# Almost strict domination and Anti-de Sitter 3-manifolds

Nathaniel Sagman  Mathematics Dept., Caltech, Pasadena, CA 91125, USA
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
<EMAIL_ADDRESS>
###### Abstract.

We define a condition called almost strict domination for pairs of
representations
$\rho_{1}:\pi_{1}(S_{g,n})\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
$\rho_{2}:\pi_{1}(S_{g,n})\to G$, where $G$ is the isometry group of a
Hadamard manifold $(X,\nu)$, and prove it holds if and only if one can find a
$(\rho_{1},\rho_{2})$-equivariant spacelike maximal surface in a certain
pseudo-Riemannian manifold, unique up to fixing some parameters. The proof
amounts to setting up and solving an interesting variational problem that
involves infinite energy harmonic maps. Adapting a construction of Tholozan,
we construct all such representations and parametrize the deformation space.

When $(X,\nu)=(\mathbb{H},\sigma)$, an almost strictly dominating pair is
equivalent to the data of an anti-de Sitter 3-manifold with specific
properties, and we can extract geometric information using maximal surfaces
and associated Higgs bundles. The results on maximal surfaces provide a
parametrization of the deformation space of such $3$-manifolds as a union of
components in a
$\operatorname{\textrm{PSL}}(2,\mathbb{R})\times\operatorname{\textrm{PSL}}(2,\mathbb{R})$
relative representation variety.

##  1\. Introduction

Recently there has been a growing interest in Lorentzian geometry in low
dimensions, with a focus on Lorentzian space forms. Among the most important
space forms is the anti-de Sitter space, which has constant curvature $-1$.
This study was originally motivated by general relativity, where anti-de
Sitter metrics are solutions to the Einstein equations, and now there are more
modern applications in physics (see the paper of Witten [46], for instance).
Another direction is the study of properly discontinuous group actions on
Lorentzian space forms, or more generally on Clifford-Klein forms, which has
its roots in some famous conjectures about affine geometry.

Some aspects of the theory of properly discontinuous actions are well
understood. However, in our previous paper [36] we found anti-de Sitter
structures on Seifert-fibered $3$-manifolds that fall outside the scope of the
usual theory. This current paper has grown out of an attempt to elaborate on
the situation from [36]. In the end, we have found a general property for
representations
$\rho_{1}:\pi_{1}(S_{g,n})\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
$\rho_{2}:\pi_{1}(S_{g,n})\to G$, where $G$ is the isometry group of a
Hadamard manifold $(X,\nu)$.

###  1.1. On AdS $3$-manifolds

In dimension $3$, anti-de Sitter space (from now on,
$\operatorname{\textrm{AdS}^{3}}$) identifies isometrically with the Lie group
$\operatorname{\textrm{PSL}}(2,\mathbb{R})$, equipped with (a constant
multiple of) its Lorentzian Killing metric. The space and time orientation
preserving component of the isometry group is
$\operatorname{\textrm{PSL}}(2,\mathbb{R})\times\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
where the action is by left and right multiplication:

$(g,h)\cdot x=gxh^{-1}.$

An AdS $3$-manifold is a Lorentzian $3$-manifold of constant curvature $-1$.
Equivalently, such a manifold is locally isometrically modelled on
$\operatorname{\textrm{PSL}}(2,\mathbb{R})$.

If an AdS $3$-manifold is geodesically complete, meaning geodesics run for all
time, then it comes from a proper quotient of
$\widetilde{\operatorname{\textrm{PSL}}}(2,\mathbb{R})$ with respect to the
lift of the action above. Goldman showed that the space of closed AdS
$3$-manifolds is larger than originally expected [17], and Kulkarni and
Raymond took up the problem of understanding all geodesically complete AdS
$3$-manifolds [28]. Among other things, they proved [28, Theorem 5.2] that any
torsion-free discrete group acting properly discontinuously on
$\operatorname{\textrm{AdS}^{3}}$ is of the form

$\Gamma_{\rho_{1},\rho_{2}}=\\{(\rho_{1}(\gamma),\rho_{2}(\gamma)):\gamma\in\Gamma\\},$

where $\Gamma$ is a the fundamental group of a surface, and
$\rho_{1},\rho_{2}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$ are
representations, with at least one of them Fuchsian (discrete and faithful).
This is generalized for actions on rank $1$ Lie groups in [24].

######  Remark 1.1.

Shortly after, Klingler [26] proved that closed Lorentzian manifolds of
constant curvature are geodesically complete. Thus, the completeness
assumption can be dropped in the work of Kulkarni and Raymond on closed AdS
$3$-manifolds.

The natural next step is to understand which $\Gamma_{\rho_{1},\rho_{2}}$ act
properly discontinuously. In the cocompact case, Salein observed it is
sufficient [37], and Kassel proved it is necessary [25] that $\rho_{1}$
strictly dominates $\rho_{2}$ (defined below). Guéritaud and Kassel extended
these results to surfaces with punctures and higher dimensional hyperbolic
spaces [19].

######  Definition 1.2.

Let $\Gamma$ be a discrete group and let
$\rho_{k}:\Gamma\to\textrm{Isom}(X_{k},g_{k})$ be representations into
isometry groups of Riemannian manifolds $(X_{k},g_{k})$, $k=1,2$. $\rho_{1}$
(strictly) dominates $\rho_{2}$ if there is a
$(\rho_{1},\rho_{2})$-equivariant $1$-Lipschitz ($\lambda$-Lipschitz,
$\lambda<1$) map $f:X_{1}\to X_{2}$.

A mapping as above is $(\rho_{1},\rho_{2})$-equivariant if for all
$\gamma\in\Gamma$ and $x\in X_{1}$,

$f(\rho_{1}(\gamma)\cdot x)=\rho_{2}(\gamma)\cdot f(x).$

When a group $\Gamma$ acts on a manifold with no reference to a representation
$\rho_{1}$, we may just write $\rho_{2}$-equivariant. By the Selberg lemma, we
only need to consider torsion-free groups.

######  Theorem 1.3 (Guéritaud-Kassel, Theorem 1.8 in [19]).

A finitely generated discrete group $\Gamma_{\rho_{1},\rho_{2}}$ acts properly
discontinuously and without torsion if and only if $\rho_{1}$ is Fuchsian and
strictly dominates $\rho_{2}$, up to interchanging $\rho_{1}$ and $\rho_{2}$.

The quotient is a Seifert-fibered AdS $3$-manifold over the hyperbolic surface
$\mathbb{H}/j(\Gamma)$ such that the circle fibers are timelike geodesics.

There was an open question: is every non-Fuchsian representation
$\pi_{1}(S_{g})\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$ strictly
dominated by a Fuchsian one? This question was answered by Deroin-Tholozan in
[12] and Guéritaud-Kassel-Wolf in [20], using different methods. Deroin-
Tholozan actually proved a more general result.

######  Theorem 1.4 (Deroin-Tholozan, Theorem A in [12]).

Let $(X,\nu)$ be a $\textrm{CAT}(-1)$ Hadamard manifold with isometry group
$G$ and $\rho:\pi_{1}(S_{g})\to G$ a representation, $g\geq 2$. Then $\rho$ is
strictly dominated by a Fuchsian representation, unless it stabilizes a
totally geodesic copy of $\mathbb{H}$ on which the action is Fuchsian.

Tholozan completed the story for closed $3$-manifolds in [41].

######  Theorem 1.5 (Tholozan, Theorem 1 in [41]).

Fix $g\geq 2$ and a $\textrm{CAT}(-1)$ Hadamard manifold $(X,\nu)$ with
isometry group $G$. The space of dominating pairs within
$T(S_{g})\times\textrm{Rep}^{nf}(\pi_{1}(S_{g}),G)$ is homeomorphic to

$\mathcal{T}(S_{g})\times\textrm{Rep}^{nf}(\pi_{1}(S_{g}),G),$

where $\textrm{Rep}^{nf}(\pi_{1}(S_{g}),G)$ is the space of representations
that do not stabilize a totally geodesic copy of $\mathbb{H}$ on which the
action is Fuchsian.

The homeomorphism is fiberwise in the sense that for each
$\rho\in\textrm{Rep}^{nf}(\pi_{1}(S_{g}),G)$, it restricts to a homeomorphism
from $\mathcal{T}(S_{g})\times\\{\rho\\}\to U\times\\{\rho\\}$, where
$U\subset\mathcal{T}(S_{g})$ is an open subset. The key point is that when
$(X,\nu)=(\mathbb{H},\sigma)$, this is the deformation space of closed AdS
quotients of $\operatorname{\textrm{PSL}}(2,\mathbb{R})$. The components of
the deformation space are thus organized according to Euler numbers.

In [3], Alessandrini-Li use results from [12] and [41] to study AdS
$3$-manifolds in the framework of non-abelian Hodge theory, finding an
alternative construction of the closed AdS circle bundles over surfaces.

###  1.2. Recent work

There is now an interest in understanding non-compact quotients of
$\operatorname{\textrm{PSL}}(2,\mathbb{R})$ (see [10] and [11], for instance).
For non-Fuchsian representations, we extended Theorem 1.4 in [36, Theorem
1.2]. Following the idea of Deroin-Tholozan, but using the infinite energy
harmonic maps, we found that (with small exceptions) every non-Fuchsian
representation is dominated by a Fuchsian one. Using strip deformations (see
[11]) we then produced AdS $3$-manifolds [36, Theorem 1.3]. Some of the same
results were proved by Gupta-Su in [22], whose methods fall more in line with
that of [20].

Near the end of the paper [36], we found something curious: pairs of
representations $\rho_{1},\rho_{2}$ that give rise to circle bundles with an
anti-de Sitter structure that do not come from properly discontinuous actions
on all of $\operatorname{\textrm{AdS}^{3}}$. At the time we did not put much
emphasis on the result; in fact, it’s buried near the end of the paper as
Proposition 7.7. The motivation for this paper is to explore representations
such as $(\rho_{1},\rho_{2})$ in more depth. The representations all satisfy a
geometric condition, which we call almost strict domination. Below, let
$(X,\nu)$ be a Hadamard manifold with isometry group $G$.

######  Definition 1.6.

Let
$\rho_{1}:\pi_{1}(S_{g,n})\to\operatorname{\textrm{PSL}}(2,\mathbb{R}),\rho_{2}:\pi_{1}(S_{g,n})\to
G$ be two representations with $\rho_{1}$ Fuchsian. We say that $\rho_{1}$
almost strictly dominates $\rho_{2}$ if

  1. (1)

for every peripheral $\zeta\in\Gamma$,
$\ell(\rho_{1}(\zeta))=\ell(\rho_{2}(\zeta))$, and

  2. (2)

there exists a $(\rho_{1},\rho_{2})$-equivariant $1$-Lipschitz map $g$ defined
on the convex hull of the limit set of $\rho_{1}(\Gamma)$ in $\mathbb{H}$ such
that the local Lipschitz constants are $<1$ inside the convex hull, and for
peripherals $\zeta$ such that $\rho_{1}(\zeta)$ is hyperbolic, $g$ takes each
boundary geodesic axis for $\rho_{1}(\zeta)$ isometrically to a geodesic axis
for $\rho_{2}(\zeta)$.

In the definition above, the global Lipschitz constant is

$\textrm{Lip}(g)=\sup_{y_{1}\neq
y_{2}}\frac{d_{\nu}(g(y_{1}),g(y_{2}))}{d_{\sigma}(y_{1},y_{2})},$

where $\sigma$ is the hyperbolic metric. The local one is

$\textrm{Lip}_{x}(g)=\inf_{r>0}\textrm{Lip}(g|_{B_{r}(x)})=\inf_{r>0}\sup_{y_{1}\neq
y_{2}\in B_{r}(x)}\frac{d_{\nu}(g(y_{1}),g(y_{2}))}{d_{\sigma}(y_{1},y_{2})},$

which by equivariance is a well-defined function on the convex core of
$\mathbb{H}/\rho_{1}(\pi_{1}(S_{g,n})$. In the language of [19], the
projection to $\mathbb{H}/\rho_{1}(\Gamma)$ of the stretch locus of an optimal
Lipschitz map is exactly the boundary of the convex core. This property is
very rare: it implies domination in the simple length spectrum (see [22]). In
view of [19], we will say a Lipschitz map $g:(\mathbb{H},\sigma)\to(X,\nu)$ is
optimal if it satisfies the properties above.

To end this introductory portion, we list some other instances where
domination has appeared. Marché and Wolff use domination to answer a question
of Bowditch and resolve the Goldman conjecture in genus $2$ [32]. Dai-Li study
domination for higher rank Hitchin representations into
$\operatorname{\textrm{PSL}}(n,\mathbb{C})$ [9]. It would be interesting to
see if the almost strict condition generalizes meaningfully to higher rank.

###  1.3. Main theorems: maximal surfaces

We first fix some notations that we will keep throughout the paper.

  * •

$\Sigma$ is surface with genus $g$ and $n$ punctures $p_{1},\dots,p_{n}$, with
$\chi(\Sigma)<0$. The deck group for the universal covering
$\pi:\tilde{\Sigma}\to\Sigma$ is denoted by $\Gamma$.

  * •

$\mathcal{T}(\Gamma)$ is the Teichmüller space of classes of complete finite
volume marked hyperbolic metrics on $\Sigma$.

  * •

$\\{\zeta_{1},\dots,\zeta_{n}\\}\subset\Gamma$ are the peripheral elements,
i.e., those representing the simple closed curves enclosing $p_{i}$. If $n=1$,
write $\zeta_{1}=\zeta$.

  * •

$(X,\nu)$ is a $\textrm{CAT}(-1)$ Hadamard manifold with isometry group $G$.

  * •

$(\mathbb{H},\sigma)$ denotes the hyperbolic space with constant curvature
$-1$.

A representation $\rho:\Gamma\to G$ is irreducible if $\rho(\Gamma)$ fixes no
point on the visual boundary $\partial_{\infty}X$. It is reductive if it is
irreducible or if $\rho(\Gamma)$ preserves a geodesic in $X$.

######  Theorem A.

Let $\rho_{1}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$ and
$\rho_{2}:\Gamma\to G$ be reductive representations with $\rho_{1}$ Fuchsian.
$\rho_{1}$ almost strictly dominates $\rho_{2}$ if and only if there exists a
complete finite volume hyperbolic metric $\mu$ on $\Sigma$ and a
$\rho_{1}\times\rho_{2}$-equivariant tame spacelike maximal immersion from

$(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H}\times X,\sigma\oplus(-\nu)).$

The maximal surfaces are not unique but are classified according to
Proposition 2.11.

We defer the definition of tame and the explanation of uniqueness to Section
2.5 once we have more background on harmonic maps. We also comment that almost
strict domination is the same as strict domination when every
$\rho_{2}(\zeta_{i})$ is elliptic (see also [36, Lemma 6.3]).

Let’s give some idea of the proof. For representations $\rho_{1},\rho_{2}$, we
define a functional $\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$
on the Teichmüller space by

$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}(\mu)=\int_{\Sigma}e(\mu,h_{\mu}^{\theta})-e(\mu,f_{\mu}^{\theta})dA_{\mu},$

where $h_{\mu}^{\theta}$, $f_{\mu}^{\theta}$ are certain harmonic maps on
$(\tilde{\Sigma},\tilde{\mu})$ that may have infinite energy, in the sense
that

$\int_{\Sigma}e(\mu,h_{\mu}^{\theta})dA_{\mu}=\int_{\Sigma}e(\mu,f_{\mu}^{\theta})dA_{\mu}=\infty.$

We show that this is always finite, provided the boundary lengths for
$\rho_{1}$ and $\rho_{2}$ agree (Section 3). We then compute the derivative
(Section 3), showing that critical points correspond to spacelike maximal
surfaces (Proposition 2.10). To anyone working with harmonic maps, this is
expected, but with no good theory of global analysis to treat infinite energy
maps on surfaces with punctures, we have to work through some thorny details
directly. In the course of our analysis, we develop a new energy minimization
result (Lemma 3.12) that may be of independent interest.

Then we show that $\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ is
proper if and only if $\rho_{1}$ almost strictly dominates $\rho_{2}$. Here is
an indication as to why this is true. Suppose we diverge along a sequence
$(\mu_{n})_{n=1}^{\infty}\subset\mathcal{T}(\Gamma)$ by pinching a simple
closed curve $\alpha$. Then there is a collar around $\alpha$ whose length
$\ell_{n}$ in $(\Sigma,\mu_{n})$ is tending to $\infty$. Almost strict
domination implies $\ell(\rho_{1}(\alpha))>\ell(\rho_{2}(\gamma)$, and the
analysis from [36] shows that the total energy of the harmonic maps in the
collar behaves like

(1)
$\ell_{n}(\ell(\rho_{1}(\alpha))^{2}-\ell(\rho_{2}(\alpha))^{2})\to\infty.$

This reasoning, however, cannot be turned into a full proof. Two problems:

  1. (1)

Along a general sequence that leaves all compact subsets of
$\mathcal{T}(\Gamma)$, the two harmonic maps could apriori behave quite
differently in a thin collar. For instance we could have twisting in one
harmonic map, which increases the energy, but no twisting in the other.

  2. (2)

For a general sequence, we also have little control over the energy outside of
thin collars.

We circumvent these issues as follows: if $g$ is an optimal map, then our
energy minimization Lemma 3.12 implies that

$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}(\mu)\geq\int_{\Sigma}e(\mu,h_{\mu}^{\theta})-e(\mu,g\circ
h_{\mu}^{\theta})dA_{\mu}.$

The integrand is positive, so now we can bound below by the energy in collars.
The contracting property of $g$ then allows us to effectively study the energy
in collars. In the end we make a rather technical geometric argument in order
to find lower bounds similar to (1) along diverging sequences.

We also remark that even in the non-compact but finite energy setting, the
result on the derivative of the energy functional was not previously contained
in the literature. Hence we record it below.

######  Proposition 1.7.

Let $\rho:\Gamma\to G$ be a reductive representation with no hyperbolic
monodromy around cusps, so that equivariant harmonic maps have finite energy.
Then the energy functional $E_{\rho}:\mathcal{T}(\Gamma)\to[0,\infty)$ that
records the total energy of a $\rho$-equivariant harmonic map from
$(\tilde{\Sigma},\mu)\to(X,\nu)$ is differentiable, with derivative at a
hyperbolic metric $\mu$ given by

$dE_{\rho}[\mu](\psi)=-4\operatorname{Re}\langle\Phi,\psi\rangle.$

Here $\Phi$ is the Hopf differential of the harmonic map at $\mu$.

See Sections 2 and 3 for notations. The proof can actually be extended to non-
positively curved settings (see Remark 3.15).

###  1.4. Main theorems: parametrizations

The next theorem concerns the space of almost strictly dominating pairs. We
denote by $\textrm{Hom}^{*}(\Gamma,G)\subset\textrm{Hom}(\Gamma,G)$ the space
of reductive representations. $G$ acts on $\textrm{Hom}^{*}(\Gamma,G)$ by
conjugation, and we define the representation space as

$\textrm{Rep}(\Gamma,G)=\textrm{Hom}^{*}(\Gamma,G)/G.$

In general this may not be a manifold, but it can have nice structure
depending on $G$. For surfaces with punctures we would like to prescribe
behaviour at the punctures.

######  Definition 1.8.

Fix a collection of conjugacy classes $\mathbf{c}=(c_{i})_{i=1}^{n}$ of
elements in $G$. The relative representation space
$\textrm{Rep}_{\mathfrak{c}}(\Gamma,G)$ is the space of reductive
representations taking $\zeta_{i}$ into $c_{i}$, modulo conjugation.

We require one technical assumption on the group $G$: that if we choose a good
covering of $\Sigma$ and let $\chi(\Gamma,G)$ denote the space of $G$-local
systems with respect to this covering that have reductive holonomy, then the
projection from $\chi(\Gamma,G)\to\textrm{Rep}(\Gamma,G)$ is a locally trivial
principal bundle. We demand the same for the relative representation space,
instead considering local systems whose holonomy representations respect
$\mathbf{c}$. This assumption is satisfied under most cases of interest in
Higher Teichmüller theory, for instance if $G$ is a linear algebraic group
(see [30, Chapter 5]).

Within $\textrm{Rep}_{\mathbf{c}}(\Gamma,G)$, we have the subset
$\textrm{Rep}_{\mathbf{c}}^{nf}(\Gamma,G)$ of representations that do not
stabilize a plane of constant curvature $-1$ on which the action is Fuchsian.
The almost strict domination condition is invariant under conjugation for both
representations, so we can define $\textrm{ASD}_{\mathbf{c}}(\Gamma,G)$ to be
the subspace of pairs of representations
$\rho_{1}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
$\rho_{2}:\Gamma\to G$ such that $\rho_{1}$ is Fuchsian and almost strictly
dominates $\rho_{2}$. Necessarily, $\rho_{1}$ lies in the Teichmüller space
$\mathcal{T}_{\mathbf{c}}(\Gamma)$ (defined in Section 2.2).

######  Theorem B.

Assume there are $m$ peripherals such that $c_{1},\dots,c_{m}$ are hyperbolic
conjugacy classes. For each choice of parameters
$\theta=(\theta_{1},\dots,\theta_{m})\in\mathbb{R}^{m}$, there exists a
homeomorphism

$\Psi^{\theta}:\mathcal{T}(\Gamma)\times\textrm{Rep}^{nf}_{\mathbf{c}}(\Gamma,G)\to\textrm{ASD}_{\mathbf{c}}(\Gamma,G).$

Moreover, the homeomorphism is fiberwise in the sense that for each
$\rho\in\textrm{Rep}_{\mathbf{c}}^{nf}(\Gamma,G)$, it restricts to a
homeomorphism

$\Psi_{\rho}^{\theta}:\mathcal{T}(\Gamma)\times\\{\rho\\}\to
U\times\\{\rho\\}\subset ASD_{\mathbf{c}}(\Gamma,G),$

where $U$ is a non-empty open subset of the Teichmüller space
$\mathcal{T}_{\mathbf{c}}(\Gamma)$.

The mappings $\Psi^{\theta}$ are defined in essentially the same way as the
map $\Psi$ from [41]. Theorem B should be compared with Theorem 1.5. We should
also remark that our results show that almost strict domination implies strict
domination if and only if all $\rho(\zeta_{i})$ are elliptic.

###  1.5. Main theorems: AdS $3$-manifolds

Concerning AdS $3$-manifolds, the following explains the relationship with
spacelike immersions.

######  Theorem C.

Given a $\rho_{1}$-invariant domain $V\subset\mathbb{H}$ on which $\rho_{1}$
acts properly discontinuously, there is a bijection between

  1. (1)

$\rho_{1}\times\rho_{2}$-equivariant continuous maps
$(h,f):V\to(\mathbb{H}\times\mathbb{H},\sigma\oplus(-\sigma))$ satisfying

$d_{\sigma}(f(x),f(y))<d_{\sigma}(h(x),h(y))$

for $x\neq y$,

  2. (2)

and Seifert fibrations $p:\Omega/(\rho_{1}\times\rho_{2}(\Gamma))\to V$, where
$\Omega\subset\operatorname{\textrm{AdS}^{3}}$ is a domain on which
$\rho_{1}\times\rho_{2}$ acts properly discontinuously and such that each
circle fiber lifts to a complete timelike geodesics in
$\operatorname{\textrm{AdS}^{3}}$.

The implication from (1) to (2) should be known to experts. The idea stems
from the work of Guéritaud-Kassel in [19].

######  Remark 1.9.

The proof in [28] that properly discontinuous subgroups of
$\operatorname{\textrm{AdS}^{3}}$ are of the form $\Gamma_{\rho_{1},\rho_{2}}$
rests on their main lemma that there is no $\mathbb{Z}^{2}$-subgroup acting
properly discontinuously. The proof is local, and one can adapt to show that
any torsion-free discrete group acting properly discontinuously on a domain in
$\operatorname{\textrm{AdS}^{3}}$ takes this form.

######  Remark 1.10.

A version of this holds more generally for geometric structures modelled on
some rank $1$ Lie groups. See Section 6.2.

Specializing to almost strict domination, we have the following.

######  Theorem D.

Let $\rho_{1},\rho_{2}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$ be
two reductive representations with $\rho_{1}$ Fuchsian. The following are
equivalent.

  1. (1)

$\rho_{1}$ almost strictly dominates $\rho_{2}$.

  2. (2)

$\rho_{1}\times\rho_{2}$ acts properly discontinuously on a domain
$\Omega\subset\operatorname{\textrm{AdS}^{3}}$ and induces a fibration from
$\Omega/(\rho_{1}\times\rho_{2}(\Gamma))$ onto the interior of the convex core
of $\mathbb{H}/\rho_{1}(\Gamma)$ such that each fiber is a timelike geodesic
circle. Moreover, when there is at least one peripheral $\zeta$ with
$\rho_{1}(\zeta)$ hyperbolic, no such domain in
$\operatorname{\textrm{AdS}^{3}}$ can be continued to give a fibration over a
neighbourhood of the convex core.

  3. (3)

There exists a complete hyperbolic metric $\mu$ on $\Sigma$ and a
$(\rho_{1},\rho_{2})$-equivariant embedded tame maximal spacelike immersion
from
$(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H}\times\mathbb{H},\sigma\oplus(-\sigma))$.

Fixing a collection of conjugacy classes $\mathbf{c}$, there is a fiberwise
homeomorphism

$\Psi:\mathcal{T}_{\mathbf{c}}(\Gamma)\times\textrm{Rep}_{\mathbf{c}}^{nf}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R}))\to\textrm{ASD}_{\mathbf{c}}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R})).$

If we restrict the domain to classes of irreducible representations, the image
identifies with a continuously varying family of AdS $3$-manifolds.

The only piece that doesn’t follow quickly from Theorems A, B, and C is the
implication from (2) to (1). To prove this part, we draw on the work of
Guéritaud-Kassel on maximally stretched laminations [19] and show that the
stretch locus (Definition 6.4) of an optimally Lipschitz map is exactly the
boundary of the convex hull of the limit set.

We comment that in (2), the domain $\Omega$ is all of
$\operatorname{\textrm{AdS}^{3}}$ if and only if every $\rho_{2}(\zeta_{i})$
is elliptic. This is a consequence of [19, Lemma 2.7] and the properness
criteria, Theorem 1.3. Also related to (2), one can get incomplete AdS
$3$-manifolds fibering over larger subsurfaces, but still not extending to the
whole surface, by doing strip deformations (see Section 6.3).

In Section 7, we adapt the work of Alessandrini and Li in [3] to construct the
circle bundle directly. This approach highlights the spacelike maximum surface
and reveals certain properties of the AdS metric, such as the volume. We
explain how the Higgs bundle construction fits in with Theorem C, reconciling
our results with that of [36, Section 7] (or if we consider the work that came
before, [19] and [12] with [3]).

###  1.6. Future Work

There are many directions to go from here. For now we point out two.

  1. (1)

In Definition 1.6, we assume that $\rho_{1}$ is Fuchsian. We can relax this,
and allow for $\rho_{1}$ to be the holonomy of a hyperbolic conifold. One can
find maximal spacelike immersions, again with applications to anti-de Sitter
geometry. We plan to address this in future work.

  2. (2)

In a similar vein, one can remove the tame hypothesis (Definition 2.7). Gupta
builds harmonic diffeomorphisms to crowned hyperbolic surfaces whose Hopf
differentials have poles of order $\geq 3$ at cusps [21]. In forthcoming work,
among other things, we plan to introduce a class of AdS $3$-manifolds that
fiber over crowned surfaces.

###  1.7. Outline

  * •

In Section 2 we give the background content for Theorem A and Theorem B:
spaces of representations, harmonic maps, and maximal surfaces. We also set up
the proof of Theorem A by defining the energy functionals
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$.

  * •

In Section 3, we treat the functional
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$: well-definedness
and the derivative.

  * •

In Section 4 we show that
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ is proper if and
only if $\rho_{1}$ almost strictly dominates $\rho_{2}$ (see 1.3 for the proof
idea).

  * •

We prove Theorem B in Section 5 by studying variations of minimizers of
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ (similar to [41,
Section 2]).

  * •

Section 6 is a change of pace. After giving an overview of the relevant
aspects of AdS geometry, we prove Theorems C and D.

  * •

As stated above, in Section 7 we explore the different construction of the AdS
$3$-manifolds via Higgs bundles.

###  1.8. Acknowledgements

I’d like to thank Qiongling Li, Vlad Markovic, and Nicolas Tholozan for
helpful discussions and correspondence. At the time of writing this second
version of the paper, I am a visiting student at the University of Oxford
while completing my PhD at the California Institute of Technology. I’d like to
thank the community at Oxford for the welcoming environment.

##  2\. Preliminaries

###  2.1. Flat bundles and Euler numbers

For any representation $\rho:\Gamma\to G$, there is an associated fiber bundle
$X_{\rho}\to\Sigma$ whose total space is the quotient of $\tilde{\Sigma}\times
X$ by the action of the deck group $\Gamma$ via

$\gamma\cdot(z,x)=(\gamma\cdot z,\rho(\gamma)x).$

This is naturally endowed with a flat $G$-connection, and upon choosing a
basepoint for the $\pi_{1}$, the holonomy representation is conjugate to
$\rho$. This mapping $\rho\mapsto X_{\rho}$ produces a bijection between the
set of conjugacy classes of representations and that of gauge equivalence
classes of flat $X$-bundles with structure group $G$.

In the case $X=(\mathbb{H},\sigma)$,
$G=\operatorname{\textrm{PSL}}(2,\mathbb{R})$ deformation retracts onto its
maximal compact subgroup $\mathbb{RP}^{1}$, and this yields a reduction of the
structure group to $\mathbb{RP}^{1}$. For closed surfaces, connected
components of $\textrm{Rep}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R}))$
are classified by the Euler number: the evaluation of the Euler class of this
bundle in $H^{2}(\Sigma,\mathbb{Z})$ against the fundamental homology class
$[\Sigma]$. More naively, this integer is computed via the volume

$\textrm{eu}(\rho)=-\frac{1}{2\pi}\textrm{Vol}(\rho)=-\frac{1}{2\pi}\int_{\Sigma}f^{*}\Big{(}\frac{dx^{2}+dy^{2}}{y^{2}}\Big{)}\in\mathbb{Z}\cap[2-2g,2g-2],$

for any $\rho$-equivariant map $f:\tilde{\Sigma}\to\mathbb{H}$.

The algebraic and topological structure of the relative representation space
depends on the isometry types of the peripheral curves. There are three types
of isometries: the translation length of an isometry $\gamma\in G$ is

$\ell(\gamma)=\inf_{x\in X}d_{\nu}(\gamma\cdot x,x),$

and $\gamma$ is

  * •

elliptic if $\ell(\gamma)=0$ and the infimum is attained,

  * •

parabolic if $\ell(\gamma)=0$ and the infimum is not attained, and

  * •

hyperbolic if $\ell(\gamma)>0$.

Elliptic isometries fix points inside $X$, parabolic isometries fix points on
$\partial_{\infty}X$, and a hyperbolic isometry stabilizes a unique geodesic
on which it acts by translation of length $\ell(\gamma)$. Around a peripheral
$\zeta$, we say a representation $\rho$ has hyperbolic, parabolic, or elliptic
mondromy if $\rho(\zeta)$ is hyperbolic, parabolic, or elliptic respectively.

For relative representation spaces, connected components are distinguished
using the relative Euler class, which is a boundeded relative cohomology class
(see [6], [33]). The relative Euler number is the evaluation against the
fundamental relative class, which happens to agree with

(2)
$\textrm{eu}(\rho)=-\frac{1}{2\pi}\textrm{Vol}(\rho)+\Theta(\rho)=-\frac{1}{2\pi}\int_{\Sigma}f^{*}\Big{(}\frac{dx^{2}+dy^{2}}{y^{2}}\Big{)}+\Theta(\rho)$

for any $f$ with a specific behaviour at the cusp (see [27]—$f^{*}$ of the
Kähler form is necessarily in an $L^{2}$-De Rham class). Here, $\Theta(\rho)$
is the total rotation, which depends on the isometry types in $\mathbf{c}$—we
omit the definition since it will be of no use to us, but we invite the reader
to see [6, Section 8]. In the relative case, the volume need not be an
integer, but the total rotation provides a correction term so that the
relative Euler number is contained in
$\mathbb{Z}\cap[\chi(\Sigma),-\chi(\Sigma)]$ [6, Theorem 12]. In analogy with
the closed case, the relative Euler number also arises from the Euler class of
a flat $\mathbb{RP}^{1}$-bundle over a closed surface that is specified by
$\rho$, constructed with special trivializations near the peripheral curves.

###  2.2. Teichmüller spaces

Fix a collection of conjugacy classes
$\mathbf{c}=(c_{i})_{i=1}^{n}\subset\operatorname{\textrm{PSL}}(2,\mathbb{R})$.

######  Definition 2.1.

Suppose $\mathbf{c}$ has no elliptic isometries. Then the Teichmüller space
$\mathcal{T}_{\mathbf{c}}(\Gamma)$ for $\mathbf{c}$ is the component of
$\textrm{Rep}_{\mathfrak{c}}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R}))$
with maximal relative Euler number $-\chi(\Sigma)$. Representations in this
component are called Fuchsian.

######  Remark 2.2.

The (relative) Euler class depends on the orientation of the surface. If we
reverse the orientation, the number multiplies by $-1$. Representations in the
minimal component are also called Fuchsian, but as a matter of convention we
work with the maximal component.

Throughout the paper, by “hyperbolic metric on $\Sigma$” we mean a complete
finite volume hyperbolic metric, unless specified otherwise. We say two such
metrices $\mu_{1},\mu_{2}$ are equivalent if there exists an orientation
preserving diffeomorphism $\psi$, isotopic to the identity, such that
$\psi^{*}\mu_{1}=\mu_{2}$. A Teichmüller space agrees with the space of
classes $[\mu]$ of hyperbolic metrics on $\Sigma$ with cusps and geodesic
boundary components whose lengths are prescribed by $\mathbf{c}$. When we make
no reference to the boundary data, we refer to the Teichmüller space
$\mathcal{T}(\Gamma)$ such that all peripheral monodromy is parabolic. For
most of the paper, when we write $\mathcal{T}(\Gamma)$, we are using the
perspective of hyperbolic metrics, and with $\mathcal{T}_{\mathbf{c}}(\Gamma)$
we are considering representations.

For the sake of convenience, we abuse notation slightly: if
$\mathbf{c}=(c_{i})_{i=1}^{n}$ has elliptic isometries, we write
$\mathcal{T}_{\mathbf{c}}(\Gamma)$ to be the Teichmüller space with boundary
length $\ell(c_{i})$ for each $c_{i}$ with positive length, and cusps for each
$c_{i}$ that is elliptic or parabolic.

We need some terminology for hyperbolic surfaces. Let $\mu$ be any metric on
$\Sigma$, with no constraint on the volume. A cusp region is a neighbourhood
surrounding a puncture of $\Sigma$ that identifies isometrically with

(3) $U(\tau):=\\{z=x+iy:(x,y)\in[0,\tau]\times[a,\infty)\\}/\langle z\mapsto
z+\tau\rangle,$

equipped with the hyperbolic metric $y^{-2}|dz|^{2}$.

######  Definition 2.3.

The convex core $C(\Sigma,\mu)$ of $(\Sigma,\mu)$ is the quotient of the
convex hull of the limit set of $\Gamma$ by the action of $\Gamma$. We denote
the convex hull of the limit set by $\tilde{C}(\Sigma,\mu)$. For quotients
$\mathbb{H}/\rho_{1}(\Gamma)$, we omit the metric from the notation.

It is the minimal convex set such that the inclusion $C(\Sigma,\mu)\to\Sigma$
is a homotopy equivalence. The convex core is a finite volume hyperbolic
surface with finitely many cusps and geodesic boundary components.
$(\Sigma,\mu)$ can be recovered from $C(\Sigma,\mu)$ by attaching infinite
funnels along the boundary components. In the language of representations, the
monodromy of the holonomy representation around a cusp is parabolic, and for a
geodesic boundary component it is hyperbolic.

###  2.3. Harmonic maps

Let $\rho:\Gamma\to G$ be a representation and $\mu$ a metric on $\Sigma$.
Given a $C^{2}$ $\rho$-equivariant map
$f:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$, the derivative $df$ defines a
section of the endomorphism bundle $T^{*}\tilde{\Sigma}\otimes f^{*}TX$. The
Levi-Civita connections on $(\tilde{\Sigma},\tilde{\mu})$ and $(X,\nu)$ induce
a connection on this bundle, and we define the tension field
$\tau(\mu,\nu,f)\in\Gamma(f^{*}TX)$ by

$\tau(\mu,\nu,f)=\textrm{trace}_{\mu}\nabla df.$

######  Definition 2.4.

A $\rho$-equivariant map $f:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ is
harmonic if $\tau(\mu,\nu,f)=0$.

We assume throughout that our metrics are smooth, which implies harmonic maps
are smooth. Existence of equivariant harmonic maps for irreducible
representations of closed surface groups to
$\operatorname{\textrm{PSL}}(2,\mathbb{C})$ was proved by Donaldson [13], who
used the heat flow developed by Eells-Sampson [15]. Corlette and Labourie
extended to more general settings [8] [29].

We simultaneously view $(\Sigma,\mu)$ as a Riemann surface with the
holomorphic structure compatible with $\mu$. In a local holomorphic coordinate
$z$ on $(\tilde{\Sigma},\tilde{\mu})$ in which $\tilde{\mu}=\mu(z)|dz|^{2}$,
the pullback metric $f^{*}\nu$ decomposes as

(4)
$f^{*}\nu=e(\mu,f)\mu(z)dzd\overline{z}+\Phi(f)(z)dz^{2}+\overline{\Phi}(f)(z)d\overline{z}^{2}.$

Since $G$ is acting by isometries and $f$ is equivariant, $f^{*}\nu$ descends
to a symmetric $2$-form on $\Sigma$. We call this the pullback metric, even
though it may have degenerate points.

$e(\mu,f)=\frac{1}{2}\textrm{trace}_{\tilde{\mu}}f^{*}\nu$

is a well-defined function on $\Sigma$ called the energy density with respect
to $\mu$, and $\Phi(f)(z)$ is a holomorphic quadratic differential on
$\Sigma$—a holomorphic section of the symmetric square of the canonical
bundle—called the Hopf differential.

The energy density can be defined for any equivariant map, harmonic or not.
When the base surface is closed, harmonic maps are critical points for the
energy functional

$f\mapsto\int_{\Sigma}e(\mu,f)dA_{\mu}$

(defined on a suitable Sobolev space), where $dA_{\mu}$ is the area form. This
“total energy” is conformally invariant and depends only on the class of $\mu$
in Teichmüller space.

###  2.4. Infinite energy harmonic maps

For surfaces with punctures, existence is proved for different families of
harmonic maps from punctured surfaces in a few places, for example, [21],
[27], [31], and [47]. In [36], we proved the most general result for
equivariant harmonic maps to negatively curved Hadamard manifolds and studied
their properties in detail. The results of [31] and [47] now correspond to the
case of Fuchsian representations. This is summarized in our result below. Set
$\Lambda(\theta)=(1-\theta^{2})-i2\theta$. Reorganize the $p_{i}$ so that
$\rho(\zeta_{i})$ is hyperbolic for $i=1,\dots,m$. For each such $p_{i}$, pick
a $\theta_{i}\in\mathbb{R}$. The vector $\theta=(\theta_{1},\dots,\theta_{m})$
is called a twist parameter.

######  Theorem 2.5 (S. Theorem 1.1 in [36]).

Let $\rho:\Gamma\to G$ be a reductive representation. Then there exists a
$\rho$-equivariant harmonic map
$f^{\theta}:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ such that if $\zeta$ is a
peripheral isometry and $\theta\in\mathbb{R}$, the Hopf differential $\Phi$
has the following behaviour at the corresponding cusp

  * •

if $\rho(\zeta)$ is parabolic or elliptic, $\Phi$ has a pole of order at most
$1$ and

  * •

if $\rho(\zeta)$ is hyperbolic, $\Phi$ has a pole of order $2$ with residue

$-\Lambda(\theta_{i})\ell(\rho(\zeta))^{2}/16\pi^{2}.$

If $\rho$ does not fix a point on $\partial_{\infty}X$, then $f^{\theta}$ is
the unique harmonic map with these properties. If $\rho$ stabilizes a
geodesic, then any other harmonic map with the same asymptotic behaviour
differs by a translation along that geodesic.

The residue of $\Phi$ is the coefficient attached to the $z^{-2}$ term of the
Laurent expansion in a holomorphic coordinate taking the cusp to
$\mathbb{D}^{*}$. This is independent of the coordinate.

The theorem says that a harmonic map is determined by the choice of twist
parameter, with an exception if $\rho$ is reducible. To elaborate on the
nomenclature, take $\zeta_{i}$ with $\rho(\zeta_{i})$ hyperbolic, and choose a
constant speed parametrization $\alpha_{i}$ for the geodesic axis $\beta_{i}$
of $\rho(\zeta_{i})$. Let $\mathcal{C}$ be the cylinder

(5) $\mathcal{C}=\\{(x,y)\in[0,1]^{2}\\}/\langle(0,y)\sim(1,y)\rangle.$

with the flat metric. There is a “model mapping”
$\tilde{\alpha}_{i}^{\theta_{i}}:\mathcal{C}\to\beta_{i}$ defined as follows.
For $\theta_{i}=0$, we set

$\tilde{\alpha}_{i}(x,y)=\alpha_{i}(x),$

a constant speed projection onto the geodesic. For $\theta_{i}\neq 0$,
$\tilde{\alpha}_{i}^{\theta}$ is defined by precomposing $\tilde{\alpha}_{i}$
with the fractional Dehn twist of angle $\theta_{i}$, the map given in
coordinates by

(6) $(x,y)\mapsto(x+\theta_{i}y,y).$

Choosing a cusp neighbourhood $U$ of a $p_{i}$, we take conformal maps
$i_{r}:\mathcal{C}\to U$ that take the boundaries linearly to
$\\{(x,y):y=r\\}$ and $\\{(x,y):y=r+1\\}$. The mappings $f^{\theta}\circ
i_{r}:\mathcal{C}\to(X,\nu)$ are harmonic, and as $r\to\infty$ they converge
in the $C^{\infty}$ sense to a harmonic mapping that differs from
$\tilde{\alpha}_{i}^{\theta_{i}}:\mathcal{C}\to\beta_{i}$ by a constant speed
translation along the geodesic (see [36, Section 5]).

When the representation is Fuchsian, the harmonic map $h^{\theta}$ descends to
a diffeomorphism from $\mathbb{H}\to C(\mathbb{H}/\rho(\Gamma))$ (see [47]).

###  2.5. Spacelike maximal surfaces

Let $\rho_{1}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
$\rho_{2}:\Gamma\to G$ be reductive representations. Then
$\rho_{1}\times\rho_{2}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})\times
G$ defines its own representation.

######  Definition 2.6.

A $\rho_{1}\times\rho_{2}$-equivariant map
$F:(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H}\times X,\sigma\oplus(-\nu))$ is
maximal if the image surface has zero mean curvature. It is spacelike if the
pullback metric $F^{*}(\sigma\oplus(-\nu))$ is non-degenerate and Riemannian.

The vanishing of the mean curvature is equivalent to the condition that $F$ is
harmonic and conformal. Using the product structure, we can write

$F=(h,f),$

where $h,f$ are $\rho_{1},\rho_{2}$-equivariant harmonic maps, and from (4),

$\Phi(F)=\Phi(h)-\Phi(f).$

Since $F$ is conformal, $\Phi(h)$ and $\Phi(f)$ agree.

######  Definition 2.7.

A spacelike maximal surface
$F:(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H}\times,\sigma\oplus(-\nu))$ is
called tame if the Hopf differentials of the harmonic maps have poles of order
at most $2$ at the cusps.

At this point, we can readily prove the easy direction of Theorem A.

######  Lemma 2.8.

If $\rho$ is Fuchsian, the existence of a tame spacelike maximal surface
implies almost strict domination.

###### Proof.

Let $F$ be such a maximal surface and split it as $F=(h,f)$. Note that
$\Phi(h)=\Phi(f)$ implies
$\ell(\rho_{1}(\zeta_{i}))=\ell(\rho_{2}(\zeta_{i}))$ for all $i$. Indeed,
$\ell(\rho_{k}(\zeta_{i}))=0$ if and only if the Hopf differential has a pole
of order at most $1$ at the cusp. And if $\ell(\rho_{k}(\zeta_{i}))>0$, this
is because the residue at each cusp is determined entirely by the choice of
twist parameter and the translation length $\ell(\rho_{k}(\zeta_{i}))$.

From [47], the harmonic map $h$ is a diffeomorphism onto the interior of the
convex hull of the limit set of $\rho_{1}(\Gamma)$. Moreover, the mapping
$f\circ h^{-1}$ is well defined. Since $F$ is spacelike (actually, by [36,
Proposition 3.13], maximal implies spacelike here), $f\circ h^{-1}$ is
strictly $1$-Lipschitz on the interior of the convex hull of the limit set for
the holonomy representation associated to $h^{*}\sigma$. Since the harmonic
maps both converge in a suitable sense to a projection onto the geodesic axis
as in Section 2.4, it has Lipschitz constant exactly $1$ on the boundary
geodesic. By definition, $\rho_{1}$ almost strictly dominates $\rho_{2}$. ∎

######  Remark 2.9.

Note that $\rho_{2}$ cannot be Fuchsian, because almost strict domination
implies $\textrm{eu}(\rho_{2})<\textrm{eu}(\rho_{1})$ by the characterization
(2).

The proof of the other direction of Theorem A will go as follows. Take a pair
$(\rho_{1},\rho_{2})$ such that $\rho_{1}$ almost strictly dominates
$\rho_{2}$. We want to show there is a hyperbolic metric $\mu$ on $\Sigma$
such that, after choosing twist parameters in some way, the associated
$(\rho_{1},\rho_{2})$-equivariant surface from
$(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H}\times X,\sigma\oplus(-\nu))$ is
spacelike and maximal. Clearly, we must have the same twist parameter $\theta$
for both $\rho_{1}$ and $\rho_{2}$, and it turns out we can choose any
$\theta$, as we will now explain. Given a metric $\mu$, let
$h_{\mu}^{\theta},f_{\mu}^{\theta}$ be harmonic maps for $\rho_{1},\rho_{2}$
respectively with twist parameter $\theta$. Define

$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}:\mathcal{T}(\Gamma)\to\mathbb{R}$

by

$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}(\mu)=\int_{\Sigma}e(\mu,h_{\mu}^{\theta})-e(\mu,f_{\mu}^{\theta})dA_{\mu}.$

This does not depend on the metric $\mu\in[\mu]$, by conformal invariance
(Section 3.1). When $\rho_{2}$ is reducible, harmonic maps are not unique, but
the energy density does not depend on the choice of harmonic map, so the
integral is well defined. Apriori, it is not given that the integral defining
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}(\mu)$ is finite, for
the harmonic maps themselves could have infinite energy. Finiteness will be
proved in Proposition 3.2. The main goal of Section 3 is to prove Proposition
2.10.

######  Proposition 2.10.

Given $(\rho_{1},\rho_{2})$ with
$\ell(\rho_{1}(\zeta_{i}))=\ell(\rho_{2}(\zeta_{i}))$ for all $i$, the
functional
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}:\mathcal{T}(\Gamma)\to\mathbb{R}$
is differentiable with derivative at a hyperbolic metric $\mu$ given by

$d\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}[\mu](\psi)=-4\operatorname{Re}\langle\Phi(h_{\mu}^{\theta})-\Phi(f_{\mu}^{\theta}),\psi\rangle.$

Thus, critical points of
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ correspond to
maximal surfaces. And by [36, Proposition 3.13], any critical point is a
spacelike immersion. In Section 4, we prove that the almost strict domination
hypothesis implies there is a metric $\mu$ that minimizes
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$, and that this is
the only critical point. Granting this, we can discuss uniqueness.

######  Proposition 2.11.

Suppose $\rho_{1}$ almost strictly dominates $\rho_{2}$ and that $\rho_{2}$ is
irreducible. Assume that for every choice of twist parameter $\theta$, each
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ admits a unique
critical point. Then every spacelike maximal immersion is of the form
$(h_{\mu_{\theta}}^{\theta},f_{\mu_{\theta}}^{\theta})$, where $\mu_{\theta}$
is the minimizer for
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$. If $\rho_{2}$ is
reducible, then for each $\theta$ there is a a $1$-parameter family of tame
maximal surfaces, and each one is found by translating $f_{\mu}^{\theta}$
along a geodesic axis.

###### Proof.

In the irreducible case, let
$F=(h^{\prime},f^{\prime}):(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ be a
spacelike maximal immersion. By the uniqueness statement in Theorem 2.5,
$h^{\prime}$ must be of the form $h_{\mu}^{\theta}$ for some parameter
$\theta$. Thus $f=f_{\mu}^{\theta}$, and by our assumption, $\mu$ is the
unique critical point for
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$.

In the reducible case, all $\rho_{2}(\zeta_{i})$ that are hyperbolic must
translate along the same geodesic. Since translating along the geodesic does
not change the energy density or the Hopf differential, the one minimizer for
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ is the only one that
yields maximal surfaces. ∎

##  3\. The derivative of the energy functional

In this section we prove Proposition 2.10. We assume $\Sigma$ has only one
puncture $p$ with peripheral $\zeta$—the analysis is local so the general case
is essentially the same. When $\rho_{1}(\zeta),\rho_{2}(\zeta)$ are
hyperbolic, we assume the twist parameter $\theta$ is $0$, and we also write
$\ell$ in place of $\ell(\rho_{k}(\zeta))$. In Section 3.7, we explain the
adjustments for the general case. With these assumptions, it causes no harm to
write $\operatorname{\mathcal{E}}$ for
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$. Not only in this
section but for the rest of the paper, we write $h_{\mu},f_{\mu}$ for
$h_{\mu}^{\theta},f_{\mu}^{\theta}$ when $\theta$ is zero.

###  3.1. Analytic preliminaries

Let $f:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ be harmonic. According to the
splitting of the complexified tangent bundle
$T\tilde{\Sigma}\otimes\mathbb{C}$ into $(1,0)$ and $(0,1)$ components, the
energy density decomposes as

$e(\mu,f)=H(\mu,f)+L(\mu,f),$

the holomorphic energy and the anti-holomorphic energy. When $(X,\nu)$ is a
surface with compatible complex structure, we can define these quantities in
the usual way as norms of holomorphic and anti-holomorphic derivatives. In
general, we set $H(\mu,f)\geq L(\mu,f)$ to be the unique solutions of the
system $x+y=e(\mu,f)$, $xy=\mu^{-2}|\Phi|^{2},$ such that $x\geq y$, which are
well-defined since $e(\mu,f)-2\mu^{-2}|\Phi|^{2}\geq 0$. When the context is
clear, we write $H=H(\mu,f)$, and the same for other analytic quantities. We
collect some notations and relations that, in the sequel, we use without
comment.

  * •

When unspecified, $C$ denotes a constant that may grow in the course of a
proof.

  * •

Setting $x=x_{1},y=x_{2}$, let
$e_{\alpha\beta}(f)=f^{*}\nu\Big{(}\frac{\partial f}{\partial
x_{\alpha}},\frac{\partial f}{\partial x_{\beta}}\Big{)}$, so that
$e(\mu,f)=\frac{1}{2}\mu^{\alpha\beta}e_{\alpha\beta}(f)$.

  * •

The Jacobian $J=J(\mu,f)$ satisfies $J=H-L$.

  * •

For a conformal metric $\mu=\mu(z)|dz|^{2}$,
$||\Phi||^{2}:=\mu^{-2}|\Phi|^{2}=HL$.

  * •

Set $|\psi|=L^{1/2}/H^{1/2}$. If $(X,\nu)=(\mathbb{H},\sigma)$ and $\rho$ is
Fuchsian, $\psi=\frac{f_{\overline{z}}}{f_{z}}$ is the Beltrami form.

We will make use of the following results.

  * •

Total energy is conformally invariant: if
$i:(\Omega,g)\to(\tilde{\Sigma},\tilde{\mu})$ is conformal, then $f\circ i$ is
harmonic.

  * •

If $f_{1},f_{2}$ are both harmonic, then the distance function $p\mapsto
d_{\nu}(f_{1}(p),f_{2}(p))$ descends to a subharmonic function on
$(\Sigma,\mu)$.

  * •

Cheng’s lemma [7]: let
$B_{\tilde{\mu}}(z,r)\subset(\tilde{\Sigma},\tilde{\mu})$ be a metric ball and
$f:B_{\tilde{\mu}}(z,r)\to(X,\nu)$ a harmonic map whose image lies in a
$\nu$-ball of radius $R_{0}$. Then

$e(\mu,f)\leq R_{0}^{2}\cdot\frac{(1+r)^{2}}{r^{2}}.$

  * •

Proposition 3.13 from [36]: if $h$ is a Fuchsian harmonic map with same Hopf
differential as $f$ (this exists by Theorem 1.4 of [36]), then
$H(\mu,f),L(\mu,f)\leq H(\mu,h)$.

We prepare notation for dealing with cusps. On a hyperbolic surface
$(\Sigma,\mu)$, there is a cusp region
$U(\tau):=\\{z=x+iy:(x,y)\in[0,\tau]\times[2,\infty)\\}/\langle z\mapsto
z+\tau\rangle,$ where $\tau=\tau_{\mu}$ depends on $\mu$. For $r\geq 2$, we
define $(\Sigma_{r},\mu)$ to be $\Sigma\backslash\\{z\in U(\tau):y>r\\}$, and
we put $(\tilde{\Sigma}_{r},\tilde{\mu})$ to be the preimage in
$\tilde{\Sigma}$. Note that, as a set of points, this depends on $\mu$. As in
Section 2.4, $\mathcal{C}$ is a conformal cylinder, and when we say a
“conformal cylinder for $\mu$,” we mean the length is adjusted to be $\tau$
and the height is $1$, so that there are conformal maps $\mathcal{C}\to
U(\tau)$. We define the maps

$i_{r}:\mathcal{C}\to\Sigma_{r+1}\backslash\Sigma_{r}\subset\Sigma$

by $(x,y)\mapsto x+i(y+r)$, which are used to study asymptotics of harmonic
maps.

It is often helpful to perturb the metric in the cusp. A metric $\mu$ is
expressed in the coordinate of $U(\tau)$ as $\mu(z)=y^{-2}|dz|^{2}$. The flat-
cylinder metric $\mu^{f}$ is defined by $\mu$ in $\Sigma_{2}=\Sigma\backslash
U$, the flat metric $|dz|^{2}$ in the cusp coordinates on
$\Sigma\backslash\Sigma_{3}$, and smoothly interpolated in between. This is
conformally equivalent to $\mu$, so harmonic maps for $\mu$ and $\mu^{f}$ are
the same.

We now recall the construction of infinite energy harmonic maps from [36,
Section 5] for a reductive representation $\rho$ such that $\rho(\zeta)$ is
hyperbolic with $\theta=0$. Denote by $\beta$ the geodesic axis of
$\rho(\zeta)$ and fix a constant speed parametrization
$\alpha:[0,\tau]\to\beta$ such that $\rho(\zeta)\alpha(0)=\alpha(\tau).$ We
define mappings $f_{r}$ to be equivariant harmonic maps on
$(\tilde{\Sigma}_{r},\tilde{\mu})$ with equivariant boundary values specified
by $\alpha$ on $\partial\tilde{\Sigma}_{r}$. In keeping with the notation of
[36], we set $\varphi=f_{2}$, extend $\varphi$ vertically into the cusp by
$\varphi(x,y)=\alpha(x)$ in a fundamental domain for $\Gamma$, and extend
equivariantly. In this cusp, $\varphi$ satisfies

(7) $e(\mu,\varphi)=\frac{\ell^{2}}{2\tau^{2}}.$

In [36, Proposition 5.1], we show there is a uniform bound

(8)
$\int_{\Sigma_{s}}e(\mu,f_{r})dA_{\mu}\leq\int_{\Sigma_{s}}e(\mu,\varphi)dA_{\mu}=\int_{\Sigma_{2}}e(\mu,\varphi)dA_{\mu}+\frac{(s-2)\ell^{2}}{2\tau}.$

Convergence on compacta in the $C^{\infty}$ topology then follows easily: the
bound on the total energy gives bounds on the energy density, and via the
elliptic theory one promotes to estimates on higher derivatives. One then use
some now well-understood arguments to show that if $\rho$ is irreducible, all
maps take a compact set into the same compact set, at which point one can use
Arzèla-Ascoli and a diagonal argument. In the reducible case we may need to
first renormalize $f_{r}$ via well-chosen translations $g_{r}$. Note that if
we choose a different constant speed parametrization $\alpha$ for our
approximation maps, we still get convergence to the same harmonic map.

Next we turn to finiteness of $\operatorname{\mathcal{E}}$. If no monodromy is
hyperbolic, then harmonic maps have finite energy. So suppose $\rho(\zeta)$ is
hyperbolic.

######  Proposition 3.1.

Let $f:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ be a $\rho$-equivariant
harmonic map. Then there exists $C,c,y_{0}>0$ such that, in the cusp
coordinates (3), for all $y\geq y_{0}$, the inequality

$|e(\mu^{f},f_{\mu})(x,y)-\ell^{2}/2\tau^{2}|<Ce^{-cy}$

holds.

###### Proof.

We implicitly work with the metric $\mu^{f}$. Write

(9)
$e=H+L=H^{1/2}L^{1/2}\Big{(}\frac{H^{1/2}}{L^{1/2}}+\frac{L^{1/2}}{H^{1/2}}\Big{)}=|\Phi|(|\psi|^{-1}+|\psi|).$

Since $\Phi$ has a pole of order at most $2$, changing coordinates to the cusp
$[0,1]\times[1,\infty)$ (see Section 3.2 below) gives the expression

(10) $|\Phi|=\frac{\ell^{2}}{4\tau^{2}}+O(e^{-2\pi y}).$

From [47, page 513], if $f$ is Fuchsian, we have the estimate

$1-|\psi|^{2}=O(e^{-\ell/2y}),$

and hence

$1-O(e^{-cy})\leq|\psi|^{2}\leq 1.$

If $\rho$ is not Fuchsian, we find the Fuchsian harmonic map
$h:(\tilde{\Sigma},\tilde{\mu})\to(\mathbb{H},\sigma)$ with the same Hopf
differential. Then from $H(f)L(f)=H(h)L(h)$ and [36, Proposition 3.13] we get

$L(h)\leq L(f),H(f)\leq H(h),$

which implies

$|\psi(h)|\leq|\psi(f)|\leq|\psi(h)|^{-1},$

and furthermore

(11) $1-O(e^{-cy})\leq|\psi|^{2}\leq\frac{1}{1-O(e^{-cy})}.$

Inserting (10) and (11) into (9) gives

$e(\mu^{f},f)=\frac{\ell^{2}}{2\tau^{2}}+O(e^{-c_{1}y}),$

as desired. ∎

The following is now evident.

######  Proposition 3.2.

For a fixed hyperbolic metric $\mu$, the integral defining
$\operatorname{\mathcal{E}}(\mu)$ is finite. That is,
$\operatorname{\mathcal{E}}:\mathcal{T}(\Gamma)\to\mathbb{R}$ is well defined.

###  3.2. Tangent vectors of Teichmüller space.

To do analysis on Teichmüller space, we need a tractable way to study tangent
vectors. Fix a metric $\mu_{0}$ on $\Sigma$ and let $z=x+iy$ be a conformal
coordinate for the compatible holomorphic structure, so that
$\mu_{0}=\mu_{0}(z)|dz|^{2}$. In this subsection we describe variations of the
metric

$\mu^{\prime}=\mu+\dot{\mu}.$

Since we are working with harmonic maps, which are conformally invariant, we
are permitted to work in a specified conformal class. In particular, we may
restrict to variations through complete finite volume hyperbolic metrics. The
hyperbolic condition is satisfied if and only if

$\dot{\mu}_{11}+\dot{\mu}_{22}=0$

and

$\phi(z)dz^{2}=(\dot{\mu}_{11}-i\dot{\mu}_{12})dz^{2}$

is a holomorphic quadratic differential. When $\Sigma$ has a puncture, the
necessary and sufficient condition on $\dot{\mu}$ to preserve the complete
finite volume property is that $\phi$ has a pole of order at most 1 at the
cusp.

######  Remark 3.3.

We have described the tangent space to Teichmüller space at $(\Sigma,\mu)$ as
the space of $L^{1}$-integrable quadratic differentials on the conjugate
Riemann surface. This characterization coincides with the one we get from the
Bers embedding into $\mathbb{C}^{3g-3+n}$. If $\mu(z)$ is a conformal metric,
the mapping

$\phi(z)dz^{2}\mapsto\mu(z)^{-1}\phi(\overline{z})\frac{d\overline{z}}{dz}$

yields the usual identification with the space of harmonic Beltrami forms.

We work out the growth condition on $\phi$ in the cusp coordinates for
$U(\tau)$. For convenience put $\tau=1$. The mapping $z\mapsto w(z)=e^{2\pi
iz}$ takes a vertical strip to a punctured disk

$\\{0\leq
x<1,y>h,y^{-2}|dz|^{2}\\}\to\\{0<|w|<e^{-h},|w|^{-2}(\log|w|)^{2}|dw|^{2}\\}$

holomorphically and isometrically. A meromorphic quadratic differential in the
disk with a pole of order at most $1$ is written

$\Phi=\phi(w)=(a_{-1}w^{-1}+\varphi(w))dw^{2},$

with $\varphi$ holomorphic. Applying the above holomorphic mapping, the
differential transforms according to

(12) $\Phi=\phi(w(z))\Big{(}\frac{\partial w(z)}{\partial
z}\Big{)}^{2}dz^{2}=\phi(e^{iz})(ie^{iz})^{2}dz^{2}=-(a_{-1}e^{iz}+e^{2iz}\varphi(e^{iz}))dz^{2}.$

Thus, any admissible variation decays exponentially in the cusp as we take
$y\to\infty$.

We also need to describe the inverse variation

$(\mu^{\prime})^{\alpha\beta}=\mu^{\alpha\beta}+\dot{\mu}^{\alpha\beta}.$

In the Einstein notation, the relation
$\mu^{\alpha\beta}\mu_{\beta\gamma}=\delta_{\alpha\gamma}$ gives
$\dot{\mu}^{\alpha\beta}\mu_{\beta\gamma}+\mu^{\alpha\beta}\dot{\mu}_{\beta\gamma}=0$,
and hence

$\dot{\mu}^{\alpha\beta}=-\mu^{\alpha\rho}\mu^{\beta\tau}\dot{\mu}_{\tau\rho}.$

When $\mu$ is conformal, this returns

$\dot{\mu}^{\alpha\beta}=-\mu^{-2}\dot{\mu}_{\alpha\beta}.$

We derive that, in the cusp coordinates,

(13) $\dot{\mu}^{\alpha\beta}=-y^{4}\dot{\mu}_{\alpha\beta},$

so the decay is still exponential. From these descriptions we deduce the
following.

######  Proposition 3.4.

Suppose $g$ is a finite energy equivariant map with respect to a finite volume
hyperbolic metric $\mu$. Then it also has finite energy for any other metric.

###### Proof.

By conformal invariance of energy, we are permitted to work with complete
finite volume hyperbolic metrics. It is enough to prove the claim for metrics
that are as close as we like to $\mu$. For then we can connect $\mu$ to any
other metric $\mu^{\prime}$ via a smooth path in the Teichmüller space, cover
this path with finitely many small balls, and argue inductively. That is, we
can assume $\mu^{\prime}=\mu+\dot{\mu}$, for some small variation $\dot{\mu}$.
In a local coordinate $z=x+iy$, we write
$|\mu+\dot{\mu}|=\textrm{det}(\mu+\dot{\mu})$, so that the volume form is

$dA_{\mu+\dot{\mu}}=\sqrt{|\mu+\dot{\mu}|}dz\wedge
d\overline{z}=\sqrt{|\mu|-|\phi|^{2}}dz\wedge d\overline{z},$

where $\phi$ is the holomorphic quadratic differential associated to
$\dot{\mu}$. For simplicity, let’s restrict to a $C^{1}$ map $g$. In a cusp,

$\displaystyle 2e(\mu+\dot{\mu},\nu,g)\sqrt{|\mu+\dot{\mu}|}$
$\displaystyle=\sqrt{|\mu|-|\phi|^{2}}(\mu+\dot{\mu}^{\alpha\beta})e_{\alpha\beta}(g)$
$\displaystyle=\frac{\sqrt{|\mu|-|\phi|^{2}}}{\sqrt{|\mu|}}\cdot\sqrt{|\mu|}e(\mu,g)+\sqrt{|\mu|-|\phi|^{2}}\dot{\mu}^{\alpha\beta}e_{\alpha\beta}(g).$

By hypothesis, the first term is uniformly bounded and converges to an
integrable quantity. For the second term, using the flat cylinder metric we
gather

$\int_{\Sigma}(e_{11}+e_{22})dA_{\mu^{f}}<\infty.$

By Cauchy-Schwarz, $e_{12}$ is integrable as well. Meanwhile, the factor
$\sqrt{|\mu|-|\phi|^{2}}\dot{\mu}^{\alpha\beta}$ decays exponentially as we go
into the cusp. ∎

The proof above shows that the bound depends only on the energy of $g$ with
respect to $\mu$ and the Teichmüller distance of the new metric to $\mu$ (see
[1, Chapter 5] for the definition).

###  3.3. Variations: finite energy harmonic maps

In the proof of Proposition 2.10, we need to know that for a variation of
hyperbolic metrics $t\mapsto\mu_{t}$, $f_{\mu_{t}}\to f_{\mu}$ pointwise as
$t\to 0$. In this subsection and the next, we show uniform convergence of
harmonic maps on compacta. To do so, we verify that analytic results for
harmonic maps can be made uniform in the source metric. We start with the case
of elliptic and parabolic monodromy at the cusp. Throughout, let
$\rho:\Gamma\to G$ be a reductive representation.

######  Proposition 3.5.

Assume the monodromy is elliptic or parabolic and fix an admissible metric
$\mu_{0}$. For all $\mu$, there exists a $C_{k}>0$ depending only on the
Teichmüller distance from $\mu_{0}$ to $\mu$ such that for all $k>0$,

$|(\nabla^{\mu^{f},\nu})^{(k)}df_{\mu}|_{\mu^{f}}\leq C_{k}.$

###### Proof.

We showed in [36, Proposition 3.8] that one can always find finite energy
harmonic maps. Take such a map $g:(\tilde{\Sigma},\tilde{\mu}_{0})\to(X,\nu)$,
which by Lemma 3.4 is finite energy for any other metric, with a bound
depending on the Teichmüller distance to $\mu_{0}$. By the energy minimizing
property in negatively curved spaces,

$\int_{\Sigma}e(\mu^{f},f_{\mu})dA_{\mu^{f}}=\int_{\Sigma}e(\mu,f_{\mu})dA_{\mu}\leq\int_{\Sigma}e(\mu,g)dA_{\mu}\leq
C.$

For the uniform bounds on the energy density, independent of $\mu$, one uses a
Harnack-type inequality, say, from [38, page 171], that only depends on
uniform quantities: the Ricci curvature and the injectivity radius. Since we
work with the flat-cylinder metric $\mu^{f}$ as opposed to the hyperbolic
metric $\mu$, we do have uniform control on the injectivity radius. The
estimates on higher order derivatives then come from the elliptic theory on
the Sobolev space adapated to $(\Sigma,\mu_{0})$ (see [35, Chapter 10]). It is
clear from the general theory that the implicit constants in these estimates
can be made uniform in $\mu$. ∎

######  Lemma 3.6.

Suppose $\rho$ is irreducible. Let $K\subset\tilde{\Sigma}$ be compact and
suppose $f:(K,\mu)\to(X,\nu)$ is $C$-Lipschitz for some $C>0$. Then there
exists a compact set $\Omega(K,C)\subset X$ that does not depend on the map
$f$ such that $f(K)\subset\Omega(K,C)$.

This is essentially carried out in the proof of Theorem 5.1 in [36] (which is
a modification of the argument from the main theorem of [13]), but with some
slightly different assumptions. We sketch a proof for the reader’s
convenience.

###### Proof.

We show the claim holds for a single point $x_{0}$, and then the Lipschitz
control promotes the result to general compact sets. Since $\rho$ is
irreducible, there exists $\gamma\in\Gamma$ such that
$\rho(\gamma)\xi\neq\xi$. For each $x\in K$, $\gamma$ may be represented by a
loop $\gamma_{x}:[0,L_{x}]\to\Sigma$ based at $\pi(x)$. Since $K$ is compact,
there is an $L>0$ such that $L_{x}\leq L$ for all $x$. Now choose a
neighbourhood $B_{\xi}\subset X\cup\partial_{\infty}X$ such that

$d_{\nu}(X\cap B_{\xi},\rho(\gamma)X\cap B_{\xi})>CL.$

This implies that for any $x\in K$, $f(x)$ cannot lie in $B_{\xi}$. Repeating
this procedure and using compactness of $\partial_{\infty}X$, we get a
neighbourhood of $\partial_{\infty}X$ in $X\cup\partial_{\infty}X$ that $f(K)$
cannot enter. We then take $\Omega(K,C)$ to be the complement of this
neighbourhood. ∎

######  Lemma 3.7.

Let $K\subset\Sigma$ be compact. Then there is a choice of harmonic maps
$f_{\mu}$ that vary continuously on lifts of $K$ inside
$(\tilde{\Sigma},\tilde{\mu})$ in the $C^{\infty}$ topology.

###### Proof.

We argue by contradiction. First if $\rho$ is irreducible, suppose there
exists $\delta>0$ and sequences $(k_{n})_{n=1}^{\infty}\subset K$,
$r_{n}\to\infty$, and $\mu_{n}\to\mu$ such that

(14) $d(f_{\mu_{n}},f_{\mu})\geq\delta$

for all $n$. Then by Lemma 3.5 we have uniform derivative bounds on each
$f_{\mu_{n}}$ and by Lemma 3.6 they all take a lift of $K$ to $\tilde{\Sigma}$
into a compact subset of $(X,\nu)$. By Arzèla-Ascoli we see the $f_{\mu_{n}}$
$C^{\infty}$-converge in $K$ along a subsequence to a limiting harmonic map
$f_{\infty}$. By equivariance, we have this same convergence on the whole
preimage of $K$.

We now show $f_{\infty}=f$, which contradicts (14). Taking a compact
exhaustion of $\Sigma$ and applying the same argument on each compact set, the
maps $f_{\mu_{n}}$ subconverge on compact subsets of $\tilde{\Sigma}$ to a
limiting finite energy harmonic map $f_{\infty}^{\prime}$ that agrees with
$f_{\infty}$ on lifts of $K$. By uniqueness for finite energy maps we get
$f_{\infty}=f$.

If $\rho$ is reducible, we can recenter the harmonic maps via translations
along the geodesic so that they take $K$ into a fixed compact set (see the
proof of [36, Propistion 5.1] for this routine procedure), and then repeat the
argument above. ∎

###  3.4. Variations: infinite energy harmonic maps

Now we treat harmonic maps for representations $\rho$ with hyperbolic
monodromy at the cusp. Recall the map $\varphi$ from Section 3.1, which for a
metric $\mu$ we now denote
$\varphi_{\mu}:(\tilde{\Sigma}_{2},\tilde{\mu})\to(X,\nu)$. We emphasize that
the boundary curve for $(\tilde{\Sigma_{2}},\tilde{\mu})$ is varying with
$\tilde{\mu}$

######  Lemma 3.8.

Near a base hyperbolic metric $\mu_{0}$, $\varphi_{\mu}$ can be chosen so that
the association $\mu\to\varphi_{\mu}$ is continuous in the $C^{\infty}$
topology .

###### Proof.

Since the metrics are varying smoothly, the lifts of $\mu$-horocycles vary
smoothly with $\mu$. Indeed, in conformal cusp coordinates as in (5), the
$\mu$-horocycles are just curves with $y_{\mu}$ constant. Smoothness here thus
comes from the regularity theory for the Beltrami equation (if we follow the
approach of Ahlfors and Bers for finding isothermal coordinates on a
hyperbolic surface).

From the standard arguments (Section 3.1), the result amounts to choosing
$\mu\to\varphi_{\mu}$ so that we have a uniform total energy bound on
compacta, independent of $\mu$, and such that the boundary data varies
continuously. It would follow that as $\mu\to\mu_{0}$, the harmonic maps
subconverge to a harmonic map, and continuity in the boundary values shows
this is exactly $\varphi$. One can then modify the contradiction argument from
Lemma 3.7 to see $C^{\infty}$ convergence on compacta along the whole
sequence.

By the energy minimizing property, it suffices to construct a family of maps
$\psi_{\mu}$ with suitably chosen boundary values and uniformly controlled
energy. To build $\psi_{\mu}$, let $f^{\mu}$ be the unique quasiconformal
diffeomorphism between $(\Sigma_{2},\mu)$ and $(\Sigma_{2},\mu_{0})$ that, in
the cusp coordinates, takes
$(i2,\tau_{\mu}+i2,\infty)\mapsto(i2,\tau_{\mu_{0}}+i2,\infty)$. Then
$\psi_{\mu}=\varphi_{\mu_{0}}\circ f^{\mu}$ has the correct boundary values.
By our choice of normalizations, $f^{\mu}$ converges to the identity as
$\mu\to\mu_{0}$. This gives an energy bound on $f^{\mu}$, and moreover we get
uniform bounds for $\psi_{\mu}$. ∎

We can now prove the analogue of Lemma 3.7 for infinite energy harmonic maps.

######  Lemma 3.9.

Let $K\subset\Sigma$ be compact. Then there is a choice of harmonic maps
$f_{\mu}$ that vary continuously on lifts of $K$ in the $C^{\infty}$ topology.

###### Proof.

From the estimate (8) and the Fatou lemma we get

$\int_{\Sigma_{s}}e(\mu,f_{\mu})dA_{\mu}\leq\int_{\Sigma_{2}}e(\mu,\varphi_{\mu})dA_{\mu}+\frac{(r-2)\ell^{2}}{2\tau_{\mu}}.$

Via the lemma above,

$\int_{\Sigma_{s}}e(\mu,f_{\mu})dA_{\mu}\leq
C+\frac{2(r-2)\ell^{2}}{2\tau_{\mu_{0}}},$

for $\mu$ close enough to $\mu_{0}$. Then from the discussion in Proposition
3.5, we get uniform control on all derivatives on $K$.

If $\rho$ is irreducible we apply Lemma 3.6, and if $\rho$ is reducible we
rescale by translations along the geodesic so that they take $K$ into a fixed
compact set. The argument from Section 3.1 then shows that for any sequence
$\mu_{n}\to\mu$, the harmonic maps subconverge on compacta in the $C^{\infty}$
sense to a limiting harmonic map $f_{\infty}$. From [36, Lemma 5.2],
$d(f_{\mu_{n}},\varphi_{\mu_{n}})$ is uniformly bounded, and an investigation
of the proof of this lemma shows it is maximized on $\Sigma_{2}$. By uniform
convergence on compacta, $d(f_{\infty},\varphi)$ is also (non-strictly)
maximized on $\Sigma_{2}$, and hence it is globally finite. According to the
classification in Theorem 2.5 and the explanation of the twist parameter,
$f_{\mu}$ is the only harmonic map with this property: if we precompose an
equivariant map with a non-trivial fractional Dehn twist, the distance between
the original map and the new map grows without bound as we limit toward a lift
of the puncture on the boundary at infinity. ∎

We deduce the following.

######  Lemma 3.10.

Given a metric $\mu_{0}$, there exists a uniform $C_{k}>0$ such that

$|(\nabla^{\mu^{f},\nu})^{(k)}df_{\mu}|_{\mu^{f}}\leq C_{k}$

everywhere on $\Sigma$, with $C_{k}$ depending on the Teichmüller distance to
$\mu_{0}$.

###### Proof.

By uniform convergence on compacta, we have uniform control on the distance to
$\varphi_{\mu}$. We then couple the uniform energy bounds on $\varphi_{\mu}$
with Cheng’s lemma to get uniform energy bounds, and then we appeal to the
elliptic theory (as we have done many times at this point). ∎

###  3.5. Energy minimizing properties

It is well known that finite energy equivariant harmonic maps minimize the
total energy among other equivariant maps. Here we show that this extends in
some sense to the infinite energy setting.

######  Definition 3.11.

Let $g:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ be $\rho$-equivariant and let
$\alpha$ be a parametrization of the geodesic axis of the image of the
peripheral, say $\beta$. We say $g$ converges at $\infty$ to $\alpha$ if,
after precomposing with the conformal mapping
$i_{r}:\mathcal{C}\to\tilde{\Sigma}_{r}\backslash\tilde{\Sigma}_{r-1}$,
$g\circ i_{r}:\mathcal{C}\to\beta$ converges in $C^{0}$ as $r\to\infty$ to a
mapping $\tilde{\alpha}:\mathcal{C}\to\beta$ given by

(15) $\tilde{\alpha}(x,y)=\alpha^{\prime}(x),$

where $\alpha^{\prime}(x)$ is some translation of $\alpha$ along $\beta$.

######  Proposition 3.12.

Let $f$ be the harmonic map from Theorem 2.5 whose Hopf differential has real
reside at the cusp. Suppose that a locally Lipschitz map
$g:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ converges at $\infty$ to $\alpha$.
Then $e(\mu,f)-e(\mu,g)$ is integrable with respect to $\mu$ and

$\int_{\Sigma}e(\mu,f)-e(\mu,g)dA_{\mu}\leq 0.$

######  Remark 3.13.

The lemma holds provided $g$ is weakly differentiable and these weak
derivatives are locally $L^{2}$. If $g$ is not harmonic, then this inequality
is strict.

###### Proof.

Let $g_{r}$ be the harmonic map with the same boundary values as $g$ on
$\Sigma_{r}$ and set $\phi:(\tilde{\Sigma},\tilde{\mu})\to\mathbb{R}$ to be
any $\Gamma$-invariant function that is equal to
$\mu^{-1}\ell^{2}/2\tau_{\mu}$ in $\mu$-conformal coordinates after
$\tilde{\Sigma}_{2}$. For example, we can take the $\mu$-energy of the
harmonic map $\varphi$. Then

$\int_{\Sigma_{r}}e(\mu,g_{r})dA_{\mu}\leq\int_{\Sigma_{r}}e(\mu,g)dA_{\mu},$

and hence

(16) $\liminf_{r\to\infty}\int_{\Sigma_{r}}e(g_{r})-\phi
dA_{\mu}\leq\int_{\Sigma}e(g)-\phi dA_{\mu}.$

Without loss of generality, $\alpha$ is the limiting parametrization of the
geodesic for $g$. Let $\varphi_{r}$ be the unique harmonic map on
$\tilde{\Sigma}_{r}$ with boundary values $\alpha$ on
$\partial\tilde{\Sigma}_{r}$, defined by using $\alpha$ on one lift and then
extending equivariantly. The distance function

$p\mapsto d(\varphi_{r}(p),g_{r}(p))$

is subharmonic and hence maximized on $\partial\Sigma_{r}$. By the convergence
property of $g$, we can thus assume that for $r$ large enough,

$d(\varphi_{r}(p),g_{r}(p))\leq 1$

for all $p$. By Cheng’s lemma, we obtain uniform bounds on the energy density
of $g_{r}$ in terms of that of $\varphi_{r}$ on compact sets. Since
$\varphi_{r}\to f$ locally uniformly on compacta, $g_{r}$ also subconverges
locally uniformly on compacta, and the limiting map is harmonic. As $g_{r}$
has bounded distance to $\varphi$, as above the uniqueness result Theorem 2.5
shows the limit must be $f$. Returning to our integrals (16), Fatou’s lemma
then yields

$\displaystyle\int_{\Sigma}e(\mu,f)-\phi dA_{\mu}$
$\displaystyle=\int_{\Sigma_{2}}e(\mu,f)-\phi
dA_{\mu}+\int_{\Sigma\backslash\Sigma_{2}}e(\mu,f)-\phi dA_{\mu}$
$\displaystyle=\int_{\Sigma_{2}}e(\mu,f)-\phi
dA_{\mu}+\int_{2}^{\infty}\Big{(}\int_{0}^{\tau_{\mu}}e(\mu^{f},f)(x,y)-\phi(x,y)dx\Big{)}dy$
$\displaystyle\leq\liminf_{r\to\infty}\int_{\Sigma_{r}}e(g_{r})-\phi
dA_{\mu}.$

(16) then gives

$\int_{\Sigma}e(\mu,f)-\phi dA_{\mu}\leq\int_{\Sigma}e(g)-\phi dA_{\mu}.$

If the integral on the right is infinite then the result of our lemma is
obvious, and if it is finite then we can rearrange to get the desired
inequality. ∎

######  Lemma 3.14.

In the setting above, work with a metric $\mu_{0}$. Then for any other $\mu$,
$f_{\mu}$ satisfies the hypothesis above.

###### Proof.

Let $w$ be a complex coordinate parametrizing a cusp as a quotient of a
vertical strip of length $1$. Recall $z$ is defined on a strip $y\geq a$,
$0\leq x\leq\tau_{\mu}$, and similar for $z_{\mu}$ (we take the same $a$). We
can assume $z=f^{\lambda_{0}}(w)$, $z_{\mu}=f^{\lambda}(w)$, where
$f^{\lambda_{0}}$, $f^{\lambda}$ are smooth quasiconformal maps with complex
dilatations $\lambda_{0},\lambda$ respectively. From the choice of
coordinates, $f^{\lambda_{0}}$ maps
$(0,1,\infty)\mapsto(0,\tau_{\mu_{0}},\infty)$ and $f^{\lambda}$ maps
$(0,1,\infty)\mapsto(0,\tau_{\mu},\infty)$.
$f^{\lambda}\circ(f^{\lambda_{0}})^{-1}$ takes
$(x,y)\mapsto(x_{\mu},y_{\mu})$, yielding a quasiconformal mapping between the
strips. Provided the metrics are close enough in Teichmüller space, $\lambda$
and $\lambda_{0}$ are related by a small variation

$\lambda=\lambda_{0}+\dot{\lambda}.$

Note that from the computation in Section 3.2, $\dot{\lambda}\to 0$ as
$y\to\infty$. It follows that $f^{\lambda}\circ(f^{\lambda_{0}})^{-1}$ has
complex dilatation tending to $0$ as $y\to\infty$. Mori’s theorem [1, Chapter
3] implies $f^{\lambda}\circ(f^{\lambda_{0}})^{-1}$ has uniform Hölder
continuity.

Upon pre-composing with cylinders that are conformal for $\mu$, the harmonic
map $f_{\mu}$ converges to a projection onto the geodesic axis $\beta$. We can
take these $\mu$-cylinders as large as we like. Using the Hölder continuity
and our normalizations, given a family $\mu_{0}$ cylinders of height $1$, we
can embed each cylinder in a $\mu$-cylinder of fixed height. $C^{0}$
convergence to the projection onto the geodesic thus follows. ∎

###  3.6. Proof of Proposition 2.10

For the remainder of this section, fix a background metric $\mu_{0}$. There is
a Serre duality pairing between holomorphic quadratic differentials on
$\Sigma$ with at most first order poles at the cusp and harmonic Beltrami
forms with appropriate decay (see Remark 3.3):

$\langle\Phi,\psi\rangle=\int_{\Sigma}\phi\psi dzd\overline{z},$

where $\Phi=\phi(z)dz^{2}$ is a coordinate expression. We recall that
Proposition 1.7 asserts that the derivative of the total energy of a finite
energy harmonic map, evaluated on a harmonic Beltrami form $\psi$, is

$dE_{\rho}[\mu](\psi)=-4\operatorname{Re}\langle\Phi(f_{\mu}),\psi\rangle.$

For finite energy harmonic maps, Proposition 2.10 is a direct corollary of
Proposition 1.7, which we prove first. For closed surfaces, Proposition 1.7 is
well known, although the history is unclear—the earliest computation may go
back to the work of Douglas [14]. The most modern version is contained in
[45]. The difference without compactness is that we must control variations at
the cusp.

###### Proof of Proposition 1.7.

Let us assume $\mu(z)$ is conformal, and let

$\mu_{t}=\mu+t\dot{\mu}+t\epsilon(t)$

be a variation through hyperbolic metrics, where $\epsilon(t)\to 0$ as $t\to
0$. We denote by $\phi_{t}$ the associated holomorphic quadratic differential,
which is given in local coordinates by

$\phi_{t}(z)=t\Big{(}(\dot{\mu}_{11}+\dot{\epsilon}_{11}(t))-i(\dot{\mu}_{12}+\dot{\epsilon}_{12}(t))\Big{)}dz^{2},$

and we set $\psi_{t}$ to be the associated harmonic Beltrami form. We put
$\phi$ to be the quadratic differential for the variation $\mu+\dot{\mu}$, and
$\psi$ the harmonic Beltrami form. Writing $f=f_{\mu}$, $f_{t}=f_{\mu_{t}}$,
our objective is to show that

$\frac{d}{dt}|_{t=0}\int_{\Sigma}e(\mu_{t},f_{t})dA_{\mu_{t}}=-4\operatorname{Re}\langle\Phi(f),\psi\rangle.$

By energy minimization, we have the inequalities

$\int_{\Sigma}e(\mu_{t},f_{t})dA_{\mu_{t}}-\int_{\Sigma}e(\mu,f)dA_{\mu}\leq\int_{\Sigma}e(\mu_{t},f)dA_{\mu_{t}}-\int_{\Sigma}e(\mu,f)dA_{\mu}$

and

$\int_{\Sigma}e(\mu_{t},f_{t})dA_{\mu_{t}}-\int_{\Sigma}e(\mu,f)dA_{\mu}\geq\int_{\Sigma}e(\mu_{t},f_{t})dA_{\mu_{t}}-\int_{\Sigma}e(\mu,f_{t})dA_{\mu}.$

Thus, it suffices to divide by $t$ and take the limit on the two expressions
on the right. We expand

(17)
$\frac{\sqrt{|\mu|_{t}}\mu_{t}^{\alpha\beta}-\sqrt{|\mu|}\mu^{\alpha\beta}}{t}=\Big{(}\frac{\sqrt{|\mu|-|\phi_{t}|^{2}}-\sqrt{|\mu|}}{t}\Big{)}\mu^{\alpha\beta}+\sqrt{|\mu|-|\phi_{t}|^{2}}(\dot{\mu}^{\alpha\beta}+\epsilon^{\alpha\beta}(t)).$

Twice the first integrand is obtained by hitting the expression above with
$e_{\alpha\beta}(f)$, and the second with $e_{\alpha\beta}(f_{t})$, which both
converge to $e_{\alpha\beta}(f)$ pointwise as $t\to 0$. The first term in (17)
converges to $0$, while the second one to
$\sqrt{|\mu|}\dot{\mu}^{\alpha\beta}$. Using the relation
$\dot{\mu}^{\alpha\beta}=-\mu^{-2}\dot{\mu}_{\alpha\beta}$, the integrand
converges pointwise to

$-\frac{1}{2\mu}(\dot{\mu}_{11}e_{11}+\dot{\mu}_{22}e_{22}+2\dot{\mu}_{12}e_{12})=-\frac{1}{2\mu}(\dot{\mu}_{11}(e_{11}-e_{22})+2\dot{\mu}_{12}e_{12})=-4\operatorname{Re}\Phi(f)\psi.$

Therefore, it suffices to show the integrands are always bounded above by an
integrable quantity, for then we can justify an application of the dominated
convergence theorem. We have the expression

$\frac{\sqrt{|\mu|-|\phi_{t}|^{2}}-\sqrt{|\mu|}}{t}=\frac{-|\phi_{t}|^{2}}{t(\sqrt{|\mu|-|\phi_{t}|^{2}}+\sqrt{|\mu|})}=\frac{-t\Big{|}(\dot{\mu}_{11}+\dot{\epsilon}_{11}(t))-i(\dot{\mu}_{12}+\dot{\epsilon}_{12}(t))\Big{|}^{2}}{\sqrt{|\mu|-|\phi_{t}|^{2}}+\sqrt{|\mu|}}.$

If $a_{t}$ is the $-1$ Laurent coefficient in the coordinates on the punctured
disk (Section 3.2) for the quadratic differential associated to
$\dot{\mu}+\epsilon(t)$, then because $\epsilon(t)\to 0$, $a_{t}$ converges to
a constant $a_{0}$. Therefore, for $t$ small enough, using the expression
(12), we see that for small $t$, the first term in (17) decays at most like

$-t(2|a_{0}|+1)^{2}y^{2}e^{-2y}y^{2}\sim y^{4}e^{-2y}.$

By similar reasoning, the second term in (17) decays like $y^{2}e^{-2y}$.
Thus, it suffices to bound $e_{\alpha\beta}(f_{t})$ by a constant. The one
obstruction to applying Lemma 3.5 for $f_{t}$ is that $e_{\alpha\beta}$
depends on the cusp coordinates for $\mu$ rather than $\mu_{t}$. The argument
of Lemma 3.14 shows that the $\mu$ and $\mu_{0}$ coordinates are related by a
mapping that is asymptotically Lipschitz—the Hölder exponent in Mori’s theorem
is the reciprocal of the quasiconformal dilatation, which is tending to $1$.
Thus, once high enough in the cusp, we have the same bound in the
$\mu_{0}$-coordinates. From the discussion above, the result follows. ∎

######  Remark 3.15.

We have worked in negative curvature, but the proof carries through in non-
positive curvature if total energies of harmonic maps to $(X,\nu)$ are unique,
and if one can locally continuously associate source metrics to harmonic maps
in the $C^{\infty}$ topology. Our proof uses the existence of a single finite
energy harmonic map and an energy minimizing property. In non-positive
curvature we have energy minimization, and for existence one can modify our
[36, Proposition 3.8] to include NPC manifolds.

Now we turn to the main result of this section. Assume the monodromy is
hyperbolic.

###### Proof of Proposition 2.10.

As above, let $\mu_{t}=\mu+t\dot{\mu}+t\epsilon(t)$ be the variation with
$\mu$. Similarly, denote by $\phi_{t}$ and $\psi$ the family of holomorphic
quadratic differentials and the harmonic Beltrami form associated to this path
respectively. Put $f=f_{\mu}$, $f_{t}=f_{\mu_{t}}$, $h=h_{\mu}$,
$h_{t}=h_{\mu_{t}}$, recalling that $f$ and $h$ correspond to the
representations $\rho_{1}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$,
$\rho_{2}:\Gamma\to G$ respectively. From the energy minimization Lemma 3.12,
we deduce

$\int_{\Sigma}e(\mu,h)-e(\mu,f_{t})dA_{\mu}\leq\operatorname{\mathcal{E}}(\mu)\leq\int_{\Sigma}e(\mu,h_{t})-e(\mu,f)dA_{\mu}$

and

$\int_{\Sigma}e(\mu_{t},h_{t})-e(\mu_{t},f)dA_{\mu_{t}}\leq\operatorname{\mathcal{E}}(\mu_{t})\leq\int_{\Sigma}e(\mu_{t},h)-e(\mu_{t},f_{t})dA_{\mu_{t}}.$

We cannot apply Lemma 3.1 to an expression like $e(\mu,f_{t})$ directly, but
from the expression for the cusp coordinates (3), it follows that when
changing $\sqrt{|\mu|}e(\mu,f)$ to $\sqrt{|\mu_{t}|}e(\mu_{t},f_{\mu})$, the
energy density is asymptotically multiplied by $\tau_{\mu_{t}}/\tau_{\mu}$.
Therefore, Lemma 3.1 does imply that every integral above is finite.
Furthermore, it makes sense to manipulate these integrals, and so the
difference
$\operatorname{\mathcal{E}}(\mu_{t})-\operatorname{\mathcal{E}}(\mu)$ is
bounded above by

$\int_{\Sigma}(\sqrt{|\mu_{t}|}e(\mu_{t},h)-\sqrt{|\mu|}e(\mu,h))-(\sqrt{|\mu_{t}|}e(\mu_{t},f_{t})-\sqrt{|\mu|}e(\mu,f_{t}))dzd\overline{z}$

and bounded below by

$\int_{\Sigma}(\sqrt{|\mu_{t}|}e(\mu_{t},h_{t})-\sqrt{|\mu|}e(\mu,h_{t}))-(\sqrt{|\mu_{t}|}e(\mu_{t},f)-\sqrt{|\mu|}e(\mu,f))dzd\overline{z}.$

Dividing by $t$, the local computations from the proof of Proposition 1.7 work
out almost the same. We write out the details for the upper bound and leave
the lower bound to the reader. The relevant quotient may be expressed

$\frac{1}{2}\Big{(}\Big{(}\frac{\sqrt{|\mu|-|\phi_{t}|^{2}}-\sqrt{|\mu|}}{t}\Big{)}\mu^{\alpha\beta}+\sqrt{|\mu|-|\phi_{t}|^{2}}(\dot{\mu}^{\alpha\beta}+\epsilon^{\alpha\beta}(t))\Big{)}(e_{\alpha\beta}(h)-e_{\alpha\beta}(f_{t})).$

The quotient converges pointwise to
$-4\operatorname{Re}(\Phi(h)-\Phi(f))\cdot\overline{\psi}$, so we are left to
bound it by an integrable quantity that does not depend on $t$. Lemma 3.10 and
the fact that the change of coordinates is asymptotically conformal shows that
the second term in the product is uniformly integrable. And it was shown in
the proof of the previous proposition that the first term in the product
decays exponentially. ∎

###  3.7. Different twist parameters

The proof of Proposition 2.10 is complete for finite energy harmonic maps, and
infinite energy harmonic maps with $\theta=0$. We sketch the necessary
adjustments for the remaining harmonic maps $f_{\mu}^{\theta}$. Fix
$\theta\in\mathbb{R}$. To construct the harmonic map $f_{\mu}^{\theta}$, we
define the fractional Dehn twist to be the map in the cusp coordinates defined
by

$x+iy\mapsto x+\theta_{i}y+iy.$

We then postcompose with the approximation maps $f_{r}$ and take limits on
these maps as in Section 3.1 (see [36, Section 5.1]).

  * •

The inequality in Lemma 3.1 becomes

$|e(\mu,f_{\mu}^{\theta})-\Lambda(\theta)\ell(\rho(\gamma))^{2}/2\tau^{2}|<Ce^{-cy}.$

The proof is the same, except the Hopf differential has expression at the cusp
according to Theorem 2.5.

  * •

For continuity of harmonic maps on compacta, the main step is to find bounds
on the total energy of $\varphi_{\mu}^{\theta}$ on compacta. This is a
consequence of the chain rule, since in the coordinates (3), the derivative
matrix of the fractional Dehn twist is simply

(18) $\begin{pmatrix}1&\theta\\\ 0&1\end{pmatrix}.$

  * •

For the energy minimizing property, we can say that a map converges at
$\infty$ to $\alpha^{\theta}$ if after pulling back to conformal cylinders
approaching the cusps, the harmonic maps converge to the composition of a
projection onto the geodesic with a fractional Dehn twist on the cylinder. We
then have the analogue of Lemma 3.12, which we can show the harmonic maps
$f^{\theta}$ satisfy.

We leave the rest of the details to the reader.

##  4\. Existence and uniqueness of maximal surfaces

With Proposition 2.10 in hand, we prove the main theorems. For convenience we
assume the twist parameter is zero—the proof has no dependence on it. The
existence result is immediate from the next proposition.

######  Proposition 4.1.

The functional
$\operatorname{\mathcal{E}}=\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}:\mathcal{T}(\Gamma)\to\mathbb{R}$
is proper if and only if $\rho_{1}$ almost strictly dominates $\rho_{2}$.

Indeed, by Proposition 2.10, properness implies the existence of a maximal
spacelike immersion. Thus, the proof of the existence result reduces to
showing that almost strict domination implies properness.

###  4.1. Mapping class groups

Preparing for the proof of Theorem A, we review some Teichmüller theory. Set
$\textrm{Diff}^{+}(\Sigma)$ to be the group of $C^{\infty}$ orientation
preserving diffeomorphisms of $\Sigma$, equipped with the $C^{\infty}$
topology. Denote by $\textrm{Diff}_{0}^{+}(\Sigma)$ the normal subgroup
consisting of maps that are isotopic to the identity. The mapping class group
of $\Sigma$ is defined

$\textrm{MCG}(\Sigma):=\pi_{0}(\textrm{Diff}^{+}(S))=\textrm{Diff}^{+}(S)/\textrm{Diff}_{0}^{+}(S).$

Given a collection of boundary lengths
$\ell=(\ell_{1},\dots,\ell_{n})\in\mathbb{R}^{n}$, we can consider the
Teichmüller space $\mathcal{T}_{\ell}(\Gamma)$ of surfaces with punctures and
geodesic boundary with lengths determined by $(\ell_{1},\dots,\ell_{n})$. This
is not a relative representation space but a union of them. The mapping class
group acts on Teichmüller space by pulling back classes of hyperbolic metrics:

(19) $[\varphi]\cdot[\mu]\mapsto[\varphi^{*}\mu].$

The action of the mapping class group is properly discontinuous and the
quotient is the moduli space of hyperbolic surfaces.

There is a wealth of metrics on Teichmüller space, and the mapping class group
acts by isometries on a number of them. In the work below, we set
$d_{\mathcal{T}}$ to be any metric distance function on Teichmüller space on
which the mapping class group acts by isometries. One example is the
Teichmüller distance.

Selecting a basepoint $z\in\Sigma$, there is an action of the mapping class
group on $\pi_{1}(\Sigma,z)$, which identifies with $\Gamma$. Given
$\varphi\in\textrm{Diff}^{+}(\Sigma)$ such that $\varphi(z)=z$, the induced
map $\varphi_{*}:\pi_{1}(\Sigma,z)\to\pi_{1}(\Sigma,z)$ is an automorphism.
While a general $\varphi$ may not fix $z$, one can find a different
$\varphi_{1}\in\textrm{Diff}^{+}(\Sigma)$ that does fix $z$ and is isotopic to
$\varphi$. The isotopy gives rise to a path $\gamma_{1}$ from $\varphi(z)$ to
$z$. If $\varphi_{2}\in\textrm{Diff}^{+}(\Sigma)$ also fixes $z$ and is
isotopic to $\varphi$, then we get another path $\gamma_{2}$ from $\varphi(z)$
to $z$. The automorphisms $(\varphi_{1})_{*}$ and $(\varphi_{2})_{*}$ of
$\pi_{1}(\Sigma,z)$ differ by the inner automorphism corresponding to the
conjugation by the class of $\gamma_{1}\cdot\overline{\gamma}_{2}$. This
association furnishes an injective homomorphism from

$\textrm{MCG}(\Sigma)\to\textrm{Out}(\pi_{1}(\Sigma))=\textrm{Aut}(\pi_{1}(\Sigma))/\textrm{Inn}(\pi_{1}(\Sigma)).$

Through this injective mapping, the mapping class group acts on representation
spaces: we can precompose a representative of a representation with a
representative of an element in $\textrm{Out}(\pi_{1}(\Sigma))$, and then take
the corresponding equivalence class. On a Teichmüller space, this agrees with
the action (19). For this action, we use the similar notation

$[\varphi]\cdot[\rho]\mapsto[\varphi^{*}\rho]=[\rho\circ\varphi_{*}].$

Note that it does not in general preserve relative representation spaces: a
mapping class may permute the punctures. Finally, we record that
$\textrm{MCG}(\Sigma)$ acts equivariantly with respect to

  * •

length spectrum: for a representation $\rho$ and
$[\varphi]\in\textrm{MCG}(\Sigma)$,
$\ell(\varphi^{*}\rho(\gamma))=\ell(\rho(\varphi_{*}(\gamma)))$. If $\mu$ is a
hyperbolic metric we set $\ell_{\mu}(\gamma)$ to be the $\mu$-length of the
geodesic representative of $\gamma$ in $(\Sigma,\mu)$. A restatement of the
above is $\ell_{\varphi^{*}\mu}(\gamma)=\ell_{\mu}(\varphi(\gamma))$.

  * •

Energy of harmonic maps: if $f:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$ is a
$\rho$-equivariant map, then

(20) $e(\mu,f_{\mu})=e(\varphi^{*}\mu,f_{\mu}\circ\varphi).$

Furthermore, if $f$ is harmonic, then
$f\circ\varphi:(\tilde{\Sigma},\tilde{\varphi}^{*}\tilde{\mu})\to(X,\nu)$ is a
$\varphi^{*}\rho$-equivariant harmonic map.

###  4.2. Existence of maximal surfaces

Before beginning the proof, we recall a basic fact from hyperbolic geometry
[48, Lemma 2.3]. Suppose a simple closed curve $\xi$ is either a geodesic
boundary component or a horocycle in a hyperbolic surface. If $\xi$ is a
boundary, let $C_{d}(\xi)$ denote the collar around $\xi$ consisting of points
with distance to $\xi$ at most $d$. If it is a horocycle, put $C_{d}(\xi)$ to
be the union of the $d$-collar and the enclosed cusp (both are defined for
suitably small $d$).

######  Lemma 4.2 (Generalized collar lemma).

Given a geometrically finite hyperbolic surface, let $\xi_{1},\dots,\xi_{n}$
be horocylces bounding cusps, and $\xi_{1}^{\prime},\dots,\xi_{m}^{\prime}$
geodesic boundary components. There exists a $d>0$ such that any non-
peripheral simple closed geodesic does not intersect $C_{d}(\xi_{i})$ or
$C_{d}(\xi_{i}^{\prime})$.

We now begin the proof of properness. Suppose $\rho_{1}$ almost strictly
dominates $\rho_{2}$. We assume there is a single peripheral curve—there are
no substantial changes in the case of many cusps—and we set $\xi$ to be either
the geodesic boundary component of the convex core of
$\mathbb{H}/\rho_{1}(\Gamma)$, or a deep horocycle of
$\mathbb{H}/\rho_{1}(\Gamma)$, depending on whether the monodromy is
hyperbolic or parabolic. Let $g$ be an optimal
$(\rho_{1},\rho_{2})$-equivariant map. If the image of the peripheral curve
under $\rho_{1}$ and $\rho_{2}$ is hyperbolic, then $g$ has constant speed on
the boundary components. Note that $g\circ h_{\mu}$ is $\rho_{2}$-equivariant,
and in the hyperbolic case, converges at $\infty$ to a projection onto the
geodesic at infinity in the sense of Definition 3.11. Thus, from Lemma 3.12 we
have

$\operatorname{\mathcal{E}}(\mu)=\int_{\Sigma}e(\mu,h_{\mu})-e(\mu,f_{\mu})dA_{\mu}\geq\int_{\Sigma}e(\mu,h_{\mu})-e(\mu,g\circ
h_{\mu})dA_{\mu}.$

One of the main advantages of replacing $f_{\mu}$ with $g\circ h_{\mu}$ is
that the integrand is now non-negative. Moving toward the proof of properness,
let $([\mu_{n}])_{n=1}^{\infty}\subset\mathcal{T}(\Gamma)$ be such that

$\operatorname{\mathcal{E}}(\mu_{n})\leq K$

for all $n$, for some large $K$. It suffices to show $[\mu_{n}]$ converges in
$\mathcal{T}(\Gamma)$ along a subsequence. Structurally, our proof is similar
to that of the classical existence result for minimal surfaces in hyperbolic
manifolds (see [39] and also the more modern paper [18]): we first prove 1)
compactness in moduli space, and then 2) extend to compactness in Teichmüller
space.

Toward 1), we show there is a $\delta>0$ such that for all non-peripheral
simple closed curves in $\Gamma$, we have $\ell_{\mu_{n}}(\gamma)\geq\delta$.
We argue by contradiction: suppose this is not the case, so that there is a
sequence of non-peripheral simple closed curves
$(\gamma_{n})_{n=1}^{\infty}\subset\Gamma$ such that

$\ell_{n}:=\ell_{\mu_{n}}(\gamma_{n})\to 0$

as $n\to\infty$. By the regular collar lemma, we know that in
$(\Sigma,\mu_{n})$, the geodesic representing $\gamma_{n}$ is enclosed by a
collar $C_{n}$ of width $w_{n}=\textrm{arcsinh}((\sinh\ell_{n})^{-1})$. Up to
a conjugation of the holonomy, $C_{n}$ is conformally equivalent to
$\\{x+iy\in\mathbb{C}:0\leq x\leq w_{n},0\leq y\leq 1\\}$ with the lines
$\\{y=0\\}$ and $\\{y=1\\}$ identified.

Let $f_{n}=f_{\mu_{n}}$, $h_{n}=h_{\mu_{n}}$. Henceforward, conformally modify
the metric $\mu_{n}$ to be flat in $C_{n}$. Our control over the energy
functional depends on our knowledge of the local Lipschitz constant of the map
$g$. This in turns relies on the image of the harmonic map $h_{n}$, in
particular how far it takes $C_{n}$ into $C_{d}(\xi)$. For all $x,0<t<d$ in
question, set $A_{x}^{t}=\\{\theta\in\\{x\\}\times S^{1}:h_{n}(x,y)\not\in
C_{t}(\xi)\\}$. We also write

$\textrm{Lip}(g,t)=\max_{x\in C(\mathbb{H}/\rho_{1}(\Gamma))\backslash
C_{t}(\xi)}\textrm{Lip}_{x}(g).$

We have the inequalities

$\displaystyle\operatorname{\mathcal{E}}(\mu_{n})(\sigma_{n})$
$\displaystyle\geq\int_{C_{n}}e(\mu_{n},h_{n})-e(\mu_{n},g\circ
h_{n})dA_{\mu_{n}}$
$\displaystyle=\int_{0}^{w_{n}}\int_{0}^{1}e(\mu_{n},h_{n})-e(\mu_{n},g\circ
h_{n})dyds$ $\displaystyle\geq
w_{n}\min_{x}\int_{A_{x}^{t}}e(\mu_{n},h_{n})-e(\mu_{n},g\circ h_{n})dy$
$\displaystyle\geq
w_{n}\min_{x}(1-\textrm{Lip}(g,t)^{2})\int_{A_{x}^{t}}e(\mu_{n},h_{n})dy$
$\displaystyle\geq\frac{w_{n}}{2}\min_{x}(1-\textrm{Lip}(g,t)^{2})\int_{A_{x}^{t}}\Big{|}\frac{\partial
h_{n}(x,y)}{\partial y}\Big{|}^{2}dy$
$\displaystyle\geq\frac{w_{n}}{2}\min_{x}(1-\textrm{Lip}(g,t)^{2})(\ell(\textrm{A}_{x}^{t}))^{-1}\Big{(}\int_{A_{x}^{t}}\Big{|}\frac{\partial
h_{n}(x,y)}{\partial y}\Big{|}dy\Big{)}^{2}$
$\displaystyle\geq\frac{w_{n}}{2}\min_{x}(1-\textrm{Lip}(g,t)^{2})\Big{(}\int_{A_{x}^{t}}\Big{|}\frac{\partial
h_{n}(x,y)}{\partial y}\Big{|}dy\Big{)}^{2}$

Here $\ell(A_{x}^{t})$ is the length of the (possibly broken) segment
$A_{x,t}$, which is clearly bounded above by $1$. Fuchsian representations
have discrete length spectrum, and hence there is a $\kappa>0$ such that
$\ell(\rho_{1}(\gamma))\geq\kappa$ for all $\gamma\in\Gamma$.

######  Lemma 4.3.

Set $t=d/2$. Then for any $x$, the inequality

$\int_{A_{x}^{d/2}}|\frac{\partial h_{n}(x,y)}{\partial
y}\Big{|}dy\geq\min\\{d/2,\kappa\\}=:k$

holds.

###### Proof.

Let $\alpha$ be any core curve of the form $\\{x\\}\times S^{1}$. If
$h_{n}(\alpha)$ always remains outside $C_{d/2}(\xi)$, then

$\int_{A_{x}^{t}}|\frac{\partial h_{n}(x,y)}{\partial
y}\Big{|}dy=\int_{0}^{1}\Big{|}\frac{\partial h_{n}(x,y)}{\partial
y}\Big{|}dy\geq\ell(\rho_{1}(\gamma_{n}))\geq\kappa.$

Thus, we assume $h_{n}(\alpha)$ intersects $C_{d/2}(\xi)$. We parametrize
$h_{n}(x,\cdot)$ by arc length, so that the integral in question measures the
hyperbolic length on $\mathbb{H}/\rho_{1}(\Gamma)$ of the segment of
$h_{n}(\alpha)$ that does not enter $C_{d/2}(\xi)$. If this length is ever
less than $d/2$, then the curve $h_{n}(\alpha)$ is contained in $C_{d}(\xi)$.
Phrased differently, it is a simple closed curve contained in an embedded
cylinder. There is only one such homotopy class of curves, namely the homotopy
class of the boundary geodesic. This situation is impossible, since
$h_{n}(\alpha)$ is homotopic to a non-peripheral simple closed curve, and
hence the length is at least $d/2$. ∎

Returning to the inequalities above, we now have

$\operatorname{\mathcal{E}}(\mu_{n})\geq\frac{w_{n}}{2}(1-\textrm{Lip}(g,d/2)^{2})k^{2}.$

We thus find $\operatorname{\mathcal{E}}(\mu_{n})\to\infty$ as $n\to\infty$,
which violates the uniform upper bound.

The lower bound on the length spectrum shows the metrics
$(\mu_{n})_{n=1}^{\infty}$ satisfy Mumford’s compactness criteria [34] and
hence project under the action of the mapping class group to a compact subset
of the moduli space.

Now we promote to compactness in Teichmüller space. By compactness in moduli
space, after passing to a subsequence of the $\mu_{n}$’s, there exists a
sequence of mapping class group representatives $(\psi_{n})_{n=1}^{\infty}$
such that $\psi_{n}^{*}\mu_{n}$ converges as $n\to\infty$ to a hyperbolic
metric $\mu_{\infty}$.

######  Lemma 4.4.

Upon passing to a further subsequence,
$([\psi_{n}])_{n=1}^{\infty}\subset\textrm{MCG}(\Sigma)$ is eventually
constant.

The main thrust of the proof is to show that for each non-peripheral
$\gamma\in\Gamma$, $\ell(\rho_{1}\circ\psi_{n}(\gamma))$ is uniformly bounded
above in $n$. Our proof of this claim is by contradiction and similar in
nature to the argument above. Suppose there is a class of curves
$\gamma\in\Gamma$ such that $\ell(\rho_{1}\circ\psi_{n}(\gamma))\to\infty$ as
$n\to\infty$. Modify the notation: set $h_{n}=h_{\mu_{n}}\circ\psi_{n}$ and
$f_{\mu_{n}}\circ\psi_{n}$. These are harmonic for $\psi_{n}^{*}\rho_{1}$ and
$\psi_{n}^{*}\rho_{2}$ respectively, and have the correct boundary behaviour
according to Theorem 2.5. From (20), we also have the equality

$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}(\mu_{n})=\operatorname{\mathcal{E}}_{\psi_{n}^{*}\rho_{1},\psi_{n}^{*}\rho_{2}}(\psi_{n}^{*}\mu_{n}).$

On each $(\Sigma,\psi_{n}^{*}\mu_{n})$, there is a collar $C_{n}$ of finite
width $w_{n}$ around $\gamma$, and the $C_{n}$’s converge to some collar
$C_{\infty}$ in $(\Sigma,\mu_{\infty})$ of width $w_{\infty}<\infty$. As
before, we perturb to a flat metric and parametrize $C_{n}$ by
$[0,w_{n}]\times S^{1}$. We note that $g$ is
$(\psi_{n}^{*}\rho_{1},\psi_{n}^{*}\rho_{2})$-equivariant. Therefore, we can
apply our previous reasoning to see that for $n$ large enough, any small
enough $x$, and $t\in[0,w_{n}]$,

(21)
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}(\mu_{n})=\operatorname{\mathcal{E}}_{\psi_{n}^{*}\rho_{1},\psi_{n}^{*}\rho_{2}}(\psi_{n}^{*}\mu_{n})\geq\frac{w_{\infty}}{4}\min_{x}(1-\textrm{Lip}(g,t)^{2})\Big{(}\int_{A_{x}^{t}}\Big{|}\frac{\partial
h_{n}(x,y)}{\partial y}\Big{|}dy\Big{)}^{2},$

where $A_{x}^{t}$ is defined as above. This leads us to the analogue of Lemma
4.3.

######  Lemma 4.5.

Set $t=d/2$. Then, independent of the choice of $x$,

$\int_{A_{x}^{d/2}}\Big{|}\frac{\partial h_{n}(x,y)}{\partial
y}\Big{|}dy\to\infty$

as $n\to\infty$.

###### Proof.

Choose any simple closed curve $\alpha$ of the form $\\{x\\}\times S^{1}$.
Parametrize $h_{n}(x,\cdot)$ by arc length so that the integral above returns
the length of the segment of $h_{n}(\alpha)$ that does not enter
$C_{d/2}(\xi)$. If $h_{n}(\alpha)$ does not enter $C_{d/2}(\xi)$, then we can
argue as we did in the proof of Lemma 4.3, bounding the length below by the
$\ell(\psi_{n}^{*}\rho_{1}(\gamma))$, which blows up. Thus, we restrict our
discussion to $\alpha$ such that $h_{n}(\alpha)$ intersects $C_{d/2}(\xi)$.
Let us first assume there is a positive integer $K$ such that every
$h_{n}(\alpha)$ enters $C_{d/2}(\xi)$ at most $K$ times. Of course,
$h_{n}(\alpha)$ cannot live entirely in $C_{d/2}(\xi)$, for then it lies in
the wrong homotopy class. We construct a new curve as follows: if we choose a
basepoint not within $C_{d/2}(\xi)$, then every time $h_{n}(\alpha)$ enters
$C_{d/2}(\xi)$, there is a corresponding point at which it exits. We plan to
erase the segment of $h_{n}(\alpha)$ that connects these two points and
replace it with the one of the two possible paths on $\partial C_{d/2}(\xi)$,
say, $\beta_{1}$ and $\beta_{2}$. If $h_{n}(\alpha)$ spends some time going in
a path along the boundary circle when it is entering or when it is exiting (we
count just touching it once as an exit), we connect the endpoint of the
entrance path to the starting point of the exiting path and concatenate with
the path going along the boundary.

The question is: which path to take? We argue there is a choice so that the
homotopy class of the new curve is the same as that of $h_{n}(\alpha)$. To
this end, orient the boundary circle so that there is a “left” and a “right”
path. Arbitrarily choose one of the two paths connecting the entrance and the
exit point, say, $\beta_{1}$, and consider the loop in the cylinder obtained
by concatenating this path with the piece of $h_{n}(\alpha)\cap C_{d/2}(\xi)$
that connects the endpoints. This is a simple closed curve in a cylinder, and
hence there are three possible homotopy classes: trivial, non-trivial and left
oriented, and non-trivial and right oriented. If the class is trivial, then
using this path $\beta_{1}$ will work. If it is non-trivial and left oriented,
then $\beta_{1}$ must be the right oriented path. Thus, if we use the opposite
path $\beta_{2}$, it will cancel the homotopy class, and therefore the new
path will indeed be homotopic to $h_{n}(\alpha)$. The right oriented case is
similar.

Each replacement adds at most $\ell(\partial C_{d/2}(\xi))$ to the length of
the path. Thus, the length of the new path is (quite crudely) bounded above by

$\ell_{d/2}(h_{n}(\alpha))+K\ell(\partial C_{d/2}(\xi)).$

Therefore,

$\ell_{d/2}(h_{n}(\alpha))\geq\ell(\psi_{n}^{*}\rho_{1}(\gamma))-K\ell(\partial
C_{d/2}(\xi))$

which tends to $\infty$ as $n\to\infty$, and moreover establishes the result
of the lemma.

We are left to consider the case of an unbounded number of crossings into
$C_{d/2}(\xi)$ as we take $n\to\infty$. The idea is that each time
$h_{n}(\alpha)$ crosses $\partial C_{d/2}(\xi)$, there is a corresponding
“down-crossing,” a curve that connects the point at which it exits to the new
point of entry. If each down crossing is denoted by $c_{j}^{n}$, then

$\int_{A_{x}^{d/2}}\Big{|}\frac{\partial h_{n}(x,y}{\partial
y}\Big{|}dy\geq\sum_{j}\ell(c_{j}^{n}).$

Clearly, if the limit of these sums is infinite, then we are done. Hence, we
assume the total length due to down-crossings is finite. This implies that
there is an even integer $K>0$ such that, as we take $n\to\infty$, there are
at most $K$ down-crossing that exit $C_{d}$. Again using our chosen basepoint,
let $b_{1}$ be the first crossing into $C_{d/2}(\xi)$, and then let $b_{2}$ be
the first crossing out of $C_{d/2}(\xi)$ that exits $C_{d}$. Set $b_{3}$ to be
the next entry point into $C_{d/2}(\xi)$, and $b_{4}$ the next exit point from
$C_{d}(\xi)$. We end up with $K_{n}\leq K$ points $b_{1},\dots b_{K_{n}}$.
Replace the segments of $h_{n}(\alpha)$ between $b_{i}$ and $b_{i+1}$ with the
correct arc on $\partial C_{d/2}(\xi)$ that does not change the homotopy
class, where $i=1,3,\dots,K_{n}-1\leq K-1$. As above, we end up with a curve
of length at most

$\ell_{d/2}(h_{n}(\alpha))+K\ell(\partial C_{d/2}(\xi))$

and homotopic to $\ell(\psi_{n}^{*}\rho_{1}(\gamma))$. Using the minimizing
property of geodesics once again, we have

$\ell_{d/2}(h_{n}(\alpha))\geq\ell(\psi_{n}^{*}\rho_{1}(\gamma))-K\ell(\partial
C_{d/2}(\xi))\to\infty,$

and the resolution of this final case completes the proof. ∎

Returning to (21), this lemma shows $\mathcal{E}(\mu_{n})\to\infty$, which is
a contradiction. We can now conclude that each sequence
$\ell(\psi_{n}^{*}\rho_{1}(\gamma))$ remains bounded above.

With this claim in hand, we observe that $\psi_{n}^{*}\rho_{1}$ converges
along a subsequence to some new Fuchsian representation
$\rho_{\infty}:\Gamma\to\textrm{PSL}_{2}(\mathbb{R})$. This is a fairly
elementary argument, so we will be brief: for any $\gamma\in\Gamma$,
$z\in\tilde{\Sigma}$, and $z_{0}$ on the geodesic axis of $\gamma$,

$d(\psi_{n}^{*}\rho_{1}(\gamma)z,z)\leq\ell(\psi_{n}^{*}\rho_{1}(\gamma))+2d(z,z_{0}).$

Hence, $\psi_{n}^{*}\rho_{1}(\gamma)z$ converges on a subsequence. Taking
$(z_{m})$ to be a countable dense sequence, by the diagonal trick we have a
subsequence on which each $\psi_{n}^{*}\rho_{1}(\gamma)z_{m}$ converges as
$n\to\infty$. This implies convergence for any $z$, and hence we have
convergence in the compact-open topology to a well defined map
$\rho_{\infty}(\gamma):\mathbb{H}\to\mathbb{H}$ that takes each point $z$ to
its limit. Letting $\gamma$ run over a (finite) generating set, by another
diagonal argument we find a subsequence along which
$\psi_{n}^{*}\rho_{1}(\gamma)$ converges to an isometry
$\rho_{\infty}(\gamma)$ for every $\gamma$. It is clear that the association
$\gamma\mapsto\rho_{\infty}(\gamma)$ defines a representation, and by relative
Euler class considerations, it must be Fuchsian.

We now restrict the $[\mu_{n}],[\psi_{n}]$ to this chosen subsequence. Since
the mapping class group acts properly discontinuously on the Teichmüller
space, it follows that $[\psi_{n}^{*}\rho_{1}]=[\rho_{\infty}]$ for all $n$
large enough. In particular, $\psi_{n}$ is isotopic to $\psi_{m}$ for all
$n,m$ large enough. This completes the proof of Lemma 4.4.

To finish the proof of Proposition 4.1, we know there is an $N$ such that
$[\psi_{n}]=[\psi_{N}]$ for all $n\geq N$. As the mapping class group acts by
isometries with respect to the metric $d_{\mathcal{T}}$,

$d_{\mathcal{T}}([\mu_{n}],[(\psi_{N}^{-1})^{*}\mu_{\infty}])=d_{\mathcal{T}}([\psi_{N}^{*}\mu_{n}],[\mu_{\infty}])\to
0,$

and hence $[\mu_{n}]$ converges to $[(\psi_{N}^{-1})^{*}\mu_{\infty}]$ as
$n\to\infty$. Proposition 4.1 is proved.

###  4.3. Uniqueness of critical points

From now on we refine our notation: when the representation $\rho$ is not
implicit, the $\rho$-equivariant harmonic map for a metric $\mu$ will be
denoted $f_{\mu}^{\rho}$. Similarly, if we consider maps
$(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$, we write $e(\mu,\nu,\cdot)$ for an
energy density rather than $e(\mu,\cdot)$ when necessary.

We move to the uniqueness statement in Theorem A, which amounts to showing
critical points of $\operatorname{\mathcal{E}}$ are unique. We can argue
similarly to [41, Theorem 1]. Suppose $\rho_{1}$ represents a point in
$\mathcal{T}_{\mathbf{c}}(\Gamma)$ and almost strictly dominates $\rho_{2}$,
and let $\mu_{1}$ be a hyperbolic metric representing a critical point in
$\mathcal{T}(\Gamma)$ for
$\operatorname{\mathcal{E}}=\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}$.
Let $\mu_{0}$ be another hyperbolic metric, representing the point in the
Teichmüller space $\mathcal{T}_{\mathbf{c}}(\Gamma)$ associated to $\rho_{1}$
and chosen so that the identity map
$\textrm{Id}:(\tilde{\Sigma},\tilde{\mu}_{1})\to(\tilde{\Sigma},\tilde{\mu}_{0})$
is harmonic. This can be achieved by replacing the target metric with its
pullback by the harmonic map, which represents the same point in
$\mathcal{T}_{\mathbf{c}}(\Gamma)$. Apriori this is the holonomy of the convex
core of a hyperbolic surface, so the universal cover is a proper subset of
$\tilde{\Sigma}$ and the identity map only goes into that subset, but we can
extend to a global metric by attaching a hyperbolic half-space along the
boundary geodesic. Let $\mu_{2}$ represent yet another point in
$\mathcal{T}(\Gamma)$, such that $[\mu_{1}]\neq[\mu_{2}]$ in Teichmüller
space, and
$\textrm{Id}:(\tilde{\Sigma},\tilde{\mu}_{2})\to(\tilde{\Sigma},\tilde{\mu}_{0})$
is harmonic—we similarly consider Id as a map into the convex hull of the
limit set. Also, let $\rho$ be the representation associated to $\mu_{2}$,
chosen so that $f_{\mu_{1}}^{\rho}:(\Sigma,\mu_{1})\to(\Sigma,\mu_{2})$ is the
unique harmonic diffeomorphism in the homotopy class (see [31, Theorem 2]). We
then have decompositions of the pullback metrics

  * •

$\mu_{0}=e(\mu_{1},\mu_{0},\textrm{Id})\mu_{1}+\Phi(f_{\mu_{1}}^{\rho_{2}})+\overline{\Phi}(f_{\mu_{1}}^{\rho_{2}}),$

  * •

$(f_{\mu_{1}}^{\rho_{2}})^{*}\nu=e(\mu_{1},\nu,f_{\mu_{1}}^{\rho_{2}})\mu_{1}+\Phi(f_{\mu_{1}}^{\rho_{2}})+\overline{\Phi}(f_{\mu_{1}}^{\rho_{2}})$,

  * •

$\mu_{2}=e(\mu_{1},\mu_{2},f_{\mu_{1}}^{\rho})\mu_{1}+\Phi(f_{\mu_{1}}^{\rho})+\overline{\Phi}(f_{\mu_{1}}^{\rho})$.

Tholozan’s computation in [41, Lemma 2.4] is purely local, and from this we
have the formula

(22)
$(e(\mu_{2},\mu_{0},\textrm{Id})-e(\mu_{2},\nu,(f_{\mu_{1}}^{\rho_{2}})^{*}\nu))dA_{\mu_{2}}=\frac{e(\mu_{1},\mu_{0},\textrm{Id})-e(\mu_{1},\nu,(f_{\mu_{1}}^{\rho_{2}})^{*}\nu)}{\sqrt{1-4\frac{||\Phi(f_{\mu_{1}}^{\rho})||_{\mu_{1}}^{2}}{e(\mu_{1},\mu_{2},f_{\mu_{1}}^{\rho})^{2}}}}dA_{\mu_{1}}.$

The denominator lies in $(0,1]$, tends to $0$ at the cusp, and is $1$ exactly
when $\overline{\Phi}(f_{\mu_{1}}^{\rho})=0$. Inputting into our integrals
yields

$\int_{\Sigma}e(\mu_{2},\mu_{0},\textrm{Id})-e(\mu_{2},\nu,(f_{\mu_{1}}^{\rho_{2}})^{*}\nu)dA_{\mu_{2}}\geq\operatorname{\mathcal{E}}(\mu_{1}).$

By Lemma 3.12, we have

$\int_{\Sigma}e(\mu_{2},\nu,(f_{\mu_{2}}^{\rho_{2}})^{*}\nu)-e(\mu_{2},\nu,(f_{\mu_{1}}^{\rho_{2}})^{*}\nu)dA_{\mu_{2}}\leq
0,$

and hence
$\operatorname{\mathcal{E}}(\mu_{2})\geq\operatorname{\mathcal{E}}(\mu_{1})$.
From (22), we have equality if and only if $[\mu_{1}]=[\mu_{2}]$ in
$\mathcal{T}(\Gamma)$. This establishes the result. And as previously
discussed, the proof of Theorem A is complete.

##  5\. Deformations of maximal surfaces

###  5.1. The maps $\Psi^{\theta}$

We now begin the proof of Theorem B. Generalizing the map $\Psi$ from [41,
subsection 2.3]), we define the map

$\Psi^{\theta}:\mathcal{T}(\Gamma)\times\textrm{Rep}_{\mathbf{c}}(\Gamma,G)\to\textrm{ASD}_{\mathbf{c}}(\Gamma,G)\subset\mathcal{T}_{\mathbf{c}}(\Gamma)\times\textrm{Rep}_{\mathbf{c}}(\Gamma,G)$

as follows. When $\mathbf{c}$ has no hyperbolic classes, we implicitly assume
there is no $\theta$. Begin with a hyperbolic surface $(\Sigma,\mu)$ and a
reductive representation
$\rho_{2}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$. Associated to
this representation, we take a $\rho_{2}$-equivariant harmonic map with twist
parameter $\theta$, $f^{\theta}:(\tilde{\Sigma},\tilde{\mu})\to(X,\nu)$. We
proved in [36, Theorem 1.4] that there exists a unique Fuchsian representation
$\rho_{1}:\Gamma\to\operatorname{\textrm{PSL}}(2,\mathbb{R})$ and a unique
$\rho_{1}$-equivariant harmonic diffeomorphism
$h^{\theta}:(\tilde{\Sigma},\mu)\to(\mathbb{H},\sigma)$ that has Hopf
differential $\Phi(h^{\theta})=\Phi(f^{\theta})$. From [36, Proposition 3.13],
the mapping $F=(h^{\theta},f^{\theta})$ is a spacelike maximal immersion into
the pseudo-Riemannian product, and hence $\rho_{1}$ almost strictly dominates
$\rho_{2}$. $\Psi^{\theta}$ is defined by

$\Psi^{\theta}([\mu],[\rho_{2}])=([\rho_{1}],[\rho_{2}]).$

It is clear that the map is fiberwise in the sense of the statement of Theorem
B. Theorem A shows $\Psi^{\theta}$ is bijection, and the content of Theorem B
is that $\Psi^{\theta}$ is a homeomorphism.

######  Remark 5.1.

We do not know if $\Psi^{\theta}=\Psi^{\theta^{\prime}}$ for
$\theta\neq\theta^{\prime}$!

###  5.2. Proof of Theorem B: continuity

The proof of continuity is almost identical for each twist parameter, so we
work only with $\Psi=\Psi^{0}$. Let us also assume for the remainder of this
section that there is a single peripheral $\zeta$. Lifting various equivalence
relations, $\Psi$ is described by a composition

$(\mu,\rho_{2})\mapsto(f_{\mu}^{\rho_{2}},\rho_{2})\mapsto(\Phi(f_{\mu}^{\rho_{2}}),\rho_{2})\mapsto(\rho_{1},\rho_{2})$

where $\rho_{1}$ is the holonomy of the hyperbolic metric $\mu^{\prime}$ on
$\Sigma$ such that the associated harmonic map from
$(\Sigma,\mu)\to(\Sigma,\mu^{\prime})$ has Hopf differential
$\Phi(f_{\mu}^{\rho_{2}})$. [36, Theorem 1.4] implies the association from
$\Phi(f_{\mu}^{\rho_{2}})\mapsto\rho_{1}$ is continuous. Here, the topology on
the space of holomorphic quadratic differentials is the Fréchet topology
coming from taking $L^{1}$-norms over a compact exhaustion. To show continuity
of $\mu\mapsto f_{\mu}^{\rho_{2}}\mapsto\Phi(f_{\mu}^{\rho_{2}})$ with respect
to this topology, note that, by the constructions of Section 3, we already
have continuity in the Teichmüller coordinate. Continuity will thus follow
from the results below.

######  Lemma 5.2.

Suppose representations $(\rho_{n})_{n=1}^{\infty}$ converge to an irreducible
$\rho$ in $\textrm{Hom}(\Gamma,G)$, and all such representations project to
$\textrm{Rep}_{\mathbf{c}}(\Gamma,G)$. Then $f_{\mu}^{\rho_{n}}$ converges to
$f_{\mu}^{\rho}$ in the $C^{\infty}$ sense on compacta as $n\to\infty$.

Below, we work in fixed local sections of the principal bundles
$\chi_{\mathbf{c}}(\Gamma,G)\to\textrm{Rep}_{\mathbf{c}}(\Gamma,G)$
$\chi_{\mathbf{c}}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R}))\to\textrm{Rep}_{\mathbf{c}}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R}))$
(having assumed in Section 1.2 that such things exist). We choose a basepoint
for the $\pi_{1}$ so that we can view these local systems through their
holonomies, which are genuine representations.

###### Proof.

We may assume each $\rho_{n}$ is irreducible and that there is a single cusp.
The parabolic and elliptic cases are much simpler than the hyperbolic case, so
we treat only the latter. Set $f_{n}=f_{\mu}^{\rho_{n}}$, $f=f_{\mu}^{\rho}$
and let $\varphi_{n}$, $\varphi$ be the approximation maps for
$f_{\mu}^{\rho_{n}}$, $f_{\mu}^{\rho}$ from Section 3.1. These maps project a
cusp neighbourhood onto a geodesic, but now the geodesic is varying with
$\rho_{n}$, and converging in the Gromov-Hausdorff sense to the geodesic for
$\rho$. If the $\varphi_{n}$’s can be chosen to vary smoothly, then we have a
uniform bound on the total energies (recall they depend on a choice of a
constant speed map onto their geodesic). As discussed in Section 3 (in
particular the proof of Lemma 3.9), this yields a uniform total energy bound
for $f_{n}$ on compacta. Via the argument described in Section 3.1, the maps
$f_{n}$ $C^{\infty}$-converge on compacta along a subsequence to some limiting
map $f^{\infty}$. The mapping is necessarily $\rho$-equivariant and harmonic.
To see that $f^{\infty}=f$, recall that $f$ is characterized by the fact that
its Hopf differential has a pole of order $2$ at the cusp and the residue has
a specific complex argument. To check this property, from the uniqueness in
Theorem 2.5 it suffices to check $d(f_{\infty},\varphi)<\infty$, and the
standard contradiction argument will also show there is no need to pass to a
subsequence. As remarked previously, the proof of [36, Lemma 5.2] shows
$d(f_{n},\varphi_{n})$ is maximized on $\partial\Sigma_{2}$. If $n_{k}$ is the
subsequence, then taking $k\to\infty$, via compactness we do win

$d(f_{\infty},\varphi_{\infty})\leq\limsup_{k\to\infty}\max_{\partial\Sigma_{2}}d(f_{n_{k}},\varphi_{n_{k}})\leq\max_{\partial\Sigma_{2}}d(f_{\infty},\varphi_{\infty})+1<\infty.$

We are left to argue that one can choose the $\varphi_{n}$ so that
$\varphi_{n}\to\varphi$ in the $C^{\infty}$ sense. We realize $\rho,\rho_{n}$
as monodromies of flat connections $\nabla,\nabla_{n}$ respectively on an
$X$-bundle $E$ over $\Sigma$ with structure group $G$. This is possible when
all $\rho_{n}$ lie in the same component of the relative representation space
([30, Corollay 6.1.2] generalizes to relative representation spaces), which we
are free to assume. The pullback bundle with respect to the universal covering
$\tilde{\Sigma}\to\Sigma$ identifies with the trivial bundle and also pulls
$\nabla,\nabla_{n}$ back to flat connections
$\tilde{\nabla},\tilde{\nabla}_{n}$. That is, up to an isomorphism, we have a
family of commutative diagrams

${(X\times\tilde{\Sigma},\tilde{\nabla}_{n})}$${(E,\nabla_{n})}$${\tilde{\Sigma}}$${\Sigma.}$

The bundle $E$ and connections $\nabla_{n}$ are constructed by choosing a good
covering of $\Sigma$ and depend on the local section of the principal bundle
$\chi_{\mathbf{c}}(\Gamma,G)\to\textrm{Rep}_{\mathbf{c}}(\Gamma,G)$ (see the
proof of [30, Lemma 6.1.1]). We can build a good covering by glueing a good
covering of a relatively compact tubular neighbourhood of $(\Sigma_{2},\mu)$
with a good covering of a tubular neighbourhood of
$(\Sigma\backslash\Sigma_{2},\mu)$. This ensures a lower bound on the
$\mu$-radius inside $\Sigma_{2}$ (note that we cannot do this on all of
$\Sigma$). With this constraint, the local systems can be prescribed so that
$\nabla_{n}\to\nabla$ in the $C^{\infty}$ sense on $\Sigma_{2}$ in the affine
space of connections.

The map $\varphi$ induces a section $s$ of the bundle $(E,\nabla)$, which can
also be seen as a section $s_{n}$ of $(E,\nabla_{n})$. With respect to the
diagram above, $s_{n}$ pulls back to a $\rho_{n}$-equivariant map
$\tilde{\varphi}_{n}:\mathbb{H}\to X$, and by our comments above,
$\tilde{\varphi}_{n}\to\varphi$ in the $C^{\infty}$ sense. The maps
$\tilde{\varphi}_{n}$ are most likely not harmonic and may not project onto
the geodesic axis of $\rho_{n}(\zeta)$. However, because they are converging
to $\varphi$, the geodesic curvature of
$\tilde{\varphi}_{n}(\partial\tilde{\Sigma}_{2})$ is tending to $0$. Moreover,
$\varphi|_{\partial\tilde{\Sigma}_{2}}$ is arbitrarily close to some
parametrization of a geodesic. So, we set $\varphi_{n}$ to be the harmonic map
with boundary data equal to this nearby geodesic projection. From energy
control on $\tilde{\varphi}_{n}$, we get the same for $\varphi_{n}$, and hence
convergence along a subsequence to a harmonic map $\varphi_{\infty}$. From the
boundary values, we must have $\varphi_{\infty}=\varphi$, and as usual we can
show there is no need to pass to a subsequence. ∎

For a reducible representation, the harmonic maps are not unique but differ by
translations along a geodesic. However, the Hopf differential is unique. We
can choose a sequence $\varphi_{n}\to\varphi$ as in the previous
proposition—boundary values are fixed so we do have uniqueness for these
harmonic maps. Then we take the harmonic maps $f_{n}$ built from using the
approximating maps $\varphi_{n}$, and the proof above goes through, with the
minor modification that we must rescale the approximating maps as in [36,
Proposition 5.1]. $C^{\infty}$ convergence implies the Hopf differentials
converge.

######  Remark 5.3.

For different twist parameters, we simply precompose every $\varphi_{n}$ with
a fractional Dehn twist. Using (18), the energies are uniformly controlled.

###  5.3. Asymmetric metrics on Teichmüller spaces

In [19, Section 8], Guéritaud-Kassel define asymmetric pseudo-metrics on
deformation spaces of geometrically finite hyperbolic manifolds. These metrics
are natural generalizations of Thurston’s asymmetric metric on Teichmüller
space. We will use such a metric in the proof of bi-continuity of
$\Psi^{\theta}$.

For $[\rho_{1}],[\rho_{2}]\in\mathcal{T}_{\mathfrak{c}}(S_{g,n})$, we set

$C(\rho_{1},\rho_{2})=\inf\\{\textrm{Lip}(f):f:\mathbb{H}\to\mathbb{H}\textrm{
is }(\rho_{1},\rho_{2})-\textrm{equivariant}\\}.$

The metric
$d_{Th}:\mathcal{T}_{\mathfrak{c}}(S_{g,n})\times\mathcal{T}_{\mathfrak{c}}(S_{g,n})\to\mathbb{R}$
is defined by

$d_{Th}([\rho_{1}],[\rho_{2}])=\log\Big{(}C(\rho_{1},\rho_{2})\frac{\delta(\rho_{1})}{\delta(\rho_{2})}\Big{)}.$

Here $\delta(\rho_{1})$ is the critical exponent of $\rho_{1}$:

$\delta(\rho_{1})=\lim_{R\to\infty}\frac{1}{R}\log\\#(j(\Gamma)\cdot p\cap
B_{p}(R)),$

where $p$ is any point in $\mathbb{H}$ and $B_{p}(R)$ is the ball of radius
$r$ centered at $p$. Alternatively, $\delta(\rho_{1})$ is the Hausdorff
dimension of the limit set of $\rho_{1}$ in
$\partial_{\infty}\mathbb{H}=S^{1}$. If all $a_{j}=0$, so that all critical
exponents are $1$ and $\mathcal{T}_{\mathfrak{c}}(S_{g,n})$ is the ordinary
Teichmüller space of a surface of finite volume, then this agrees with
Thurston’s asymmetric metric introduced in [43]. Here asymmetric means that in
general,

$d_{Th}([\rho_{1}],[\rho_{2}])\neq d_{Th}([\rho_{2}],[\rho_{1}]).$

######  Proposition 5.4 (Guéritaud-Kassel, Lemma 8.1 in [19]).

The function
$d_{Th}:\mathcal{T}_{\mathbf{c}}(\Gamma)\times\mathcal{T}_{\mathbf{c}}(\Gamma)\to\mathbb{R}$
is a continuous asymmetric metric.

######  Remark 5.5.

The correction factor $\delta(\rho_{1})/\delta(\rho_{2})$ is needed for non-
negativity. For a general hyperbolic $n$-manifold, continuity may fail and the
generalization may be just an asymmetric pseudo-metric ($d(x,y)$ need not
imply $x=y$).

This metric allows us to control translation lengths nicely. Consider a left
open ball around $\rho_{2}$:

$B_{Th}(\rho_{2},C)=\\{[\rho_{1}]\in\mathcal{T}_{\mathbf{c}}(\Gamma):d_{Th}(\rho_{1},\rho_{2})<C\\}.$

If $\rho_{1}\in B_{Th}(\rho_{2},C)$, there is a
$(\rho_{1},\rho_{2})$-equivariant Lipschitz $g$ map with Lipschitz constant
$<e^{C}$. Therefore, for every $\gamma\in\Gamma$,

$d(\rho_{2}(\gamma)g(z),g(z))=d(g(\rho_{1}(\gamma)z),g(z))<e^{C}d(\rho_{1}(\gamma)z,z),$

and it follows that $\ell(\rho_{2}(\gamma))<e^{C}\ell(\rho_{1}(\gamma))$.

###  5.4. Proof of Theorem B: bi-continuity

The inverse mapping of $\Psi^{\theta}$, on the Teichmüller side, takes as
input a class of an almost strictly dominating pair $(\rho_{1},\rho_{2})$ and
returns the unique minimizer of
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$. We can follow the
same approach as in [41, subsection 2.4] here, adapted to our infinite energy
setting.

Let $X,Y$ be metric spaces and $(F_{y})_{y\in Y}$ a family of continuous
functions $F_{y}:X\to\mathbb{R}$ depending continuously on $y$ in the compact-
open topology. $(F_{y})_{y\in Y}$ is said to be uniformly proper if for any
$C\in\mathbb{R}$, there exists a compact subset $C\subset X$ such that for all
$y\in Y$ and $x\not\in K$, we have $F_{y}(x)>C$. We say that the family
$(F_{y})_{y\in Y}$ is locally uniformly proper if for all $y_{0}\in Y$, there
is a neighbourhood $U$ of $y_{0}$ such that $(F_{y})_{y\in
U}\subset(F_{y})_{y\in Y}$ is uniformly proper.

######  Lemma 5.6 (Proposition 2.6 in [41]).

Let $X$ and $Y$ be two metric spaces and $(F_{y})_{y\in Y}$ a locally
uniformly proper family of continous functions from $X$ to $\mathbb{R}$
depending continuously on $Y$ (for the compact open topology). Assume that
each $F_{y}$ achieves its minimum at a unique point $x_{m}(y)\in X$. Then the
function

$y\mapsto x_{m}(y)$

is continuous.

We verify the conditions for
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$, with
$(\rho_{1},\rho_{2})$ living in the space of almost strictly dominating
representations. For the remainder of this subsection, we are working over
local sections for our bundles of local systems over representation spaces.

######  Lemma 5.7.

The association
$(\rho_{1},\rho_{2})\mapsto\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$
is continuous for the compact-open topology.

In the proof, we require control on the energy of the harmonic maps at the
cusp, as we vary the source metric and the representation. We defer the proof
of the lemmas below to the next subsection. In these lemmas, let $\rho_{0}$ be
a reductive representation and $\mu_{0}$ a hyperbolic metric.

######  Lemma 5.8.

Suppose $\rho_{0}$ has parabolic monodromy. Then for every representation
$\rho$ in the same representation space that is close enough to $\rho_{0}$,
and metric $\mu$ close to $\mu_{0}$, there is a function $\tilde{e}$ that is
integrable in the flat metric and such that

$\sqrt{|\mu|}e(\mu,f_{\mu}^{\rho})\leq\tilde{e}$

everywhere.

######  Lemma 5.9.

Suppose $\rho_{0}$ has hyperbolic monodromy. Then for every representation
$\rho$ in the same representation variety that is close enough to $\rho_{0}$,
and metric $\mu$ close to $\mu_{0}$, working in the cusp coordinates for $\mu$
there is a $y_{0}>0$, $C,c>0$ such that for all $y\geq y_{0}$,

$\frac{\Lambda(\theta)\ell^{2}}{2\tau_{\mu}^{2}}-Ce^{-cy}\leq\sqrt{\mu}e(\mu,f_{\mu}^{\rho})\leq\frac{\Lambda(\theta)\ell^{2}}{2\tau_{\mu}^{2}}+Ce^{-cy},$

where $f_{\mu}^{\rho}$ is the harmonic map with twist parameter $\theta$.

###### Proof.

We want to show that if $(i_{n},j_{n})\to(\rho_{1},\rho_{2})$ in
$\textrm{Rep}_{\mathbf{c}}(\Gamma,\operatorname{\textrm{PSL}}(2,\mathbb{R})\times
G)$ and $K\subset\mathcal{T}(\Gamma)$ is compact, then
$\operatorname{\mathcal{E}}_{n}:=\operatorname{\mathcal{E}}_{i_{n},j_{n}}^{\theta}\to\operatorname{\mathcal{E}}=\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$
uniformly on $K$ as $n\to\infty$. The finite energy case is easy: from our
previous results, if $\mu_{n}\to\mu$ and $\rho_{n}\to\rho$, then the energy
densities of the harmonic maps converge pointwise to $e(\mu,f_{\mu}^{\rho})$
(recall this is independent of the harmonic map if $\rho$ is reducible). Lemma
5.8 then justifies an application of the domination convergence theorem, so
that the total energies converge.

Going forward, we assume the monodromoy is hyperbolic. Fixing a metric $\mu$,
set $h_{n}=h_{\mu}^{i_{n}}$, $f_{n}=f_{\mu}^{j_{n}}$, $h=h_{\mu}^{\rho_{1}}$,
$f=f_{\mu}^{\rho_{2}}$. It suffices to show
$\operatorname{\mathcal{E}}_{n}(\mu)\to\operatorname{\mathcal{E}}(\mu)$, and
that the rate only depends on the Teichmüller distance from $\mu$ to a base
metric $\mu_{0}$. By Lemma 5.9, for $r>y_{0}$,

$\displaystyle|\operatorname{\mathcal{E}}(\mu)-\operatorname{\mathcal{E}}_{n}(\mu)|$
$\displaystyle=\Big{|}\int_{\Sigma_{r}}(e(\mu,h)-e(\mu,f))-(e(\mu,h_{n})-e(\mu,f_{n}))dA_{\mu}\Big{|}$
$\displaystyle+\Big{|}\int_{\Sigma\backslash\Sigma_{r}}(e(\mu,h)-e(\mu,f))-(e(\mu,h_{n})-e(\mu,f_{n}))dA_{\mu}\Big{|}$
$\displaystyle\leq\Big{|}\int_{\Sigma_{r}}(e(\mu,h)-e(\mu,f))-(e(\mu,h_{n})-e(\mu,f_{n}))dA_{\mu}\Big{|}+\int_{\Sigma\backslash\Sigma_{r}}Ce^{-cy}dA_{\mu}$

holds for every $\mu$ close enough to $\mu_{0}$. Fixing $\epsilon>0$, for
every $r\geq y_{0}$ we can find $N_{r}>0$, depending only on Teichmüller
distance to $\mu_{0}$, such that for all $n\geq N_{r}$, the first integral is
$<\epsilon/2$. Hence, for such $n$,

$|\operatorname{\mathcal{E}}(\mu)-\operatorname{\mathcal{E}}_{n}(\mu)|<\epsilon/2+\frac{2\pi
C}{c}e^{-cr}.$

Taking $r=c^{-1}\log(4\pi C\epsilon^{-1})$, we get
$|\operatorname{\mathcal{E}}(\mu)-\operatorname{\mathcal{E}}_{n}(\mu)|<\epsilon$.
∎

We now show that the functionals
$\operatorname{\mathcal{E}}_{\rho_{1},\rho_{2}}^{\theta}$ are locally
uniformly proper. From here we assume $\theta=0$, because the proof is
identical for every $\theta$. We essentially show that the bounds from the
proof of Proposition 4.1 depend continuously on $([\rho_{1}],[\rho_{2}])$. For
$([\rho_{1}],[\rho_{2}])\in\textrm{ASD}_{\mathbf{c}}(\Gamma,G)$, we choose an
open neighbourhood $U\subset\textrm{ASD}_{\mathbf{c}}(\Gamma,G)$ containing
$([\rho_{1}],[\rho_{2}])$ with compact closure. We intersect it with a product
open set $U_{1}\times U_{2}$, where $U_{1}$ is a left $d_{Th}$-open ball
around $\rho_{1}$. We then lift via some section to the space of local
sections to view these points as representations. Picking a boundary geodesic
or a horocycle for $C(\mathbb{H}/\rho_{1}(\Gamma))$, as we perturb the
representations we get a continuously varying family of such curves in the new
metric. We write $C_{d}^{j}(\xi)$ for the collar neighbourhood of such a curve
in $\mathbb{H}/j(\Gamma)$, $j\in U_{2}$. By choosing $U$ even small enough, we
can assume we have a fixed presentation for our fundamental group, and the
collar neighbourhood $C_{d_{j}}^{j}(\xi)$ has uniform upper and lower bounds
$\delta_{1}\leq d_{j}\leq\delta_{2}$.

Set $\delta=(\delta_{1}+\delta_{2})/2$. We can choose a neighbourhood
$C_{\delta}$ containing every $C_{\delta}^{j}(\xi)$ for all $j\in U_{2}$. For
a $(j,\rho)$-equivariant map $g$, we put

$\textrm{Lip}_{\delta}(g)=\max_{x\in C(\mathbb{H}/j(\Gamma))\backslash
C_{\delta}}\textrm{Lip}_{x}(g).$

######  Lemma 5.10.

Shrinking $U$ if necessary, there exists an $\epsilon>0$ such that for every
$(j,\rho)\in U$, there is a $(j,\rho)$-equivariant map $g_{j,\rho}$ that
satisfies $\textrm{Lip}(g_{j,\rho})\leq 1$ and
$\textrm{Lip}_{\delta}(g_{j,\rho})<(1-\epsilon)^{1/2}$. Moreover, we can
choose it so that if $\rho(\zeta)$ is hyperbolic, then $g_{j,\rho}$ translates
the geodesic axis of $j(\zeta)$ along the axis of $\rho(\zeta)$ with constant
speed $1$.

###### Proof.

We first define $g_{j,\rho}$ on the complement of $C_{\delta}$. One at a time,
we vary $j$ and then $\rho$. By our choice of $U_{1}$, there is an
$\epsilon_{0}>0$ such that for every $j\in U_{1}$, there is a
$(j,\rho_{1})$-equivariant $(1+\epsilon_{0})$-Lipschitz map. Composing with
our original optimal $(\rho_{1},\rho_{2})$-equivariant map, we get a
$(j,\rho_{2})$-equivariant map with nice control. Choosing $U_{1}$ small
enough, we can shrink $\epsilon_{0}$ so as to ensure the right behaviour
outside of $C_{\delta}$.

Now we fix a base surface $\mathbb{H}/j(\Gamma)$ and vary $\rho$ around
$\rho_{2}$. We can use flat connections as in Lemma 5.2. For any continuous
path of classes with initial point $\rho_{2}$, the procedure detailed there
gives a path of equivariant maps starting at $g$. From compactness of the
complement of the collar and cusp neighbourhoods, the local Lipschitz
constants vary upper semicontinuously. In particular, we can achieve an upper
bound $\textrm{Lip}_{\delta}(\cdot)<(1-\epsilon)^{1/2}$ when close enough to
$([\rho_{1}],[\rho_{2}])$.

Now we extend in $C_{\delta}$ and above. Note that while the local Lipschitz
constants of $g_{j,\rho}$ are uniformly controlled, this can be strictly below
the global Lipschitz constant, and in the hyperbolic case this global
Lipschitz constant is exactly $1$. To see this, we do have
$d_{\nu}(g_{j,\rho}(x),g_{j,\rho}(y))<d_{\sigma}(x,y)$ for every $x\neq y$, so
$\textrm{Lip}(g_{j,\rho})\leq 1$ certainly. But in the case of hyperbolic
monodromy, for any two points $x,y$ that are connected by a segment that
mostly fellow-travels the geodesic axis of $j(\zeta)$,

$d_{\nu}(g_{j,\rho}(x),g_{j,\rho}(y))=d_{\sigma}(x,y)+O(1),$

where the implied constant depends only on the position of $x$ and $y$ in
$\mathbb{H}/j(\Gamma)$. Since $x,y$ can be taken as far as we like, in the
limit the ratio of distances becomes $1$. Using the equivariant Kirszbraun-