{i}$ satisfying this condition is clearly unique.)
Indeed, fix $i$, then because of the choice of $U$ there exists
$\mathscr{C}_{i}^{\prime}$ refining $\mathscr{C}_{0}$ and a sequence of
positive numbers $t_{j}^{\prime}$ converging to zero such that
$\mathbf{p}_{i,t_{j}^{\prime}}\in\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})\subset\mathcal{A}(\mathscr{C}_{i}^{\prime})$.
Since $\mathbf{p}_{i,t}$ is an analytic curve, it belongs to
$\mathcal{A}(\mathscr{C}_{i}^{\prime})$. The set
$\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})$ is an open subset of
$\mathcal{A}(\mathscr{C}_{i}^{\prime})$, and $\mathbf{p}_{i,0}=\mathbf{p}_{0}$
belongs to the closure of $\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})$. If
$\mathbf{p}_{i,t}$ does not belong to
$\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})$ for all small enough positive
$t$, then for every $t_{j}$ consider the maximal open segment in
$[0,\varepsilon]$ containing $t_{j}^{\prime}$ such that $\mathbf{p}_{i,t}$ is
in $\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})$ over this segment (because
$\mathcal{P}_{i}(S,\mathscr{C}_{i}^{\prime})$ is open in
$\mathcal{A}(\mathscr{C}_{i}^{\prime})$, such a segment exists). Let
$t_{j}^{\prime\prime}$ be the leftmost endpoint of this segment. Then
$t_{j}^{\prime\prime}$ is a non-increasing sequence converging to zero and up
to passing to a subsequence there exists a cellulation
$\mathscr{C}^{\prime\prime}_{i}$ such that
$\mathbf{p}_{i,t_{j}^{\prime\prime}}\in\mathcal{P}_{i}(S,\mathscr{C}^{\prime\prime}_{i})$.
Moreover, by construction of $\\{t^{\prime\prime}_{j}\\}$, the cellulation
$\mathscr{C}_{i}^{\prime}$ is a strict subdivision of the cellulation
$\mathscr{C}^{\prime\prime}_{i}$. Then the curve $\mathbf{p}_{i,t}$ belongs to
the analytic variety $\mathcal{A}(\mathscr{C}^{\prime\prime}_{i})$. If
$\mathbf{p}_{i,t}$ does not belong to
$\mathcal{P}_{i}(S,\mathscr{C}^{\prime\prime}_{i})$ for all small enough
positive $t$, we can continue this process with a strictly larger cellulation.
Since we cannot pass to a strictly larger cellulation infinitely many times,
eventually we stop. Thus, we get a cellulation $\mathscr{C}_{i}$ such that
$\mathbf{p}_{i,t}$ belongs to $\mathcal{P}_{i}(S,\mathscr{C}_{i})$ for all
small enough positive $t$. (We remark that by uniqueness our resulting
cellulation $\mathscr{C}_{i}$ actually coincides with our starting cellulation
$\mathscr{C}_{i}^{\prime}$.)

We note now that for each $t$ there is $i$ such that
$\mathbf{p}_{t}=\mathbf{p}_{i,t}$ by Lemma 2.12. Since any two analytic curves
either coincide, or intersect in a discrete set of points, we see that
$[-\varepsilon/2,\varepsilon/2]$ can be decomposed into finitely many closed
segments, over which $\mathbf{p}_{t}=\mathbf{p}_{i,t}$ for some fixed $i$ for
this segment. Thereby, the second statement of the lemma holds, and the first
statement for the curve $\mathbf{p}_{t}$ follows from the statement above for
the curves $\mathbf{p}_{i,t}$. ∎

###  4.2. The total length functional

####  4.2.1. Definition

We saw in the preceding section how to associate to any flat metric with
negative singular curvatures $d$ on $S$ a vector field $X_{d}$ over
$\mathcal{T}(S)$. We now associate to $d$ a real-valued function over
$\mathcal{T}(S)$, the _total length function_ , which can be defined in two
ways.

First, for $\rho\in\mathcal{T}(S)$, if $\mathscr{C}$ is the face cellulation
of ${\rm conv}(\mathbf{p}_{d}(\rho))$, with
$\mathbf{p}_{d}(\rho)=\mathcal{I}^{-1}(d)\cap\pi^{-1}(\rho)$,

$\operatorname{L}_{d}(\rho)=\sum_{e\in E(\mathscr{C})}\theta_{e}l_{e}$

where $\theta_{e}$ is the dihedral angle at the edge $e$ of ${\rm
conv}(\mathbf{p}_{d}(\rho))$, and $l_{e}$ is its $d$-length.

On the other hand, for $\rho\in\mathcal{T}(S)$, the Gauss image of
$\partial{\rm conv}(\mathbf{p}_{d}(\rho))$ defines a balanced convex
cellulation $(\mathscr{G},w)$ over $(S,h)$, where $h$ is the hyperbolic metric
on $S$ with holonomy $\rho$ and the weights are the dihedral angles, and we
have

$\operatorname{L}_{d}(\rho)=\sum_{e\in E(\mathscr{G})}w_{e}\ell_{e}$

where $\ell_{e}$ is the $h$-length of the edge. In fact, if $e^{*}$ is dual to
$e$, $l_{e}=w_{e^{*}}$ and $\theta_{e}=\ell_{e^{*}}$.

We first prove in Section 4.2.2 that $\operatorname{L}_{d}$ is $C^{1}$. This
result is based on the Schläfli Formula. In Section 4.2.3 we show that
$\operatorname{L}_{d}$ is proper. After preliminary work in Section 4.3, in
Section 4.4 we compute the first derivatives of $\operatorname{L}_{d}$. By
investigating its second derivatives, in Section 4.5 we show that it is
strictly convex.

####  4.2.2. Smoothness

Before discussing our functional, we have to make a short sidestep to discuss
a useful tool, the Schläfli formula in the Minkowski space. We will need it
only in a very restricted case: for deformations of tetrahedra close to
spacelike degenerate ones. Thereby, we will make our discussion as short as
possible, and we refer to [Sou04, RS00] for a more advanced exposition.

Let $T\subset\mathbb{R}^{2,1}$ be a tetrahedron with all spacelike faces.
Particularly, all its edges are spacelike, but are divided into two types, if
the exterior normals to the adjacent faces belong (1) to the same component of
the unit pseudo-sphere, or (2) to the opposite components. By a (non-oriented)
_dihedral angle_ $\theta_{e}$ of an edge $e$ we mean (1) the non-oriented
Lorentzian angle between the exterior normals to the adjacent faces, or (2)
the non-oriented Lorentzian angle between the exterior normal of one face and
the opposite to the exterior normal of the other face, depending respectively
if $e$ is of type (1) or (2).

Suppose that the vertices of $T$ undergo an infinitesimal deformations,
inducing infinitesimal deformations $\dot{\theta}_{e}$ on all the dihedral
angles. Then the Schläfli formula is the following result:

$\sum_{e\in E(T)}\dot{\theta}_{e}l_{e}=0~{}.$ (17)

This formula is proven in a much greater generality in [Sou04, Theorem 2].
Note, however, that Souam excludes from the consideration the degenerate case,
which is what we actually need: when two adjacent faces of an oriented
polyhedron have coinciding normals. However, the formula easily extends to our
case by continuity. Souam makes his exclusion due to the impossibility to
define coherently his _signed dihedral angle_ in this situation. However, when
the branches of values of dihedral angles are given to every edge during a
deformation, the variational formula anyway holds. We note that other proofs
can be extracted from the articles [Ale03], [RS00], though the desired result
is also not explicitly formulated there.

Let $\Sigma$ be a spacelike polyhedral Cauchy surface in
$\Omega^{+}(\rho,\tau)$, not necessarily convex. For an edge $e$ of $\Sigma$,
its _oriented dihedral angle_ is the non-oriented Lorentzian angle between the
future normals of the two adjacent sides, taken with plus sign if the edge is
future-convex, and with minus sign otherwise. Then we define _the total mean
curvature_ of $\Sigma$ as

$M(\Sigma)=\sum_{e\in E(\Sigma)}\theta_{e}l_{e}~{},$

where the sum is over all edges of $\Sigma$. We now use the Schläffli formula
to show

######  Lemma 4.5.

$\operatorname{L}_{d}$ is $C^{1}$.

###### Proof.

Pick $\mathbf{p}=(\rho,\tau,f)\in\mathcal{P}_{c}(S,V)$, let $U$ be its
neighborhood and $\mathscr{C}$ be its face cellulation. Let
$\mathscr{T}_{1},\ldots,\mathscr{T}_{r}$ be all subtriangulations of
$\mathscr{C}$. For every $i=1,\ldots,r$, an element
$\mathbf{p}^{\prime}=(\rho^{\prime},\tau^{\prime},f^{\prime})\in U$ and a
triangulation $\mathscr{T}_{i}$ determine a polyhedral surface $\Sigma_{i}$ in
$\Omega^{+}(\rho^{\prime},\tau^{\prime})$, which, provided that $U$ is small
enough, are spacelike Cauchy surfaces. Define $M_{i}:=M(\Sigma_{i})$, which is
then an analytic function over $U$. For
$\mathbf{p}^{\prime}\in(U\cap\overline{\mathcal{P}}_{c}(S,\mathscr{T}_{i}))$
the surface $\partial{\rm conv}(\mathbf{p}^{\prime})$ coincides with
$\Sigma_{i}$. Thus, over
$\overline{\mathcal{P}}_{c}(S,\mathscr{T}_{i})\cap\mathcal{I}^{-1}(d)$, the
function $\operatorname{L}_{d}\circ\pi$ coincides with $M_{i}$. Hence, it is
enough to show that at the initial configuration $\mathbf{p}$ the
differentials of $M_{i}$ coincide.

We assume that $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$ differ by a flip of one
diagonal in a quadrilateral $Q$, and will show the coincidence of the
differentials of $M_{1}$ and $M_{2}$. Since any two triangulations under our
consideration can be connected by a sequence of such flips, this will finish
the proof.

Consider the quadrilateral $Q$ as a degenerate spacelike tetrahedron $T$. Let
$\dot{\mathbf{p}}$ be a tangent vector at $\mathbf{p}$ at the initial
position. By means of local development of $T$ in $\mathbb{R}^{2,1}$,
$\dot{\mathbf{p}}$ induces a variation on the dihedral edges of $T$. By (17),
we have

$\sum_{e\in E(T)}\dot{\theta}_{e}l_{e}=0~{}.$

On the other hand, as shown in Claim 2 in the proof of Lemma 2.13, the
differentials of all edge-lengths for two surfaces coincide in the initial
position. It is easy to see then, that

$\dot{M}_{1}-\dot{M}_{2}=\pm\sum_{e\in E(T)}\dot{\theta}_{e}l_{e}=0~{}.$

This finishes the proof. ∎

####  4.2.3. Properness

We prove properness of $\operatorname{L}_{d}$ in Lemma 4.10.

######  Lemma 4.6.

For every $\varepsilon,c_{1},c_{2}>0$ there exists $c_{3}>0$ such that if a
Euclidean triangle $T$ has ${\rm area}(T)\leq c_{1}$, all side-lengths $\geq
c_{2}$ and one side-length $\geq c_{3}$, then it has an angle
$\geq\pi-\varepsilon$.

###### Proof.

Let $T$ be a triangle that satisfy our conditions, with side-lengths
$l_{1}\leq l_{2}\leq l_{3}$ and with respective angles $\alpha_{1}$,
$\alpha_{2}$ and $\alpha_{3}$. We have

$\sin(\alpha_{2})=\frac{2{\rm
area}(T)}{l_{1}l_{3}}\leq\frac{2c_{1}}{c_{2}c_{3}}~{}.$

Since $\alpha_{1}\leq\alpha_{2}$ by the Sine Law, we see that for sufficiently
large $c_{3}$ we get $\alpha_{1}+\alpha_{2}\leq\varepsilon$, as desired. ∎

######  Corollary 4.7.

For every $\varepsilon>0$ there exists $c=c(\varepsilon,d)>0$ such that if a
geodesic triangulation $\mathscr{T}$ of $(S,V,d)$ has an edge of length $\geq
c$, then it has an angle $\geq\pi-\varepsilon$.

######  Lemma 4.8.

For every $c_{1},c_{2}>0$ there exists $\varepsilon$, $0<\varepsilon<\pi$,
such that if a hyperbolic polygon $P$ has ${\rm area}(P)\geq c_{1}$ and ${\rm
diam}(P)\leq c_{2}$, then all its angles are $\geq\varepsilon$.

###### Proof.

Note that $P$ belongs to a circular sector of radius $c_{2}$ centered at any
its vertex, with sides of the sector containing the sides of the polygon
adjacent to the vertex. If an angle of the vertex goes to zero, then the area
of such sector monotonically decreases to zero. This implies the desired
conclusion. ∎

We now introduce few more notions. Let $d\in\mathcal{\mathcal{M}}_{-}(S,V)$.
We will say that a geodesic cellulation $\mathscr{C}$ of $(S,V,d)$ is _strict_
if all the faces are (isometric to) strictly convex Euclidean polygons. By
strictly convex we mean that the angle of each polygon at every point of $V$
is strictly less than $\pi$. Let
$\theta:E(\mathscr{C})\rightarrow\mathbb{R}_{\geq 0}$ be a weight function. We
will say that $(\mathscr{C},\theta)$ is _$H$ -admissible_, if $\mathscr{C}$ is
strict, all weights are positive, and for every vertex there exists a
hyperbolic polygon with edges determined by $\theta$ and with angles
complementary to the respective face angles of $\mathscr{C}$. This polygon is
called the _dual polygon_ to $v\in V$. We will say that $(\mathscr{C},\theta)$
is _weakly $H$-admissible_ if it is a subdivision of a $H$-admissible
$(\mathscr{C}^{\prime},\theta^{\prime})$ (without adding new vertices and with
new edges having zero weight). The _total length_ of the weight function is
$\sum_{e\in E(\mathcal{C})}\theta_{e}l_{e}$, where $l_{e}$ is the length of
$e$ in $d$.

By the dualization construction, every $H$-admissible weighted geodesic
cellulation determines a hyperbolic metric $h$. When $\mathscr{C}$ is given,
we will be calling a function $\theta$ (weakly) $H$-admissible, if it makes a
(weakly) $H$-admissible pair with $\mathscr{C}$.

######  Lemma 4.9.

For every $c>0$ there are finitely many triangulations $\mathscr{T}$ of
$(S,V,d)$ supporting a weakly $H$-admissible weight function of total length
$\leq c$.

###### Proof.

Consider a triangulation $\mathscr{T}$ with a weakly $H$-admissible weight
function $\theta$ such that $L(\mathscr{T},\theta)\leq c$, where
$L(\mathscr{T},\theta)$ is the total length. Let $c_{1}$ be the minimal
distance between two points of $V$ in $d$. Then for every $e\in
E(\mathscr{T})$ we have $\theta_{e}\leq c/c_{1}$. Because the number of edges
of $\mathscr{T}$ is fixed, the degree of each vertex is bounded from above,
hence for every $v\in V$ the diameter of the dual polygon is bounded from
above. Note that its area is minus the curvature of $v$. It follows from Lemma
4.8 that there exists $\varepsilon$, $0<\varepsilon<\pi$, such that all angles
in $\mathscr{T}$ are $\leq\pi-\varepsilon$. Corollary 4.7 implies that then
all the lengths of edges of $\mathscr{T}$ are $\leq c_{2}$ for some $c_{2}>0$.
Since the lengths of geodesics between points of $V$ form a discrete set,
there are finitely many such triangulations. ∎

######  Lemma 4.10.

$\operatorname{L}_{d}$ is proper.

###### Proof.

Let $\rho_{i}\in\mathcal{T}(S)$ be a sequence such that
$\operatorname{L}_{d}(\rho_{i})$ converges to $c\in\mathbb{R}_{>0}$. Due to
Lemma 4.9, up to a subsequence, we can assume that all the corresponding
cellulations $\mathscr{C}_{i}$ admit a subdivision to the same triangulation
$\mathscr{T}$. Up to passing to a subsequence, all weights on $\mathscr{T}$
converge to a weight function $\theta:E(\mathscr{T})\rightarrow[0,c]$. We need
to check that $\theta$ is weakly $H$-admissible. Note that the space of
hyperbolic polygons with marked sides and fixed angles is closed (we allow the
degenerations of edges to zero, but the result is still a convex hyperbolic
polygon as its area is fixed). Hence, by continuity, for every $v\in V$, the
weights on the edges incident to $v$ determine a dual hyperbolic polygon. It
remains only to check that the decomposition $\mathscr{C}$ obtained from
$\mathscr{T}$ by deleting all edges of zero length, is a strict cellulation
(particularly, that all faces are simply connected). Since all the dual
polygons are convex hyperbolic polygons, there are at least three edges in
$\mathscr{C}$ from every vertex, and all face angles in $\mathscr{C}$ are
$<\pi$. Suppose that a face of $\mathscr{C}$ is not simply connected. That
would mean that there is a non-simply connected compact flat surface with
strictly convex boundary, which cannot happen due to the Gauss–Bonnet formula.
Hence, $\mathscr{C}$ is a strict cellulation and $(\mathscr{C},\omega)$
determines $\rho\in\mathcal{T}(S)$, which is the limit of $\rho_{i}$. ∎

###  4.3. Pairing of Codazzi tensors and balanced convex cellulations

The aim of the present Section is to prove Formula (24), which will be used to
relate the differential of the total length function to the Weil–Petersson
metric.

####  4.3.1. Background on Codazzi tensors

Let $b$ be a (smooth) symmetric $(0,2)$ tensor over the hyperbolic surface
$(S,h)$. Recall that if $\nabla$ is the Levi-Civita connection of $h$, then

$\nabla_{X}b(Y,Z)=X.b(Y,Z)-b(\nabla_{X}Y,Z)-b(\nabla_{X}Z,Y)~{}.$ (18)

Since $b$ is symmetric, we have

$\nabla_{X}b(Y,Z)=\nabla_{X}b(Z,Y)~{}.$

We will say that a symmetric $(0,2)$ tensor $b$ is _Codazzi_ if

$\nabla_{X}b(Y,Z)=\nabla_{Y}b(X,Z)~{}.$

Recall that the divergence $\operatorname{div}b$ of $b$ is the $1$-form
defined as

$\operatorname{div}(b)(X)=\operatorname{tr}_{h}(\nabla_{.}b(X,\cdot))~{}.$

######  Lemma 4.11.

If $b$ is a symmetric traceless $(0,2)$ tensor over $(S,h)$, then Codazzi is
equivalent to divergence-free.

###### Proof.

The almost-complex structure $J$ defined by $h$ in conformal coordinates at a
point is given by $\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$. So it is
immediate to check that $bJ$ and $Jb$ are symmetric and traceless and that

$bJ=-Jb~{}.$ (19)

In particular,

$b(X,JY)=(bJ)(X,Y)=(bJ)(Y,X)=b(Y,JX)~{},$

and also

$b(JX,JY)=b(Y,JJX)=-b(X,Y)~{}.$ (20)

It is easy to see that $J$ is parallel for $\nabla$, for example by expressing
$\nabla$ in conformal coordinates. In turn, $J$ and $\nabla$ commutes:

$0=(\nabla_{X}J)(Y)=\nabla_{X}(JY)-J\nabla_{X}Y~{}.$

From the equations above and from (18) we easily check that

$\nabla_{X}(bJ)(Y,Z)=\nabla_{X}b(Y,JZ)~{}.$

We first prove that if $b$ is Codazzi, then $bJ$ divergence-free. Indeed, if
$e_{i}$ is an orthonormal frame for $h$, we compute for example

$\operatorname{div}(bJ)(e_{1})=\nabla_{e_{1}}b(e_{1},e_{2})-\nabla_{e_{2}}b(e_{1},e_{1})$

which is equal to zero because $b$ is symmetric and Codazzi. The same holds
for $e_{2}$.

We also have that if $b$ Codazzi, then $bJ$ is Codazzi. Indeed,

$\nabla_{X}(bJ)(Y,Z)=\nabla_{X}b(Y,JZ)=\nabla_{Y}b(X,JZ)=\nabla_{Y}(bJ)(X,Z)~{}.$

Now, if $b$ is Codazzi, $bJ$ is Codazzi, hence $bJJ=-b$ is divergence free.

In the same way, it is immediate to check that if $b$ is divergence free, then
$bJ$ is divergence free and Codazzi. ∎

For future reference let us note the following consequence of the Divergence
Theorem.

######  Lemma 4.12.

Let $f,\varphi$ be smooth functions on a hyperbolic surface with boundary
$(M,h)$ and suppose that $b$ is symmetric and divergence free. Then

$\int_{M}\mbox{\rm tr}_{h}(\varphi b\nabla^{2}f)=-\int_{M}b(\nabla
f,\nabla\varphi)+\int_{\partial M}\varphi b(\nabla f,N)$

where $N$ is the unit outward normal of $\partial M$.

The integrations are implicitly done with respect to the volumes forms of $M$
and $\partial M$ respectively. Also, for a $(0,2)$ symmetric tensor $b$, we
will denote by $b^{\sharp}$ the self-adjoint $(1,1)$ tensor such that
$b(X,Y)=h(b^{\sharp}(X),Y)$, and hence for two such tensors $b_{1}$ and
$b_{2}$, $\mbox{\rm tr}_{h}(b_{1}b_{2})$ means $\mbox{\rm
tr}(b_{1}^{\sharp}b_{2}^{\sharp})$.

###### Proof.

Recalling that $\nabla_{X}b^{\sharp}=\left(\nabla_{X}b\right)^{\sharp}$ (see
e.g. [GHL04, 2.66]), we have

$\operatorname{div}(b)(X)=\operatorname{tr}(\nabla_{.}b^{\sharp}(X))=\operatorname{tr}(\nabla_{.}(b^{\sharp}(X)))-\operatorname{tr}(b^{\sharp}(\nabla_{.}X))=\operatorname{div}(b^{\sharp}(X))-\operatorname{tr}(b^{\sharp}(\nabla_{.}X))$

so that

$\operatorname{div}(b)(\nabla f)=\operatorname{div}(b^{\sharp}(\nabla
f))-\operatorname{tr}(b^{\sharp}(\nabla_{.}\nabla
f))=\operatorname{div}(b^{\sharp}(\nabla
f))-\operatorname{tr}(b^{\sharp}(\nabla^{2}f)^{\sharp})=\operatorname{div}(b^{\sharp}(\nabla
f))-\operatorname{tr}_{h}(b\nabla^{2}f)$

hence if $b$ is divergence free, multiplying by $\varphi$ and reordering

$\operatorname{tr}_{h}(\varphi
b\nabla^{2}f)=\varphi\operatorname{div}(b^{\sharp}(\nabla
f))=\operatorname{div}(\varphi b^{\sharp}(\nabla f))-b^{\sharp}(\nabla
f).\varphi~{}.$

Integrating and using the Divergence Theorem leads to the result. ∎

####  4.3.2. Traceless Codazzi tensor associated to a balanced convex
cellulation

Let us consider $\mathbf{p}_{d}(h)\in\mathcal{P}_{c}(S,V)$, and let
$s_{\mathbf{p}}$ be the support function of $P=\widetilde{\rm
conv}(\mathbf{p}_{d}(h))$. We already used support functions in the proof of
Lemma 2.4, but here we assume that the domain of a support function is the
hyperboloid $\mathbb{H}^{2}$. In more details, for $x\in\mathbb{H}^{2}$,

$s_{\mathbf{p}}(x):=\sup\\{\langle p,x\rangle:p\in P\\}~{}.$

It is known to be well-defined over $\mathbb{H}^{2}$, see [Bon05b]. For more
details on support functions in this setting we refer to [FV16, BF17]. The
function $s_{\mathbf{p}}$ in general is not $\rho(\pi_{1}S)$ invariant, but as
for $x\in\mathbb{H}^{2}$ there exists a point $p$ on $P$ such that
$s_{\mathbf{p}}(x)=\langle p,x\rangle$, then
$s_{\mathbf{p}}(\rho(\gamma)(x))=\langle\rho_{\tau}(\gamma)(p),\rho(\gamma)(x)\rangle$,
hence we have

$s_{\mathbf{p}}(\rho(\gamma)(x))=s_{\mathbf{p}}(x)+\langle\rho^{-1}(\gamma)(\tau(\gamma)),x\rangle~{}.$
(21)

If $(\mathscr{G},w)$ is the balanced geodesic cellulation of $(S,h)$ defined
by $\mathbf{p}_{d}(h)\in\mathcal{P}_{c}(S,V)$, over any face $F$ of the lift
of $\mathscr{G}$ on $\mathbb{H}^{2}$, $s_{\mathbf{p}}$ is the restriction of
the linear map $x\mapsto\langle x,v\rangle$, where $v$ is the vertex of $P$
sent by the Gauss map to $F$.

######  Proposition 4.13.

There exists a unique smooth function $s_{d}$ on $\mathbb{H}^{2}$ such that it
satisfies (21) and $\nabla^{2}s_{d}-s_{d}\tilde{h}$ is a traceless symmetric
Codazzi tensor which is $\rho$-invariant, where $\tilde{h}$ is the metric over
$\mathbb{H}^{2}$.

We denote by $b_{d}$ the projection of this tensor onto the hyperbolic surface
$(S,h)$. We refer to [FV16, BS16, BF17, OS83] for details.

######  Remark 4.14.

To simplify computations, it is worth to note (see e.g. [FV16]) that if $F$ is
the one-homogeneous extension to the future cone of the origin of a smooth
function $f$ over the hyperboloid, then at $x\in\mathbb{H}^{2}$,

$\operatorname{grad}F(x)=\nabla f(x)-f(x)x~{};$ (22)
$\operatorname{Hess}F(x)=(\nabla^{2}-\tilde{h})(x)f~{}.$ (23)

######  Lemma 4.15.

Let $b$ be a traceless divergence-free tensor on $(S,h)$. Let $\\{O_{i},i\in
I\\}$ be a finite covering of $S$ by discs and let $s$ be a $C^{2}$ function
on $\mathbb{H}^{2}$ satisfying (21). For any $i$, let us denote by $s_{i}$ the
function over $S$ with support in $O_{i}$ defined by the restriction of $s$ to
an arbitrary lift of $O_{i}$. Let $\\{\varphi_{i},i\in I\\}$ be a partition of
unity subordinated to $\\{(O_{i})_{i\in I}\\}$. Then

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=-\sum_{i}\int_{S}b(\nabla
s_{i},\nabla\varphi_{i})~{}.$

Note that by (21) two different choices of lifts of $O_{i}$ will result by
adding to $s_{i}$ the restriction to $\mathbb{H}^{2}$ of a linear map.

###### Proof.

By (21) the function $s-s_{d}$ is $\rho$-invariant, hence it defines a
function $f$ over $S$. Let $s_{0,i}=s_{i}-f$. Then

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=\sum_{i\in I}\int_{S}\varphi_{i}\mbox{\rm
tr}_{h}(bb_{d})=\sum_{i}\int_{S}\mbox{\rm
tr}_{h}(\varphi_{i}b(\nabla^{2}-h)s_{0,i})$

and as $b$ is traceless we obtain

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=\sum_{i}\int_{S}\mbox{\rm
tr}_{h}(\varphi_{i}b\nabla^{2}s_{0,i})~{}.$

Lemma 4.12 gives (because the support of $\varphi_{i}$ is in $O_{i}$, there is
no boundary term)

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=-\sum_{i}\int_{S}b(\nabla
s_{0,i},\nabla\varphi_{i})~{},$

and finally

$\sum_{i}\int_{S}b(\nabla s_{0,i},\nabla\varphi_{i})=\sum_{i}\int_{S}b(\nabla
s_{i},\nabla\varphi_{i})$

because

$\sum_{i}\int_{S}b(\nabla f,\nabla\varphi_{i})=\int_{S}b(\nabla f,\nabla
1)=0~{}.$

∎

######  Lemma 4.16.

Let $b$ be a traceless symmetric Codazzi tensor on $(S,h)$, and let
$(\mathscr{G},w)$ be the balanced convex cellulation of $(S,h)$ given by
$\mathbf{p}_{d}(h)$. Then

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=-\sum_{e\in
E(\mathscr{G})}w_{e}\int_{e}b(U_{e},U_{e})~{},$ (24)

where $U_{e}$ is a unit tangent vector of $e$.

###### Proof.

In the proof we will denote $s_{\mathbf{p}}$ by $s$. Let $\\{\varphi_{i},i\in
I\\}$ be a partition of unity subordinated to an open cover $\\{O_{i},i\in
I\\}$ of $S$ such that

  * •

on each $O_{i}$ we have well-defined maps $s_{i}$ and $s_{0,i}$,

  * •

$I=I_{f}\sqcup I_{e}\sqcup I_{v}$ where if $i\in I_{f}$, $O_{i}$ is contained
in a face of $\mathscr{G}$, if $i\in I_{e}$, $O_{i}$ meets exactly one edge of
$\mathscr{G}$, if $i\in I_{v}$, $O_{i}$ meets exactly one vertex of
$\mathscr{G}$ and no edges but the edges emanating from this vertex.

Lemma 4.15 was proved for $C^{2}$ functions satisfying (21). However, as $s$
is a support function, it can be approximated by $C^{2}$ support functions
satisfying (21) [BF17, Appendix A], and on $S$ minus the edges and vertices,
the gradients also converge by convexity. Actually the proof of Lemma 4.15
could be directly implemented in the present proof without using any
approximation, but we have preferred to state a separate statement.

By Lemma 4.11 $b$ is divergence-free. Denoting $A_{i}=\int_{S}b(\nabla
s_{i},\nabla\varphi_{i})$, from Lemma 4.15

$\int_{S}\mbox{\rm tr}_{h}(bb_{d})=-\sum_{i}A_{i}~{}.$ (25)

Let us compute $A_{i}$ with respect to the different sets of indices $i$.

  1. (1)

Let $i\in I_{f}$. Then on $O_{i}$ the function $s_{i}$ is $C^{\infty}$, so by
Lemma 4.12

$A_{i}=-\int_{O_{i}}\mbox{\rm tr}_{h}(\varphi_{i}b\nabla^{2}s_{i})$

(there is no boundary term as the support of $\varphi_{i}$ is included in
$O_{i}$). As $b$ is traceless,

$A_{i}=-\int_{O_{i}}\mbox{\rm tr}_{h}(\varphi_{i}b(\nabla^{2}s_{i}-s_{i}h))$

and as $s_{i}$ is the restriction of a linear map, by (23)
$\nabla^{2}s_{i}-s_{i}h=0$. Hence $A_{i}=0$.

  2. (2)

Let $i\in I_{e}$. We have an edge $e$ which separates $O_{i}$:

$O_{i}=O_{i}^{-}\sqcup O_{i}^{+}\sqcup(O_{i}\cap e)$

and let $N_{i}^{-}=-N_{i}^{+}$ be the outward unit normal of $\partial
O_{i}^{\pm}$ (the choice of $+$ or $-$ being arbitrary). Let $s_{i}^{\pm}$ be
the extensions of $s_{i}|_{O_{i}^{\pm}}$ to a neighborhood of $O_{i}^{\pm}$,
coming from the restriction of linear maps. By Lemma 4.12

$\begin{split}\ A_{i}=&-\int_{O^{+}_{i}}\mbox{\rm
tr}_{h}(\varphi_{i}b(\nabla^{2}s_{i}^{+}))-\int_{O^{-}_{i}}\mbox{\rm
tr}_{h}(\varphi_{i}b(\nabla^{2}s_{i}^{-}))\\\ &+\int_{\partial
O_{i}^{+}}\varphi_{i}b(\nabla s_{i}^{+},N_{i}^{+})+\int_{\partial
O_{i}^{-}}\varphi_{i}b(\nabla s_{i}^{-},N_{i}^{-})~{}.\end{split}$

By the same reason as in the $i\in I_{f}$ case, $\int_{O^{\pm}_{i}}\mbox{\rm
tr}_{h}(\varphi_{i}b(\nabla^{2}s_{i}^{\pm}))=0$, so

$A_{i}=\int_{\partial O_{i}^{+}}\varphi_{i}b(\nabla
s_{i}^{+},N_{i}^{+})+\int_{\partial O_{i}^{-}}\varphi_{i}b(\nabla
s_{i}^{-},N_{i}^{-})~{}.$

Now $\varphi_{i}$ has compact support in $O_{i}$, hence $\partial O_{i}^{\pm}$
reduces to $e\cap O_{i}$. It will be convenient to write them as integrals on
$e$:

$A_{i}=\int_{e}\varphi_{i}b(\nabla
s_{i}^{+},N_{i}^{+})+\int_{e}\varphi_{i}b(\nabla s_{i}^{-},N_{i}^{-})~{},$

and clearly on $e\cap O_{i}$, $N_{i}^{-}=-N_{i}^{+}$. So

$A_{i}=\int_{e}\varphi_{i}b(\nabla s_{i}^{+}-\nabla s_{i}^{-},N_{i}^{+})~{}.$

It is easy to compute that over $e$,

$\nabla s_{i}^{+}-\nabla s_{i}^{-}=-w_{e}N_{i}^{+}~{}.$ (26)

Indeed, there are vertices $v_{\pm}$ of $\widetilde{\rm
conv}(\mathbf{p}_{d}(h))$ on a same edge $e^{*}$, whose Gauss image is a lift
of $e$, such that $s_{i}(x)=\langle x,v_{\pm}\rangle$ (here we abuse notation
identifying $x$ as a point in $O_{i}$ and as a point in $\mathbb{H}^{2}$). By
definition, $w_{e}$ is the length of $e$, and $v_{+}-v_{-}=w_{e}N_{i}^{-}$
(here we also abuse the notation and consider $N_{i}^{-}$ as the unit vector
in $\mathbb{R}^{2,1}$ normal to the plane defining the edge $e$). From (22),
and because for all $x\in e$, $\langle x,N_{i}^{+}\rangle=0$, we obtain (26).
So we are left with

$A_{i}=-w_{e}\int_{e}\varphi_{i}b(N_{i}^{+},N_{i}^{+})~{}.$

  3. (3)

Let $i\in I_{v}$. Around a vertex $v$, number the emanating edges in a
clockwork fashion $e_{1},\ldots,e_{n}$, and also in a cyclic manner,
$e_{n+1}=e_{1}$ etc. Let us denote by $O_{i,k}$ the part of $O_{i}$ contained
between the edges $e_{k},e_{k+1}$. Now we can proceed exactly as in point 2.
The only difference is now that the boundary of $O_{i,k}$ on which
$\varphi_{i}$ is not zero is the union of two geodesic segments, so we need to
use the version of the Divergence Theorem for manifolds with corners. But for
such manifolds, only the codimension one boundary component enters Stokes
theorem, so we may apply the same formula, see e.g. [Lee13]. In turn,

$A_{i}=-\sum_{k=1}^{n}w_{e_{k}}\int_{e_{k}}\varphi_{k}b(N_{i}^{+},N_{i}^{+})~{}.$

Returning to (25), to compute $\int_{S}\mbox{\rm tr}_{h}(bb_{d})$ we need to
sum all the $A_{i}$. This will lead to the result by definition of a partition
of unity. Indeed, let $e$ an edge of the cellulation. Suppose that for example
$e=(e\cap O_{1})\cup(e\cap O_{2})\cup(e\cap O_{3})$ i.e. $O_{1}$ and $O_{2}$
will each contain a (different) vertex of the cellulation, while $O_{2}$
don’t. So

$\int_{e}\varphi_{1}b(N_{e},N_{e})+\int_{e}\varphi_{2}b(N_{e},N_{e})+\int_{e}\varphi_{3}b(N_{e},N_{e})=\int_{e}b(N_{e},N_{e}),$

where $N_{e}$ is any unit normal to $e$.

Hence from (25) we have $\mbox{\rm
tr}_{h}(bb_{d})=\sum_{e}w_{e}\int_{e}b(N_{e},N_{e})$ and the result follows
because as $U_{e}=JN_{e}$, from (20) we get $b(N_{e},N_{e})=-b(U_{e},U_{e})$.
∎

####  4.3.3. Weil–Petersson metric

From now on we consider $\mathcal{T}(S)$ as a subspace of the space of
hyperbolic metrics on $S$, transverse to the action by the group of isotopies.
For this point of view, the tangent space of $\mathcal{T}(S)$ at $h$ is the
space of symmetric traceless Codazzi $(0,2)$ tensors over $(S,h)$, see e.g.
[Tro92, Yam14, BS16] (note Lemma 4.11). In the following we will consider
implicitly paths $h(t)$ such that $h^{\prime}(0)$ is a non-zero symmetric
traceless Codazzi tensor.

By a folklore result, known as _Hopf Lemma_ (see e.g. [KT21, Lemma 3.1]), $b$
is a symmetric traceless Codazzi tensor if and only if

$Q(X)=b(X,X)+iJb(X,X)~{}$ (27)

is a holomorphic quadratic differential over the Riemann surface underlying
$(S,h)$.

In a local chart a holomorphic quadratic differential is expressed as
$Q=q(z)dz^{2}$. If the metric is expressed as $h=\chi(z)dzd\bar{z}$, then for
two holomorphic quadratic differentials $Q_{1}$ and $Q_{2}$ the function
$\frac{q_{1}\bar{q}_{2}}{\chi}$ is independent on the choice of coordinates
and is denoted by $\frac{Q_{1}\bar{Q}_{2}}{h^{2}}$. The _Weil–Petersson
cometric_ is defined as

$\langle
Q_{1},Q_{2}\rangle_{\textsc{wp}}=\mathrm{Re}\int_{S}\frac{Q_{1}\bar{Q}_{2}}{h^{2}}~{}.$

With (27), one compute easily (see [Tro92]) that for two symmetric transverse
Codazzi tensors $b_{1},b_{2}$ over $(S,h)$,

$\langle X(b_{1}),X(b_{2})\rangle_{\textsc{wp}}=\frac{1}{2}\int_{S}\mbox{\rm
tr}_{h}(b_{1}b_{2})~{},$ (28)

where $X(b)$ is the tangent vector of Teichmüller space associated to the
Codazzi tensor $b$, as a first order deformation of the hyperbolic metric.

Now recall the tangent vector field $X_{d}$ over $\mathcal{T}(S)$ defined by
the flat metric $d$ (Notation 4.1). The vector $X_{d}(h)$ defines a balanced
geodesic cellulation $(\mathscr{G},w)$, which in turn defines a map
$s_{\mathbf{p}}$ and hence a symmetric traceless Codazzi tensor $b_{d}$, as
explained in the preceding section. Here, the symmetric traceless Codazzi
tensor $b_{d}$ is not identified with a first-order deformation of hyperbolic
metrics, as in (28), but rather with an element of $\mathcal{H}^{1}(\rho)$.
Hence there are two ways to identify $b_{d}$ with a tangent vector of
Teichmüller space, but they differ only by the almost-complex structure
$\mathcal{J}$ of $\mathcal{T}(S)$ [BS16, Theorem B]:

$\mathcal{J}X(b_{d})=X_{d}(h)~{}.$ (29)

This almost complex structure for holomorphic quadratic differential is the
left multiplication by the almost complex structure $J$ of $(S,h)$ and then
$\mathcal{J}$ acts on symmetric traceless Codazzi tensors as $b\mapsto bJ$
(recall (19)). At the end of the day, using (29) to implicitly identify
$X_{d}(h)$ with a tangent vector of Teichmüller space as a space of metrics,
(28) and (24), we get that if $\omega$ is the Weil–Petersson symplectic form,
then

$\begin{split}\omega(X(b),X_{d}(h))&=\omega(X(b),\mathcal{J}X(b_{d}))=\langle
X(b),X(b_{d})\rangle_{\textsc{wp}}=\\\ &=\frac{1}{2}\int_{S}\mbox{\rm
tr}_{h}(bb_{d})=-\frac{1}{2}\sum_{e}w_{e}\int_{e}b(U_{e},U_{e})~{}.\end{split}$
(30)

######  Remark 4.17.

Formula (30) above is classical when the balanced geodesic cellulation is
replaced by a simple closed geodesic, see e.g. [Wol82, Wol83, Wol87]. We also
mention that the Weil–Petersson symplectic form was also computed in [FS20a]
in terms of two balanced geodesic cellulations $(\mathscr{G},w)$ and
$(\mathscr{G}^{\prime},w^{\prime})$ over $(S,h)$. Namely, if $d$ and
$d^{\prime}$ are the respective dual flat metrics,

$w(X_{d}(h),X_{d^{\prime}}(h))=\frac{1}{2}\sum_{p\in e\cap
e^{\prime}}w_{e}w_{e}^{\prime}\cos\theta_{ee^{\prime}}$

where $\theta_{ee^{\prime}}$ is the angle of intersection between $e$ and
$e^{\prime}$ according to the orientation of $S$.

For future reference let us consider a symmetric traceless Codazzi tensor $b$
which is a first-order deformation of a path of hyperbolic metrics $h(t)$ with
$h(0)=h$ along a Weil–Petersson geodesic. From [Yam14, Theorem 3.8] we have

$h^{\prime\prime}(0)=\left(\frac{1}{4}-\frac{1}{2}(\Delta_{h}-2)^{-1}\right)\|b\|^{2}h~{},$
(31)

with $\|b\|^{2}=\mbox{\rm tr}_{h}(bb)$ and $\Delta_{h}$ the Laplacian of $h$.
This formula is also proved in [Wol12, (3.4)]. In this reference
$h^{\prime}(0)$ is given in term of a holomorphic quadratic differential that
corresponds to 2 times our Codazzi tensor $b$, and with this difference taken
into account, both formula are the same. Note that the Weil–Petersson metrics
they both consider differ by a factor $2$, but that does not change the
parametrization of the Weil–Petersson geodesics with the same tangent vector.

###  4.4. First derivatives

Let $h(t)$, $t\in(-\varepsilon,\varepsilon)$, be an analytical path of
hyperbolic metrics in $\mathcal{T}(S)$, sufficiently small so that the
conclusion of Lemma 4.4 occurs. For each $t>0$, let
$P(t)=\partial\widetilde{\rm conv}(\mathbf{p}_{d}(h))$. The quotient of the
image by the Gauss map of $\partial P(t)$ gives a weighted graph
$(\mathscr{G}_{t},w_{t})$ over $S$. More precisely, there exists a graph $G$
and a piecewise smooth mapping $f(t)$ from $G$ to $\mathscr{G}_{t}$.

From Lemma 4.4 it follows that the dependence on $t>0$ of the vertices of
$P(t)$ in Minkowski space is analytic. As the combinatorics of $P(t)$ is
fixed, and as the Gauss map can be written as a cross product of vertices, for
a vertex $v\in G$ the map $t\mapsto f(t)(v)$ is smooth (note that the graph is
sent to the smooth surface $S$, as the hyperbolic metric changes over $S$, we
cannot keep the conformal structure over $S$).

By construction for each $t>0$, $(\mathscr{G}_{t},w_{t})$ is a balanced convex
cellulation over $(S,h(t))$, whose dual metric is (isometric to) $d$. In turn,
the edges lengths of $P(t)$ are constant, so the weights $w_{t}$ does not
depend on $t$, and we will write them $w$.

Let us insist that for $t>0$, for all edges $e$ of $G$, the image of
$f_{e}(t)$ does not collapse to a point. Also for $t=0$ if a connected
subgraph of $G$ collapses to a point, then it is a tree. The same conclusion
occurs for $t<0$, but the underlying graph could be different from $G$. We
will call _admissible_ such a path $(\mathscr{G}_{t},w,h(t))$.

The fact that the dual metric is prescribed allow to define unit tangent
vector also for edges reduced to a point, as explained in the lemma below.

######  Lemma 4.18.

Let $(\mathscr{G}_{t},w,h(t))$ be an admissible path. For $t>0$ let
$U_{e,v}(t)$ be the $h(t)$-unit normal vector at the vertex $f_{v}(t)$ of
$f_{e}(t)$. Then $U_{e,v}(t)$ converges to a $h(0)$-unit tangent vector
$U_{e,v}(0)$ at $f_{v}(0)$. Moreover, if $f_{e}(0)$ does not collapse to a
point, then $U_{e,v}(0)$ is its unit tangent vector at $f_{v}(0)$.

###### Proof.

Let $G^{\prime}\subset G$ be a connected subgraph collapsing to a point at
$t=0$ and $v$ be its leaf. Then $v$ has an edge $e$ that does not collapse to
a point. Hence we have the well-defined tangent vector $U_{e,v}(0)$. The dual
Euclidean metric determines all angles between the tangent vectors to the
edges at $v$ as the supplementary angles between the dual edges in $d$, see
Figure 4. Hence we can reconstruct the tangent vectors for any edge of
$G^{\prime}$ at $v$. For an edge $e^{\prime}$ of $G^{\prime}$ adjacent to $v$
and $v^{\prime}$ the tangent vector $U_{e^{\prime},v^{\prime}}(0)$ is
determined just as $-U_{e^{\prime},v}(0)$. Then again we can reconstruct all
tangent vectors at $v^{\prime}$ with the help of the angles of $d$. Proceeding
this way we reconstruct all tangent vectors at the edges of $G^{\prime}$. It
is easy to see that the result is independent from the sequence, in which we
are doing the reconstruction, and every obtained tangent vector is the limit
of the respective tangent vectors as $t\rightarrow 0+$. ∎

In particular, for $t\geq 0$, the following balance condition holds: for any
vertex $v$ of $G$,

$\sum_{E\ni e\ni v}w_{e}U_{e,v}(t)=0~{}.$ (32)

Moreover, let $G_{0}$ be the graph obtained from $G$ by contracting the
components of collapsed edges at $t=0$, so $\mathscr{G}_{0}$ is its geodesic
realization in $(S,h(0))$. By $E_{0}$ we denote the set of its edges (we may
refer to them as to _visible edges_ and to the vertices of $G_{0}$ as to
_visible vertices_). Then at $t=0$ for a vertex $v$ of $G_{0}$ we still have
the balance condition

$\sum_{E_{0}\ni e\ni v}w_{e}U_{e,v}(0)=0~{}.$

as for a collapsed edge $e\in E$ with vertices $v$ and $v^{\prime}$ we have
$U_{e,v}(0)=-U_{e,v^{\prime}}(0)$.

Of course, a similar result holds for $t\leq 0$, but the vectors $U_{e,v}(0)$
defined by continuity are not necessarily the same for $t>0$ or $t<0$, see
Figure 4.

Figure 4. To Lemma 4.18: vectors $U_{e,v}(0)$ for $t>0$ and $t<0$ are
different.

For any edge $e\in E(G)$ and $t\in[0,\varepsilon)$ we consider a
parameterization of the $h(t)$-geodesic $f_{e}(t)$ over $[0,1]$ proportionally
to the arc length, and we will denote $f_{e}(t)(s)$ by $f_{e}(t,s)$, so
$f_{e}$ is now a continuous function from $[0,\varepsilon)\times[0,1]$ to $S$.
In the case of collapsed edges at $t=0$ the map $f(0,s)$ sends $[0,1]$ to a
point. By considering that the displacement of lifts of points in the
hyperbolic plane is smooth, it follows that $f$ is smooth over
$(0,\varepsilon)\times[0,1]$. We denote by $V_{e}$ the image by the tangent
map of $f_{e}$ of $\frac{\partial}{\partial t}$ and by $T_{e}$ the image by
the tangent map of $f_{e}$ of $\frac{\partial}{\partial s}$. We will denote by
$U_{e}(t,s)$ the $h(t)$-unit tangent vector of $f_{e}(t)$ at $f_{e}(t,s)$.
Comparing with notation of Lemma 4.18, if $f_{e}(t,0)=f_{v}(t)$, then
$U_{e,v}(t)=U_{e}(t,0)$. In particular, if $f_{e}(t)$ is not reduced to a
point (which does not occur for $t>0$),

$T_{e}(t,s)=\ell_{e}(t)U_{e}(t,s)~{},$ (33)

where $\ell_{e}(t)$ is the $h(t)$-length of the edge $f_{e}(t)$ (which may be
zero).

Note that in general, if $\nabla^{t}$ is the Levi-Civita connection of $h(t)$,
$\nabla^{t}_{V}T$ is meaningless. However, there is a notion of connection
defined by pull-back over the variation $f$, which allows formally the same
computations as if $V$ and $T$ were defined in a neighborhood, so we will
abuse the notation and write $\nabla^{t}_{V}T$, see e.g. Section 3.B.1 in
[GHL04] or Section 6.1 in [Jos17]. In particular, as $V$ and $T$ are images of
coordinates fields,

$\nabla^{t}_{V}T=\nabla^{t}_{T}V~{}.$ (34)

######  Remark 4.19.

A remark that we will often use is that at a vertex $v$, the vector $V$ does
not depend on the edges starting at $v$, conversely to $U$ and $T$. In turn,
(32) implies, for any $t_{1}$ and $t_{2}$,

$\sum_{e\ni v}w_{e}h(t_{2})(U_{e}(t_{1},0),V(t_{1},0))=0~{}.$ (35)

For an admissible path $(\mathscr{G}_{t},w,h)$ we will look at the total
length of the balanced convex cellulation at $t$:

$L(t):=\operatorname{L}_{d}(h(t))=\sum_{e\in E}w_{e}\ell_{e}(t)~{},$ (36)

which is of course continuous over $(-\varepsilon,\varepsilon)$. (In the
present section $E$ is the set of edges of $G$.)

######  Lemma 4.20.

Along an admissible path

$L^{\prime}(t)=\frac{1}{2}\sum_{e\in
E}w_{e}\ell_{e}\int_{0}^{1}h^{\prime}(t)(U_{e}(t,s),U_{e}(t,s))ds~{}.$ (37)

We will often skip the points $(t,s)$ from $U$ and $V$ when the context makes
them clear.

###### Proof.

We recall that $L$ is $C^{1}$ (because $\operatorname{L}_{d}$ is $C^{1}$). We
prove the formulas for $t\not=0$, for which there are no collapsed edges, and
notice that the formula extends continuously to $t=0$. (It is actually in
general true that a continuous function over $(-\varepsilon,\varepsilon)$ with
continuous derivative over $(-\varepsilon,\varepsilon)\setminus\\{0\\}$ that
extends continuously at $0$ is $C^{1}$.)

Let $t\not=0$. Let us denote by $\operatorname{L}_{h(u)}(f(\alpha))$ the total
length for the metric $h(u)$ of the the graph at time $\alpha$, so that
$f(\alpha)$ is geodesic for $h(\alpha)$ and
$L(t)=\operatorname{L}_{h(t)}(f(t))$. We have

$L^{\prime}(t)=\frac{d}{du}\operatorname{L}_{h(u)}(f(t))|_{u=t}+\frac{d}{d\alpha}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}~{}.$

The point is that
$\frac{d}{d\alpha}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}$ is a
weighted sum of variations of geodesics, which is equal for each $e$ (see e.g.
[Jos17, (6.1.4)]), to
$\frac{1}{\ell_{e}(t)}h(t)(T,V)|_{s=0}^{s=1}=h(t)(U,V)|_{s=0}^{s=1}$, whose
weighted sum is equal to zero by Remark 4.19. Hence,

$\frac{d}{d\alpha}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}=0~{}.$ (38)

Now

$\begin{split}L^{\prime}(t)&=\frac{d}{du}\operatorname{L}_{h(u)}(f(t))|_{u=t}=\frac{1}{2}\sum_{e\in
E}w_{e}\int_{0}^{1}\frac{h^{\prime}(t)(T,T)}{h(t)(T,T)^{1/2}}ds\\\
&=\frac{1}{2}\sum_{e\in
E}w_{e}\int_{0}^{1}h^{\prime}(t)(U,U)h(t)(T,T)^{1/2}ds=\frac{1}{2}\sum_{e\in
E}w_{e}\ell_{e}\int_{0}^{1}h^{\prime}(t)(U,U)ds\end{split}$

and this last expression converges to the same expression when $t\to 0^{\pm}$.
∎

Recall the tangent vector field $X_{d}$ over $\mathcal{T}(S)$ defined by the
flat metric $d$ (Notation 4.1).

######  Proposition 4.21.

The vector field $X_{d}$ is minus the Weil–Petersson symplectic gradient of
$\operatorname{L}_{d}$.

###### Proof.

Let $b$ be a symmetric traceless Codazzi tensor over $(S,h)$, which is tangent
to the analytical path $h(t)$ at $h(0)=h$ in $\mathcal{T}(S)$. From (37) if
$f_{e}(0)$ is a point, then the corresponding integral does not contribute to
the sum. In particular,

$\operatorname{L}_{d}(h(t))^{\prime}|_{t=0}=\frac{1}{2}\sum_{e\in
E_{0}}w_{e}\int_{e}b(U_{e},U_{e})~{}.$

As $\operatorname{L}_{d}$ is $C^{1}$ by Lemma 4.5, we have
$d\operatorname{L}_{d}(h)(b)=\frac{1}{2}\sum_{e\in E_{0}}w_{e}\int_{e}b$ and
the result follows from (30). ∎

######  Corollary 4.22.

$\operatorname{L}_{d}$ is $C^{2}$.

###### Proof.

This is because $X_{d}$ is $C^{1}$ (Lemma 4.2). ∎

###  4.5. Second derivative along Weil–Petersson geodesic

We continue to derivate the length function $\operatorname{L}_{d}$ along an
admissible path, starting from the first derivative obtained in Lemma 4.20.

######  Lemma 4.23.

Along an admissible path

$L^{\prime\prime}(0)=X-Y$

with

$\begin{split}&X=\sum_{e\in
E_{0}}w_{e}\ell_{e}\int_{0}^{1}\frac{1}{2}h^{\prime\prime}(0)(U,U)-\frac{1}{4}h^{\prime}(0)(U,U)^{2}ds~{},\\\
&Y=\sum_{e\in
E_{0}}w_{e}\ell_{e}\int_{0}^{1}\|\nabla_{U}V^{\bot}\|^{2}+\|V^{\bot}\|^{2}ds~{}.\end{split}$

Here $\|\cdot\|$ is the $h(0)$ norm and $V^{\bot}$ is the component of $V$
orthogonal to $U$.

###### Proof.

With the same argument as for the proof of Lemma 4.20, for $t\not=0$

$L^{\prime\prime}(t)=\frac{d^{2}}{du^{2}}\operatorname{L}_{h(u)}(f(t))|_{u=t}+\frac{d}{d\alpha}\left(\frac{d}{du}\operatorname{L}_{h(u)}(f(\alpha))|_{u=t}\right)|_{\alpha=t}~{}.$

It is easy to see that

$\frac{d^{2}}{du^{2}}\operatorname{L}_{h(u)}(f(t))|_{u=t}=\sum_{e\in
E}w_{e}\ell_{e}\int_{0}^{1}\frac{1}{2}h^{\prime\prime}(t)(U(t),U(t))-\frac{1}{4}h^{\prime}(t)(U(t),U(t))^{2}ds~{},$

which extends continuously when $t\to 0^{\pm}$.

The derivative of (38) with respect to $t$ says that

$\frac{d}{du}\left(\frac{d}{d\alpha}\operatorname{L}_{h(u)}(f(\alpha))|_{\alpha=t}\right)|_{u=t}=-\frac{d^{2}}{d\alpha^{2}}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}~{}.$

From [Jos17, (6.1.7)]

$\frac{d^{2}}{d\alpha^{2}}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}=\sum_{e}\frac{w_{e}}{\ell_{e}}\left(\int_{0}^{1}\|\nabla_{T}V^{\bot}\|^{2}-h(R(T,V^{\bot})V^{\bot},T)ds+h(\nabla_{V}V,T)|^{s=1}_{s=0}\right)~{},$

where $R$ is the Riemann curvature tensor of $h=h(0)$. From Remark 4.19,
$\sum_{e}\frac{w_{e}}{\ell_{e}}h(\nabla_{V}V,T)|^{s=1}_{s=0}=0$. Also, as the
sectional curvature of $h$ is constant equal to $-1$, we obtain

$\frac{d^{2}}{d\alpha^{2}}\operatorname{L}_{h(t)}(f(\alpha))|_{\alpha=t}=\sum_{e}w_{e}\ell_{e}\int_{0}^{1}\|\nabla_{U}V^{\bot}\|^{2}+\|V^{\bot}\|^{2}ds~{},$

that leads to the result, up to justify that we can change the order between
$\frac{d}{du}$ and $\frac{d}{d\alpha}$. Indeed, one may write explicitly
$\frac{d^{2}}{d\alpha du}\operatorname{L}_{h(u)}(f(\alpha))$ and see that all
the second partial derivatives of $(u,t)\mapsto\operatorname{L}_{h(u)}(f(t))$
are continuous. ∎

The following result is an adaptation of the corresponding one in [Wol12]
(where the second variation of length of closed geodesics is studied) and of
some arguments of [KT21] (where the second variation of energy of weighted
graphs is studied).

######  Proposition 4.24.

If the admissible path follows a Weil–Petersson geodesic up to second order,
then $L^{\prime\prime}(0)>0.$

###### Proof.

We denote the respective integrals over an edge $e$ in Lemma 4.23, divided by
$\ell_{e}$, by $X_{e}$ and $Y_{e}$, so

$L^{\prime\prime}(0)=\sum_{e\in E_{0}}\omega_{e}\ell_{e}(X_{e}-Y_{e})~{}.$
(39)

Let $b$ be the traceless symmetric Codazzi tensor that corresponds to
$h^{\prime}(0)$. Pick an edge $e\in E_{0}$. By
$\alpha:[0,1]\rightarrow\mathbb{R}$ and by $\beta:[0,1]\rightarrow\mathbb{R}$
we denote the functions, if $U^{\perp}$ is the rotation by $\pi/2$ of $U$,

$\alpha:=\frac{1}{2}b(U,U),~{}~{}~{}~{}~{}\beta:=\frac{1}{2}b(U,U^{\bot})~{}.$

(In more details, e.g., $\alpha(s)$, $s\in[0,1]$, is defined as
$\frac{1}{2}b_{f_{e}(s)}(U_{f_{e}(s)},U_{f_{e}(s)})$.)

Let us first bound $X_{e}$. We have

$X_{e}=\int_{0}^{1}\frac{1}{2}h^{\prime\prime}(0)(U,U)-\frac{1}{4}h^{\prime}(0)(U,U)^{2}ds~{}.$

As we are following up to second order a Weil–Petersson geodesic starting from
$h$ and directed by the Codazzi tensor $b$, (31) applies with, as $b$ is
symmetric and traceless,

$\|b\|^{2}=\mbox{\rm
tr}_{h}(bb)=2(b(U,U)^{2}+b(U,U^{\bot})^{2})=8\alpha^{2}+8\beta^{2}~{}.$ (40)

Now we get

$X_{e}=\int_{0}^{1}-\frac{1}{4}(\Delta-2)^{-1}\|b\|^{2}+\beta^{2}ds~{}.$ (41)

We note that [Wol12, Lemma 5.1], [KT21, Formula (3.13)]

$-\frac{1}{4}(\Delta-2)^{-1}\|b\|^{2}\geq\frac{1}{24}\|b\|^{2}.$ (42)

So we substitute it in (41) and get

$X_{e}\geq\int_{0}^{1}\frac{1}{3}\alpha^{2}+\frac{4}{3}\beta^{2}ds~{}.$ (43)

Now we deal with $Y_{e}$. Take the deformation vector field $V$ and decompose
it into the tangential and orthogonal parts, $V^{\top}$ and $V^{\bot}$, to the
edge $e$. By $\zeta:[0,1]\rightarrow\mathbb{R}$ and
$\eta:[0,1]\rightarrow\mathbb{R}$ we denote the oriented lengths of $V^{\top}$
and $V^{\bot}$ respectively, pulled back to $[0,1]$. In other terms,

$\zeta:=h(V,U),~{}~{}~{}~{}~{}\eta:=h(V,U^{\perp})~{}.$

In the following, we will denote by $\dot{\phantom{x}}$ the derivative with
respect to the parameter $s$ of the respective functions. As $\zeta$ is the
oriented length of $V^{\top}$, $\dot{\zeta}$ is the oriented length of
$\ell_{e}\nabla_{U}V^{\top}$ and because $\nabla_{U}U=0$, $\nabla_{U}V^{\top}$
is collinear with $U$, hence

$\dot{\zeta}=\ell_{e}h(\nabla_{U}V,U)=\ell_{e}h(\nabla_{U}V^{\top},U)~{},$
(44) $\zeta\dot{\zeta}=\ell_{e}h(V^{\top},\nabla_{U}V^{\top})~{}.$ (45)

Similarly, $\dot{\eta}$ is the oriented length of
$\ell_{e}\nabla_{U}V^{\bot}$. Then $\nabla_{U}V^{\bot}$ is orthogonal to $U$,
i.e., collinear to $U^{\bot}$ hence

$\dot{\eta}=\ell_{e}h(\nabla_{U}V,U^{\bot})=\ell_{e}h(\nabla_{U}V^{\bot},U^{\bot})~{},$
(46)
$\eta\dot{\eta}=\ell_{e}h(V^{\perp},\nabla_{U}V^{\perp})=\ell_{e}h(V,\nabla_{U}V)-\ell_{e}h(V^{\top},\nabla_{U}V^{\top})=\ell_{e}h(V,\nabla_{U}V)-\zeta\dot{\zeta}~{}.$
(47)

The key points are connections between $\alpha$ and $\zeta$ and between
$\beta$ and $\eta$. We start from the first pair. Note that by (34) and (33),
along an edge,

$\nabla_{U}V=\nabla_{V}U+\frac{\ell^{\prime}}{\ell}U~{}.$ (48)

From this and $h_{t}(U_{t},U_{t})=1$ we get by differentiating at $t=0$

$0=b(U,U)+2h(\nabla_{V}U,U)=b(U,U)+2h(\nabla_{U}V,U)-2\frac{\ell^{\prime}}{\ell}~{}.$

From this

$\dot{\zeta}=\ell
h(\nabla_{U}V,U)=-\frac{\ell}{2}b(U,U)+{\ell^{\prime}}=-\ell\alpha+{\ell^{\prime}}~{}.$
(49)

For the pair $\eta$ and $\beta$ we have the following differential equation

$\ddot{\eta}-\ell^{2}\eta=-\ell\dot{\beta}~{}.$ (50)

This formula is proved in Section 4.1 in [Wol12] and also in Lemma 3.12 in
[KT21]. Note that in [Wol12], what is denoted by $V$ is the normal component
of the deformation, as there Wolf is interested in the case of closed
geodesic, thus neglecting the impact of the tangent component. We, however,
have to care about the impact of $V^{\top}$ on the boundary terms. Also, the
geodesics are parameterized by arc-length in [Wol12], while they are
parameterized proportionally to arc-length in our case. In contrast, the
parametrization in [KT21] is the same as ours.

It is easy to check (as in [Wol12]) that there exists a primitive $\theta$ of
$\eta$ satisfying

$\ddot{\theta}-\ell^{2}\theta=-\ell\beta~{},$ (51)

and, as a primitive, $\dot{\theta}=\eta$.

Using (46) the term $Y_{e}$ in (39) can be written as

$Y_{e}=\int_{0}^{1}(\ell^{-1}\dot{\eta})^{2}+\eta^{2}ds~{}.$ (52)

We now partially follow computations from [Wol12] keeping a special track on
the boundary terms (which do not appear in [Wol12] as only the case of a
closed geodesic is considered there):

$\begin{split}Y_{e}&=\int_{0}^{1}(\ell^{-1}\dot{\eta})^{2}+\eta^{2}ds\\\
&\stackrel{{\scriptstyle\mathrm{IBP}}}{{=}}\ell^{-2}\eta\dot{\eta}|^{1}_{0}+\int^{1}_{0}(\eta^{2}-\ell^{-2}\eta\ddot{\eta})ds\\\
&\stackrel{{\scriptstyle\eqref{eq:la
equation}}}{{=}}\ell^{-2}\eta\dot{\eta}|_{0}^{1}+\ell^{-1}\int_{0}^{1}\eta\dot{\beta}ds\\\
&\stackrel{{\scriptstyle\mathrm{IBP}}}{{=}}\ell^{-2}\eta\dot{\eta}|^{1}_{0}+\ell^{-1}\eta\beta|^{1}_{0}-\ell^{-1}\int^{1}_{0}\dot{\eta}\beta
ds~{}.\end{split}$ (53)

We denote $\ell^{-1}\int^{1}_{0}\dot{\eta}\beta ds$ by $Y^{*}_{e}$. We first
see what happens with the boundary term
$\ell^{-2}\eta\dot{\eta}|^{1}_{0}+\ell^{-1}\eta\beta|^{1}_{0}$ in the total
sum, and then bound $Y^{*}_{e}$.

We substitute (47) instead of $\eta\dot{\eta}|^{1}_{0}$. Then, in the total
sum the term $\ell_{e}h(V,\nabla_{U}V)$ gets eliminated by the balance
condition, so we have

$\sum_{e\in E_{0}}\omega_{e}\ell^{-1}_{e}\eta\dot{\eta}|^{1}_{0}=-\sum_{e\in
E_{0}}\omega_{e}\ell^{-1}_{e}\zeta\dot{\zeta}|^{1}_{0}.$ (54)

Using integration by parts and (49) we get

$\zeta\dot{\zeta}|^{1}_{0}=\int_{0}^{1}(\dot{\zeta}^{2}+\ddot{\zeta}\zeta)ds=\int_{0}^{1}(\dot{\zeta}^{2}-\ell\dot{\alpha}\zeta)ds=-\ell\alpha\zeta|^{1}_{0}+\int^{1}_{0}\dot{\zeta}(\dot{\zeta}+\ell\alpha)ds~{}.$
(55)

By definition,

$\eta\beta+\alpha\zeta=\frac{1}{2}(h(V,U^{\perp})b(U,U^{\perp})+h(V,U)b(U,U))=\frac{1}{2}b(U,h(V,U^{\perp})U^{\perp}+h(V,U)U)=\frac{1}{2}b(U,V)~{}.$
(56)

When we take its weighted sum, it disappears due to the balance condition.

From (49)

$\dot{\zeta}(\dot{\zeta}+\ell\alpha)=\left({\ell^{\prime}}\right)^{2}-\alpha{\ell}{\ell^{\prime}}=\left({\ell^{\prime}}-\frac{{\ell}\alpha}{2}\right)^{2}-\frac{{\ell}^{2}\alpha^{2}}{4}\geq-\frac{{\ell}^{2}\alpha^{2}}{4}~{}.$
(57)

Using (53), (54), (55), (56) and (57) we then get

$-Y=-\sum_{e\in E_{0}}w_{e}\ell_{e}Y_{e}\geq\sum_{e\in
E_{0}}w_{e}\ell_{e}\left(Y^{*}_{e}-\int_{0}^{1}\frac{\alpha^{2}}{4}ds\right)~{}.$

Introducing the notations $Y^{*}:=\sum_{e}w_{e}\ell_{e}Y^{*}_{e}$,
$A:=\sum_{e}w_{e}\ell_{e}\int_{0}^{1}\alpha^{2}$, the equation above writes

$-Y\geq Y^{*}-\frac{1}{4}A~{}.$ (58)

Now we write

$Y^{*}_{e}=\ell^{-1}\int^{1}_{0}\dot{\eta}\beta
ds=\int_{0}^{1}\ell^{-1}\left(\left(\sqrt{\frac{\ell^{-1}}{4}}\dot{\eta}+\sqrt{\ell}\beta\right)^{2}-\frac{\ell^{-1}}{4}\dot{\eta}^{2}-\ell\beta^{2}\right)ds~{}.$

By adding to the integrand $\pm\eta^{2}/4$, with (52) we have that

$Y^{*}_{e}=-\frac{Y_{e}}{4}+\int_{0}^{1}\frac{\eta^{2}}{4}+\ell^{-1}\left(\sqrt{\frac{\ell^{-1}}{4}}\dot{\eta}+\sqrt{\ell}\beta\right)^{2}ds-\int_{0}^{1}\beta^{2}ds$
(59)

We denote $B:=\sum_{e}w_{e}\ell_{e}\int_{0}^{1}\beta^{2}$,

$C:=\sum_{e\in
E_{0}}w_{e}\ell_{e}\int_{0}^{1}\frac{\eta^{2}}{4}+\ell^{-1}\left(\sqrt{\frac{\ell^{-1}}{4}}\dot{\eta}+\sqrt{\ell}\beta\right)^{2}ds.$
(60)

Taking the weighted sum of (59), we get

$Y^{*}=-\frac{1}{4}Y-B+C~{}.$ (61)

Using this with (58),

$-Y\geq-\frac{1}{4}Y-\frac{1}{4}A-B+C$

i.e.

$-Y\geq-\frac{1}{3}A-\frac{4}{3}B+\frac{4}{3}C~{}.$

With our current notation (43) gives

$X\geq\frac{1}{3}A+\frac{4}{3}B~{},$

hence

$L^{\prime\prime}(0)=X-Y\geq\frac{4}{3}C~{}.$

Looking at (60) we see that $L^{\prime\prime}(0)$ is non-negative. Moreover,
$L^{\prime\prime}(0)=0$ implies $C=0$, hence on every edge $\eta=0$. Thereby,
also $\dot{\eta}=0$ on every edge, and then $C=0$ also implies that $\beta=0$
on every edge, so $B=0$. From the definition of $Y_{e}^{*}$ and $Y^{*}$ we
then get that $Y^{*}=0$. Thus (58) gives us

$-Y\geq-\frac{1}{4}A$

and we have

$L^{\prime\prime}(0)=X-Y\geq\frac{1}{12}A.$

It follows that $L^{\prime\prime}(0)=0$ also implies that $\alpha=0$ at every
edge. Then $b$ is zero along every edge. But $b$ is the real part of a
holomorphic quadratic differential, see (27). By holomorphy, it follows that
$b=0$ over $S$, which is excluded by our choice of the path. ∎

The Weil–Petersson metric is not complete over $\mathcal{T}(S)$, but it has an
exhaustion by compact convex sets for the Weil–Petersson metric [Wol87], so we
may speak about Weil–Petersson convex funtions.

######  Corollary 4.25.

The Weil–Petersson Hessian of $\operatorname{L}_{d}$ is positive definite and
$\operatorname{L}_{d}$ is strictly convex.

###### Proof.

The Weil–Petersson metric is not complete over $\mathcal{T}(S)$, but however
the exponential map is a homeomorphism [Wol87], hence we may consider a
Weil–Petersson geodesic tangent to a traceless symmetric Codazzi tensor $b$.
We then consider an admissible path of $\mathcal{T}(S)$ that coincides with
this Weil–Petersson geodesic up to second order. Recall that
$\operatorname{L}_{d}$ is $C^{2}$ by Corollary 4.22. Hence from Proposition
4.24, the Hessian of $\operatorname{L}_{d}$ is positive definite. As any pair
of points in $\mathcal{T}(S)$ can be joined by a Weil–Petersson geodesic
[Wol87], it follows that $\operatorname{L}_{d}$ is strictly convex. ∎

###  4.6. Proof of Theorem II’

Having the results of the preceding sections in hand, one may mimic the
argument in [Bon05a] to conclude the proof of Theorem II’.

Let $d_{1}$ and $d_{2}$ be two flat metrics with negative singular curvatures
over $S$. The function $L_{d_{1}}+L_{d_{2}}$ is $C^{1}$ and proper (Lemma 4.5
and Lemma 4.10), hence it has a critical point $h\in\mathcal{T}(S)$. As
$X_{d_{i}}$ is minus the Weil–Petersson symplectic gradient of $L_{d_{i}}$
(Proposition 4.21), it follows that $X_{d_{1}}(h)+X_{d_{2}}(h)=0$. As
$L_{d_{1}}+L_{d_{2}}$ is strictly convex (Corollary 4.25), this critical point
is unique. As the Hessian of $L_{d_{1}}+L_{d_{2}}$ is positive definite
(Corollary 4.25), the intersection is transverse.

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IMAG, Université de Montpellier, CNRS, Montpellier, France

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University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1,
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