# Infinite energy harmonic maps from Riemann surfaces to CAT(0) spaces

Georgios Daskalopoulos  Department of Mathematics
Brown Univeristy
Providence, RI<EMAIL_ADDRESS>and  Chikako Mese  Johns Hopkins
University
Department of Mathematics
Baltimore, MD<EMAIL_ADDRESS>
###### Abstract.

We present some results about harmonic maps with possibly infinite energy from
punctured Riemann surfaces to CAT(0) spaces. In particular, we give precise
estimates of their energy growth near the punctures and prove uniqueness.

GD supported in part by NSF DMS-2105226, CM supported in part by NSF
DMS-2005406.

Key words: harmonic maps, uniqueness, CAT(0), Riemann surface, infinite energy

Table of Contents

  * §1 Introduction p. 1

  * §2 Preliminaries p. 2

  * §3 Harmonic maps from a punctured disk p. 3

  * §4 Existence of infinite energy harmonic maps p. 4

  * §5 Uniqueness of infinite energy harmonic maps p. 5

##  1\. Introduction

This is the first in a series of papers where we develop the techniques to
study non-abelian Hodge theory on quasi-projective varieties. The sequel is
[DM2] and [BDDM]. More specifically, in this first paper, we study harmonic
maps with possibly infinite energy from punctured Riemann surfaces into
complete CAT(0) spaces and we a give precise estimate of their energy growth
near the punctures. We also prove a uniqueness result. Both results are
crucial for the next papers.

Throughout this paper, we use the following notation unless explicitly stated
otherwise:

  * •

$\tilde{X}$ is a complete CAT(0)-space

  * •

$\bar{\mathcal{R}}$ is a compact Riemann surface

  * •

$\mathcal{R}=\bar{\mathcal{R}}\backslash\\{p^{1},\dots,p^{n}\\}$ is a
punctured Riemann surface

  * •

$\Pi:\tilde{\mathcal{R}}\rightarrow\mathcal{R}$ is the universal covering map

  * •

$\rho:\pi_{1}(\mathcal{R})\rightarrow\mathsf{Isom}(\tilde{X})$ is a
homomorphism.

For each puncture $p^{j}$, $j=1,\dots,n$,

  * •

$\mathbb{D}^{j}$ is a conformal disks in $\mathcal{R}$ centered at the
puncture $p^{j}$ ($\mathbb{D}^{j}\cap\mathbb{D}^{i}=\emptyset$ for $i\neq j$)

  * •

$\mathbb{D}^{j}_{r}=\\{z\in\mathbb{D}:|z|<r\\}$

  * •

$\lambda^{j}\in\pi_{1}(\mathcal{R})$ is the conjugacy class associated to the
loop in $\mathcal{R}$ around $p^{j}$,

  * •

$I^{j}:=\rho(\lambda^{j})\in\mathsf{Isom}(\tilde{X})$, and

  * •

$\Delta_{I^{j}}:=\inf_{P\in\tilde{X}}d(I^{j}(P),P)$ is the translation length
of $I^{j}$.

Recall that $I^{j}$ is said to be semisimple if there exists
$P_{*}\in\tilde{X}$ such that $d(I^{j}(P_{*}),P_{*})=\Delta_{I^{j}}$.

We prove the following two theorems.

######  Theorem 1.1 (Existence).

Assume

  * (A)

The action of $\rho(\pi_{1}(\mathcal{R}))$ does not fix a point at infinity.

  * (B)

The isometry $I^{j}$ is semisimple or there exist a geodesic ray
$c:[0,\infty)\rightarrow\tilde{X}$ and constants $a,b>0$ such that

$d^{2}(I^{j}(c(t)),c(t))\leq\Delta_{I^{j}}^{2}(1+be^{-at}).$

Then there exists a $\rho$-equivariant harmonic map
$u:\tilde{\mathcal{R}}\rightarrow\tilde{X}$ with logarithmic energy growth
towards the punctures, i.e.

(1.1) $\sum_{j=1}^{n}\frac{\Delta^{2}_{I^{j}}}{2\pi}\log\frac{1}{r}\leq
E^{u}[\mathcal{R}_{r}]\leq\sum_{j=1}^{n}\frac{\Delta^{2}_{I^{j}}}{2\pi}\log\frac{1}{r}$

where
$\mathcal{R}_{r}=\mathcal{R}\backslash\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r}$ and
$E^{u}[\mathcal{R}_{r}]$ is the energy of $u$ in $\mathcal{R}_{r}$.

For smooth targets, the existence essentially follows from [Loh], [Wo] and
[KM]. The new statement in Theorem 1.1 is the precise growth estimate towards
the punctures of the harmonic map.

We also prove the following uniqueness result.

######  Theorem 1.2 (Uniqueness).

Assume (A) and (B) of Theorem 1.1. If the $\rho$-equivariant harmonic map
$\tilde{u}:\tilde{\mathcal{R}}\rightarrow\tilde{X}$ has logarithmic energy
growth (cf. (1.1)), then it is the unique harmonic map with logarithmic energy
growth in the following cases:

  * (i)

$\tilde{X}$ is a negatively curved space (i.e. for any $P\in\tilde{X}$, there
exists a neighborhood $U$ of $P$ which is $CAT(-\kappa)$ for some $\kappa>0$),

  * (ii)

$\tilde{X}$ is an irreducible symmetric space of non-compact type, or

  * (iii)

$\tilde{X}$ is an irreducible locally finite Euclidean building such that the
action of $\rho(\pi_{1}(\mathcal{R}))$ does not fix an unbounded closed convex
strict subset of $\tilde{X}$.

In the case when the harmonic map has finite energy, uniqueness follows from
previous works of the authors and Corlette ([Me], [Co], [DM1] for cases (i),
(ii), (iii) respectively). In [DM2], we use the results of this paper to
develop non-Abelian Hodge theory techniques extending the works by Gromov-
Schoen [GS] from the finite to the infinite energy case and Mochizuki [M] from
the Archimedean case to the non-Archimedean case. The upcoming joint paper
[BDDM] contains an application of this technique to non-Abelian Hodge theory.

Summary of the paper. For the purpose of explaining the main ideas, assume
that $\tilde{X}$ is a universal cover of a compact manifold $X$ and let
$\gamma:\mathbb{S}^{1}\rightarrow X$ be an arc length parameterization of a
minimizing closed geodesic in $X$. Denote the energy of $\gamma$ by
$E^{\gamma}$. A prototype map $v$ corresponding to $\gamma$ is

$v:[0,\infty)\times\mathbb{S}^{1}\rightarrow X$

which is equal to $\gamma$ on slices $\\{t\\}\times\mathbb{S}^{1}$ for all
sufficiently large $t$. (We note that the idea of the prototype map first
appears in [Loh].) The energy of $v$ on the finite cylinder
$[0,T]\times\mathbb{S}^{1}$ is bounded from above by $T\cdot E^{\gamma}+C$
where $C$ is a constant independent of $T$. Let

$u_{T}:[0,T]\times\mathbb{S}^{1}\rightarrow X$

be the Dirichlet solution on the finite cylinder $[0,T]\times\mathbb{S}^{1}$
with boundary values equal to $v$. Since $u_{T}$ is an energy minimizing map,
the energy of $u_{T}$ is bounded from above by $T\cdot E^{\gamma}+C$. Since
$\gamma$ is also an energy minimizing map, the energy of $u_{T}$ in any finite
cylinder $[0,t]\times\mathbb{S}^{1}$ for $0<t\leq T$ is bounded from below by
$t\cdot E^{\gamma}$. From this, we conclude that the energy of $u_{T}$ on any
finite cylinder $[t_{1},t_{2}]\times\mathbb{S}^{1}$ has an upper bound
independent of $T$; namely, $(t_{2}-t_{1})\cdot E^{\gamma}+C$. Thus, the
regularity theory for harmonic maps implies that the family of maps
$\\{u_{T}\\}_{T\geq 1}$ has a uniform Lipschitz bound on any compact subset of
$[0,\infty)\times\mathbb{S}^{1}$. This implies that there exists a sequence
$T_{i}\rightarrow\infty$ such that $u_{T_{i}}$ converges locally uniformly to
a harmonic map $u:[0,\infty)\times\mathbb{S}^{1}\rightarrow X$. The conformal
equivalence of the punctured disk $\mathbb{D}^{*}$ and the infinite cylinder
$[0,\infty)\times\mathbb{S}^{1}$ thus defines a harmonic map on
$\mathbb{D}^{*}$ with lower and upper energy bound

(1.2) $E^{\gamma}\log\frac{1}{r}\leq E^{u}[\mathbb{D}_{r}]\leq
E^{\gamma}\log\frac{1}{r}+C.$

We extend this idea to prove the existence of equivariant harmonic maps from
any punctured Riemann surface $\Sigma$.

For finite energy harmonic maps, the uniqueness can be derived from the fact
that the energy functional is a convex function along a geodesic
interpolation. Namely, if $u_{t}$ is the geodesic interpolation map (i.e.
$t\mapsto u_{t}(x)$ is a geodesic), then
$E^{u_{t}}\leq(1-t)E^{u_{0}}+tE^{u_{1}}$. If $u_{0}$ and $u_{1}$ are energy
minimizing maps, then the convexity implies that $t\mapsto E^{u_{t}}$ is a
constant function. This idea does not directly generalize to our situation
because we deal with infinite energy maps. The main idea to prove the
uniqueness assertion in our setting is to introduce a modified energy
functional by subtracting off the logarithmic energy growth near the
punctures. We then prove that the modified energy is constant along the
geodesic interpolation of harmonic maps. This allows us to apply the argument
used by the authors in [Me], [DM1].

##  2\. Preliminaries

###  2.1. Maps into CAT(0) spaces

######  Definition 2.1.

A say that $\tilde{X}$ is a complete CAT(0) space if it is a complete geodesic
space that satisfies the following condition: For any three points
$P,R,Q\in\tilde{X}$ and an arclength parameterized geodesic
$c:[0,l]\rightarrow\tilde{X}$ with $c(0)=Q$ and $c(l)=R$,

$d^{2}(P,Q_{t})\leq(1-t)d^{2}(P,Q)+td^{2}(P,R)-t(1-t)d^{2}(Q,R)$

where $Q_{t}=c(tl)$. (We refer to [BH] for more details.)

######  Notation 2.2.

_It follows immediately from Definition 2.1 that, given $P,Q\in\tilde{X}$ and
$t\in[0,1]$, there exists a unique point with distance from $P$ equal to
$td(P,Q)$ and the distance from $Q$ equal to $(1-t)d(P,Q)$. We denote this
point by_

$(1-t)P+tQ.$

Fix a smooth conformal Riemannian metric $g$ on $\mathcal{R}$ (i.e. $g$ is
given by $\rho(z)\,dzd\bar{z}$ for some smooth function $\rho$ in any
holomorphic coordinate $z$ of $\mathcal{R}$). We refer to [KS1] for the notion
of energy $E^{f}$ and energy density function $|\nabla f|^{2}$ of a map
$f:\mathcal{R}\rightarrow\tilde{X}$. The (Korevaar-Schoen) energy of $f$ in a
domain $\Omega\subset\mathcal{R}$ is

$E^{f}[\Omega]=\int_{\Omega}|\nabla f|^{2}d\mbox{vol}_{g}.$

It is well-known that $E^{f}[\Omega]$ and $|\nabla f|^{2}d\mbox{vol}_{g}$ is
invariant of the choice of the smooth conformal Riemannian metric $g$.

We say a continuous map $u:\mathcal{R}\rightarrow\tilde{X}$ is harmonic if it
is locally energy minimizing; more precisely, at each $p\in\mathcal{R}$, there
exists a neighborhood $\Omega$ of $p$ so that all continuous comparison maps
which agree with $u$ outside of this neighborhood have no less energy.

For $V\in\Gamma\Omega$ where $\Gamma\Omega$ is the set of Lipschitz vector
fields on $\Omega$, $|f_{*}(V)|^{2}$ is similarly defined. The real valued
$L^{1}$ function $|f_{*}(V)|^{2}$ generalizes the norm squared on the
directional derivative of $f$. The generalization of the pull-back metric is
the continuous, symmetric, bilinear, non-negative and tensorial operator

$\pi_{f}(V,W)=\Gamma\Omega\times\Gamma\Omega\rightarrow L^{1}(\Omega,{\bf R})$

where

$\pi_{f}(V,W)=\frac{1}{2}|f_{*}(V+W)|^{2}-\frac{1}{2}|f_{*}(V-W)|^{2}.$

We refer to [KS1] for more details.

Fix a local chart $(U;z)$. By the conformal invariance of the energy, we can
assume that the metric $g$ the Eucliean metric $dzd\bar{z}$ in $U$. We extend
$\pi_{f}$ linearly cover $\mathbb{C}$. Then energy density function of $f$ is
given by

$\frac{1}{4}|\nabla f|^{2}=\pi_{f}(\frac{\partial f}{\partial
z^{i}},\frac{\partial f}{\partial\bar{z}^{j}}).$

Furthermore, set $z=re^{i\theta}$ to define polar coordinates $(r,\theta)$. We
define

$\left|\frac{\partial u}{\partial
r}\right|^{2}:=\pi_{f}(\frac{\partial}{\partial r},\frac{\partial}{\partial
r})\ \mbox{ and }\ \left|\frac{\partial
u}{\partial\theta}\right|^{2}:=\pi_{f}(\frac{\partial}{\partial\theta},\frac{\partial}{\partial\theta}).$

###  2.2. Equivariant maps and section of the flat $\tilde{X}$-bundle

######  Definition 2.3.

Let $\rho:\pi_{1}(M)\rightarrow\mathsf{Isom}(\tilde{X})$. A map
$\tilde{f}:\tilde{M}\rightarrow\tilde{X}$ is said to be $\rho$-equivariant if

$\tilde{f}(\gamma p)=\rho(\gamma)\tilde{f}(p),\ \ \forall\gamma\in\pi_{1}(M),\
p\in\tilde{M}.$

Following Donaldson [Do], we will replace equivariant maps with sections of an
associated fiber bundle. The quotient under the action of $\pi_{1}(M)$ on the
product $\tilde{M}\times\tilde{X}$ is the twisted product

$\tilde{M}\times_{\rho}\tilde{X}.$

In other words, $\tilde{M}\times_{\rho}\tilde{X}$ is the set of orbits
$[(p,x)]$ of a point $(p,x)\in\tilde{M}\times\tilde{X}$ under the action of
$\gamma\in\pi_{1}(M)$ via the deck transformation on the first component and
the isometry $\rho(\gamma)$ on the second component. The fiber bundle

$\tilde{M}\times_{\rho}\tilde{X}\rightarrow M$

with fibers over $p\in M$ is isometric to $\tilde{X}$ is the flat
$\tilde{X}$-bundle over $M$ defined by $\rho$.

There is a one-to-one correspondence between sections of this fibration and
$\rho$-equivariant maps

$\tilde{f}:\tilde{M}\rightarrow\tilde{X}\ \ \ \ \longleftrightarrow\ \ \ \
f:M\rightarrow\tilde{M}\times_{\rho}\tilde{X}$

satisfying the relationship

$[(\tilde{p},\tilde{f}(\tilde{p}))]\leftrightarrow f(p)\ \mbox{ where
$\Pi(\tilde{p})=p$.}$

Since the energy density function $|\nabla\tilde{f}|^{2}$ of $\tilde{f}$ is a
$\rho$-invariant function, we can define

$|\nabla f|^{2}(p):=|\nabla\tilde{f}|^{2}(\tilde{p}).$

We can similarly define the pullback inner product and directional energy
density functions of $f$ by using the corresponding notions for $\tilde{f}$
given in Subsection 2.1. For $U\subset M$, the (vertical) energy of a section
$f$ is

(2.1) $E^{f}[U]=\int_{U}|\nabla f|^{2}d\mbox{vol}_{g}.$

Furthermore, for sections $f_{1}$, $f_{2}$, we define

(2.2)
$d(f_{1}(p),f_{2}(p)):=d(\tilde{f}_{1}(\tilde{p}),\tilde{f}_{2}(\tilde{p}))$

where $\tilde{f}_{1}$, $\tilde{f}_{2}$ are the associated $\rho$-equivariant
maps to sections $f_{1}$, $f_{2}$ respectively.

###  2.3. Sublogarithmic growth

We will

(2.3) fix a conformal disk $\mathbb{D}^{j}\subset\bar{\mathcal{R}}$ centered
at each puncture $p^{j}$

such that $\mathbb{D}^{i}\cap\mathbb{D}^{j}=\emptyset$ for $i\neq j$.
Furthermore, let

$\mathbb{D}^{j*}=\mathbb{D}^{j}\backslash\\{0\\}$

.

Fix $P_{0}\in\tilde{X}$ and a fundamental domain $F$ of $\tilde{\mathcal{R}}$.
Let $f_{0}$ be the section of the fiber bundle
$\tilde{\mathcal{R}}\times_{\rho}\tilde{X}\rightarrow\mathcal{R}$ such that,
for any $p\in\mathcal{R}\cap\Pi(F)$, $f_{0}(p)=[(\tilde{p},P_{0})]$ where
$\tilde{p}=\Pi^{-1}(p)\cap F$. (Note that $\Pi(F)$ is of full measure in
$\mathcal{R}$.)

For a given section
$f:\mathcal{R}\rightarrow\mathcal{R}\times_{\rho}\tilde{X}$, define

(2.4) $\delta_{j}:\mathbb{D}^{j*}\rightarrow[0,\infty),\ \ \ \
\delta_{j}(z)=\operatorname*{ess\,inf}_{\\{z\in\mathbb{D}^{j*}\\}}d(f(z),f_{0}(z)).$

Recall that $d(f(z),f_{0}(z))$ is defined by (2.2).

######  Definition 2.4.

We say a section
$f:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ (or its
associated equivariant map) has _sub-logarithmic growth_ if for any
$j=1,\dots,n$

$\lim_{|z|\rightarrow 0}\delta_{j}(z)+\epsilon\log|z|=-\infty\mbox{ in
}\mathbb{D}^{j*}.$

By the triangle inequality, this definition is independent of the choice of
$P_{0}\in\tilde{X}$. We say that $f$ has logarithmic energy growth if near the
punctures satisfies condition (1.1) in Theorem 1.1.

##  3\. Harmonic maps from a punctured disk

Before we consider the domain to be a general punctured Riemann surface, we
first start by studying harmonic maps from a punctured disk
$\bar{\mathbb{D}}^{*}$. Denote

$\displaystyle\mathbb{D}_{r}^{*}$ $\displaystyle=$
$\displaystyle\\{z\in\mathbb{D}:|z|<r\\}$ $\displaystyle\mathbb{D}_{r,r_{0}}$
$\displaystyle=$ $\displaystyle\mathbb{D}_{r_{0}}\backslash\mathbb{D}_{r}.$

Identify the boundary of the disk with the circle; i.e.

$\partial\mathbb{D}=\mathbb{S}^{1},$

This induces a natural identification

$2\pi\mathbb{Z}\simeq\pi_{1}(\mathbb{S}^{1})\simeq\pi_{1}(\bar{\mathbb{D}}^{*}).$

The main result of this section is the following.

######  Theorem 3.1 (Existence and Uniqueness of the Dirichlet solution on
$\mathbb{D}^{*}$).

Assume the following:

  * •

$\rho:2\pi\mathbb{Z}\simeq\pi_{1}(\mathbb{S}^{1})\simeq\pi_{1}(\bar{\mathbb{D}}^{*})\rightarrow\mathsf{Isom}(\tilde{X})$
is a homomorphism

  * •

$k:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
is a locally Lipschitz section

  * •

$I:=\rho([\mathbb{S}^{1}])$ satisfy condition (B) of Theorem 1.1 with $I^{j}$
replaced by $I$.

Then there exists a harmonic section

$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}\mbox{
with }u|_{\mathbb{S}^{1}}=k|_{\mathbb{S}^{1}}.$

Furthermore, there exists a constant $C>0$ that depends only on
$\mathsf{E}_{\rho}=\frac{\Delta_{I}^{2}}{2\pi}$, $a$, $b$ from (B) if $I$ is
not semisimple and the section $k$ satisfying the following properties:

  * (i)

$\displaystyle{\mathsf{E}_{\rho}\log\frac{1}{r}\leq
E^{u}[\mathbb{D}_{r,1}]\leq\mathsf{E}_{\rho}\log\frac{1}{r}+C,\ \ \ 0<r\leq
1}$

  * (ii)

$\displaystyle{\left|\frac{\partial u}{\partial
r}\right|^{2}\leq\frac{C}{r^{2}(-\log r)}\ \ \mbox{ and }\ \
\left(\left|\frac{\partial
u}{\partial\theta}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)\leq\frac{C}{-\log
r}}$ in $\mathbb{D}_{\frac{1}{4}}^{*}$

  * (iii)

$u$ has sub-logarithmic growth.

Moreover, $u$ is the only harmonic section satisfying
$u|_{\partial\mathbb{D}}=k|_{\partial\mathbb{D}}$ and property (iii).

###### Proof.

Combine Lemma 3.5, Lemma 3.7, Lemma 3.9 and Lemma 3.10 below. ∎

The purpose for the rest of this subsection is to prove the lemmas that
comprise the proof of Theorem 3.1. We start with the following preliminary
lemma about subharmonic functions.

######  Lemma 3.2.

For $r>0$, let $\nu:\mathbb{D}_{r}^{*}\rightarrow\mathbb{R}$ be a subharmonic
function that extends as a continuous function to
$\bar{\mathbb{D}}_{r}^{*}=\overline{\mathbb{D}_{r}}\backslash\\{0\\}$. If we
assume that

(3.1) $\lim_{z\rightarrow 0}\ \nu(z)+\epsilon\log|z|=-\infty,\ \
\forall\epsilon>0,$

then

$\sup_{z\in\mathbb{D}_{r}^{*}}\nu(z)\leq\max_{\zeta\in\partial\mathbb{D}_{r}}\nu(\zeta).$

In particular, $\nu$ extends to a subharmonic function on $\mathbb{D}_{r}$
(cf. [HK]).

###### Proof.

Let $h:\mathbb{D}_{r}\rightarrow\mathbb{R}$ be the unique Dirichlet solution
with $h|_{\partial\mathbb{D}_{r}}=\nu|_{\partial\mathbb{D}_{r}}$. By the
maximum principle, $h$ is non-negative and bounded on $\mathbb{D}_{r}$. The
function

$f_{\epsilon}(z)=\nu(z)-h(z)+\epsilon(\log|z|-\log r)$

is subharmonic on $\mathbb{D}_{r}^{*}$ with
$f_{\epsilon}|_{\partial\mathbb{D}_{r}}\equiv 0$ on $\partial\mathbb{D}_{r}$.
Furthermore, since $h$ is a bounded function and $\nu$ satisfies (3.1),

$\lim_{z\rightarrow 0}f_{\epsilon}(z)=-\infty.$

By the maximum principle, $f_{\epsilon}\leq 0$. By letting
$\epsilon\rightarrow 0$, we conclude $\nu(z)-h(z)\leq 0$ for all
$z\in\mathbb{D}_{r}^{*}$. Thus,

$\sup_{z\in\mathbb{D}_{r}^{*}}\nu(z)\leq\sup_{z\in\mathbb{D}_{r}}h(z)\leq\max_{\zeta\in\partial\mathbb{D}_{r}}h(\zeta)=\max_{\zeta\in\partial\mathbb{D}_{r}}\nu(\zeta).$

∎

######  Lemma 3.3.

If $\rho$ and $I$ are as in Theorem 3.1, then for any section
$f:\mathbb{D}_{r,r_{0}}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$,

$\mathsf{E}_{\rho}\log\frac{r_{0}}{r}\leq\int_{\mathbb{D}_{r,r_{0}}}\frac{1}{r^{2}}\left|\frac{\partial
f}{\partial\theta}\right|^{2}rdr\wedge d\theta\leq
E^{f}[\mathbb{D}_{r,r_{0}}],\ \ 0<r<r_{0}<1.$

###### Proof.

By the definition of $\mathsf{E}_{I}$, any section
$c:\mathbb{S}^{1}\rightarrow\mathbb{R}\times_{\rho}\tilde{X}$ satisfies

$\mathsf{E}_{\rho}\leq\int_{0}^{2\pi}\left|\frac{\partial
c}{\partial\theta}\right|^{2}d\theta.$

After identifying $\mathbb{S}^{1}$ with $\partial\mathbb{D}_{r}$, we can view
the restriction $f|_{\partial\mathbb{D}_{r}}$ as a section
$\mathbb{S}^{1}\rightarrow\mathbb{R}\times_{\rho}\tilde{X}$. Thus,

$\displaystyle\mathsf{E}_{\rho}\log\frac{r_{0}}{r}$ $\displaystyle=$
$\displaystyle\mathsf{E}_{\rho}\int_{r}^{r_{0}}\frac{1}{r^{2}}rdr\wedge
d\theta$ $\displaystyle\leq$
$\displaystyle\int_{0}^{2\pi}\int_{r}^{r_{0}}\frac{1}{r^{2}}\left|\frac{\partial
f}{\partial\theta}(r,\theta)\right|^{2}rdr\wedge d\theta$ $\displaystyle\leq$
$\displaystyle E^{f}[\mathbb{D}_{r,r_{0}}].$

∎

######  Lemma 3.4.

If $\rho$, $I$ and
$k:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
are as in Theorem 3.1, then there exists a constant $C>0$ and a Lipschitz
section
$v:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
with $v|_{\partial\mathbb{D}}=k|_{\partial\mathbb{D}}$ such that

(3.2) $\mathsf{E}_{\rho}\log\frac{r_{0}}{r}\leq
E^{v}[\mathbb{D}_{r,r_{0}}]\leq\mathsf{E}_{\rho}\log\frac{r_{0}}{r}+C,\ \ \
0<r<r_{0}\leq\frac{1}{2}.$

Moreover, the Lipschitz constant of $v$ and $C$ are dependent only on
$\mathsf{E}_{\rho}=\frac{\Delta_{I}^{2}}{2\pi}$ and $a$, $b$ form assumption
(B) if $I$ is not semisimple and $k$.

###### Proof.

We will first construct a one-parameter family of sections
$\gamma_{s}:\mathbb{S}^{1}\mapsto\mathbb{R}\times_{\rho}\tilde{X}$ as follows:

  * •

If $I$ is semisimple, let $P_{\star}\in\tilde{X}$ be a point where the infimum
$\Delta_{I}$ is attained. For any $s\in[0,\infty)$, define
$\tilde{\gamma}_{s}:\mathbb{R}\rightarrow\tilde{X}$ to be the
$\rho$-equivariant geodesic (a constant map if $I$ is elliptic) such that
$\tilde{\gamma}_{s}(0)=P_{\star}$ and $\tilde{\gamma}_{s}(2\pi)=I(P_{\star})$
and let
$\gamma_{s}:\mathbb{S}^{1}\rightarrow\tilde{\mathbb{S}}^{1}\times_{\rho}\tilde{X}$
be the associated section.

  * •

Otherwise, let $c:[0,\infty)\rightarrow\tilde{X}$ be a geodesic ray defined in
assumption (B). Define the $\rho$-equivariant curve
$\tilde{\gamma}_{s}:\mathbb{R}\rightarrow\tilde{X}$ such that
$\tilde{\gamma}_{s}|_{[0,2\pi]}$ is the geodesic from $c(s)$ and $I(c(s))$ and
let
$\gamma_{s}:\mathbb{S}^{1}\rightarrow\tilde{\mathbb{S}}^{1}\times_{\rho}\tilde{X}$
be the associated section.

We note that by the quadrilateral comparison of CAT(0) spaces (cf. [Re]), for
$\theta\in[0,2\pi]$,

$d(\tilde{\gamma}_{s_{1}}(\theta),\tilde{\gamma}_{s_{2}}(\theta))\leq(1-\frac{\theta}{2\pi})d(c(s_{1}),c(s_{2}))+\frac{\theta}{2\pi}d(I\circ
c(s_{1})),I\circ c(s_{2})))=|s_{1}-s_{2}|.$

Thus

(3.3) $\left|\frac{d}{ds}(\tilde{\gamma}_{s}(\theta))\right|\leq 1,\ \ \
\forall s\in(0,\infty),\ \theta\in\mathbb{S}^{1}.$

By the assumption on $c(s)$, we have

(3.4)
$\int_{\mathbb{S}^{1}}\left|\frac{\partial\gamma_{s}}{\partial\theta}\right|^{2}d\tau\leq\mathsf{E}_{\rho}(1+2\pi
be^{-as}).$

Define a $\rho$-equivariant map
$\tilde{v}:\widetilde{\bar{\mathbb{D}}^{*}}\rightarrow\tilde{X}$ as follows:
First, for $(r,t)\in\widetilde{\bar{\mathbb{D}}^{*}}$, let

$\tilde{v}(r,t):=\tilde{\gamma}_{(-\log r-\log 2)^{\frac{1}{3}}}(t)\mbox{ for
}r\in(0,\frac{1}{2}]\ \mbox{ and }\ \tilde{v}(1,t):=\tilde{k}(1,t).$

Next, for each $t\in\mathbb{R}$, let

$r\mapsto\tilde{v}(r,t)\mbox{ for }r\in[\frac{1}{2},1]$

to be the arclength parameterization of the geodesic between
$\tilde{v}(\frac{1}{2},t)$ and $\tilde{k}(1,t)$. Finally, let
$v:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
be the section associated to the $\rho$-equivariant map $\tilde{v}$.

For $0<r<r_{0}<\frac{1}{2}$, we have

$\displaystyle\mathsf{E}_{\rho}\log\frac{r_{0}}{r}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{2\pi}\int_{r}^{r_{0}}\frac{1}{r^{2}}\left|\frac{\partial
v}{\partial\theta}\right|^{2}rdr\wedge d\theta\ \ \ \ \ \mbox{(by
Lemma~{}\ref{lowerbd})}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{2\pi}\int_{r}^{r_{0}}\frac{1}{r^{2}}\left|\frac{\partial(\gamma_{(-\log
r-\log 2)^{\frac{1}{3}}})}{\partial\theta}\right|^{2}rdr\wedge d\theta$
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}\int_{-\log r_{0}}^{-\log
r}\left|\frac{\partial(\gamma_{(t-\log
2)^{\frac{1}{3}}})}{\partial\theta}\right|^{2}dtd\theta$ $\displaystyle\leq$
$\displaystyle\int_{-\log r_{0}}^{-\log r}(\mathsf{E}_{\rho}+be^{-a(t-\log
2)^{\frac{1}{3}}})dt\ \ \ \ \ \ \ \ \mbox{(by (\ref{gammatbd}))}$
$\displaystyle=$ $\displaystyle\mathsf{E}_{\rho}\log\frac{r_{0}}{r}+C.$

By (3.3),

$\displaystyle\left|\frac{\partial v}{\partial
r}\right|^{2}(r,\theta)=\left|\frac{\partial}{\partial s}\Big{|}_{s=(-\log
r-\log
2)^{\frac{1}{3}}}\gamma_{s}(\theta)\right|^{2}\left|\frac{\partial((-\log
r-\log 2)^{\frac{1}{3}})}{\partial r}\right|^{2}\leq\frac{1}{r^{2}(-\log
r-\log 2)^{\frac{4}{3}}},$

and thus

$\displaystyle\int_{0}^{2\pi}\int_{r}^{r_{0}}\left|\frac{\partial v}{\partial
r}\right|^{2}rdr\wedge d\theta$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{r}^{r_{0}}\frac{1}{r(-\log r-\log
2)^{\frac{4}{3}}}dr\wedge d\theta\leq C.$

∎

######  Lemma 3.5 (Existence and property (i) of Theorem 3.1).

If $\tilde{X}$, $\rho$, $\mathsf{E}_{\rho}$ and $k$ as in Theorem 3.1, then
there exists a harmonic section
$u:\mathbb{D}^{*}\rightarrow\widetilde{\mathbb{D}^{*}}\times_{\rho}\tilde{X}$
with $u|_{\partial\mathbb{D}}=k|_{\partial\mathbb{D}}$ and a constant $C>0$
that depends only on $\mathsf{E}_{\rho}=\frac{\Delta_{I}^{2}}{2\pi}$, $a$, $b$
from (B) if $I$ is not semisimple and the section $k$ such that

$\mathsf{E}_{\rho}\log\frac{1}{r}\leq
E^{u}[\mathbb{D}_{r,1}]\leq\mathsf{E}_{\rho}\log\frac{1}{r}+C,\ \ \ 0<r\leq
1.$

###### Proof.

Let $v$ be as in Lemma 3.4. By [KS1, Theorem 2.7.2] (or more specifically, by
the proof of this theorem which solves the equivariant problem), there exists
a unique harmonic section

$u_{r}:\mathbb{D}_{r,1}\rightarrow\widetilde{\mathbb{D}_{r,1}}\times_{\rho}\tilde{X}\mbox{
with }u_{r}|_{\partial\mathbb{D}_{r,1}}=v|_{\partial\mathbb{D}_{r,1}}.$

We have that

$\displaystyle\mathsf{E}_{\rho}\log\frac{r_{0}}{r}+E^{u_{r}}[\mathbb{D}_{r_{0},1}]$
$\displaystyle\leq$ $\displaystyle
E^{u_{r}}[\mathbb{D}_{r,r_{0}}]+E^{u_{r}}[\mathbb{D}_{r_{0},1}]\ \ \ \ \
\mbox{(by Lemma~{}\ref{lowerbd})}$ $\displaystyle=$ $\displaystyle
E^{u_{r}}[\mathbb{D}_{r,1}]$ $\displaystyle\leq$ $\displaystyle
E^{v}[\mathbb{D}_{r,1}]\ \ (\mbox{since $u_{r}$ is minimizing})$
$\displaystyle=$ $\displaystyle
E^{v}[\mathbb{D}_{r,r_{0}}]+E^{v}[\mathbb{D}_{r_{0},1}]$ $\displaystyle\leq$
$\displaystyle\mathsf{E}_{\rho}\log\frac{r_{0}}{r}+C+E^{v}[\mathbb{D}_{r_{0},1}]\
\ \ \ \ \mbox{(by (\ref{glog}))}.$

Therefore,

(3.5) $E^{u_{r}}[\mathbb{D}_{r_{0},1}]\leq C+E^{v}[\mathbb{D}_{r_{0},1}].$

Note that the right hand side of the above is independent of
$r\in(0,\frac{1}{2}]$. Thus, $\\{u_{r}|_{\mathbb{D}_{2r_{0},1}}\\}$ is an
equicontinuous family of sections (cf. [KS1, Theorem 2.4.6]). Consequently,
there exists a sequence $r_{i}\rightarrow 0$ and a harmonic section

(3.6)
$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}\mbox{
with }u|_{\partial\mathbb{D}}=k|_{\partial\mathbb{D}}.$

such that the sequence $\\{u_{r_{i}}\\}$ converges uniformly on every compact
subsets of $\mathbb{D}^{*}$ to $u$.

To prove property (i), we note that for $0<r<r_{1}<1$,

$\displaystyle
E^{u_{r}}[\mathbb{D}_{r,r_{1}}]+E^{u_{r}}[\mathbb{D}_{r_{1},1}]$
$\displaystyle=$ $\displaystyle E^{u_{r}}[\mathbb{D}_{r,1}]$
$\displaystyle\leq$ $\displaystyle E^{v}[\mathbb{D}_{r,1}]$ $\displaystyle=$
$\displaystyle E^{v}[\mathbb{D}_{r,r_{1}}]+E^{v}[\mathbb{D}_{r_{1},1}]$

which, combined with Lemma 3.3 and (3.2), imply that

$E^{u_{r}}[\mathbb{D}_{r_{1},1}]\leq E^{v}[\mathbb{D}_{r_{1},1}]+C.$

Letting $r\rightarrow 0$ and applying the lower semicontinuity of energy [KS1,
Theorem 1.6.1], we obtain

$E^{u}[\mathbb{D}_{r_{1},1}]\leq E^{v}[\mathbb{D}_{r_{1},1}]+C.$

Applying Lemma 3.3 and (3.2), we obtain

$\mathsf{E}_{\rho}\log\frac{1}{r_{1}}\leq E^{u}[\mathbb{D}_{r_{1},1}]\leq
E^{v}[\mathbb{D}_{r_{1},1}]+C\leq\mathsf{E}_{\rho}\log\frac{1}{r_{1}}+C.$

∎

######  Lemma 3.6.

If
$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
is a harmonic section with logarithmic energy growth, then

$\int_{\mathbb{D}^{*}}\left|\frac{\partial u}{\partial r}\right|^{2}rdr\wedge
d\theta\leq C\ \mbox{ and }\
\int_{\mathbb{D}^{*}}\frac{1}{r^{2}}\left(\left|\frac{\partial
u}{\partial\theta}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)rdr\wedge
d\theta\leq C$

where $C$ that depends $C_{0}$ from (1.1).

###### Proof.

Follows immediately from the inequality (1.1) in the Definition 2.4 and Lemma
3.3. ∎

######  Lemma 3.7 (Property (ii) of Theorem 3.1).

If
$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
is a harmonic section with logarithmic energy growth, then

(3.7) $\displaystyle\left(\left|\frac{\partial
u}{\partial\theta}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{(-\log r)}\ \text{ in
}\,\mathbb{D}_{\frac{1}{4}}^{*}$ (3.8) $\displaystyle\lim_{r\rightarrow
0}(-\log r)\left(\left|\frac{\partial
u}{\partial\theta}\right|^{2}(r,\theta)-\frac{\mathsf{E}_{\rho}}{2\pi}\right)$
$\displaystyle=$ $\displaystyle 0$ (3.9) $\displaystyle\left|\frac{\partial
u}{\partial r}\right|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C}{r^{2}(-\log r)}\ \mbox{ in
}\,\mathbb{D}_{\frac{1}{4}}^{*}$ (3.10) $\displaystyle\lim_{r\rightarrow
0}(-\log r)r^{2}\left|\frac{\partial u}{\partial r}\right|^{2}(r,\theta)$
$\displaystyle=$ $\displaystyle 0.$

where $C$ that depends $C_{0}$ from (1.1).

###### Proof.

Consider the cylinder

${\mathcal{C}}=(0,\infty)\times\mathbb{S}^{1}$

and let

(3.11) $\Phi:{\mathcal{C}}\rightarrow\mathbb{D}^{*},\ \ \ \
(t,\psi)=(r=e^{-t},\theta=\psi).$

Since $\Phi$ is a conformal map, $u\circ\Phi$ is harmonic. Thus, the
directional energy density functions
$\left|\frac{\partial(u\circ\Phi)}{\partial t}\right|^{2}$ and
$\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}$ are subharmonic
by [KS1, Remark 2.4.3]. Furthermore,
$\left|\frac{\partial(u\circ\Phi)}{\partial t}\right|^{2}$ and
$\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}$
are integrable in ${\mathcal{C}}=(0,\infty)\times\mathbb{S}^{1}$ by Lemma 3.6
and the chain rule.

The subharmonicity of the directional energy density functions implies

$0\leq\int_{(t_{1},t_{2})\times\mathbb{S}^{1}}\triangle\varphi\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}dtd\psi,\
\ \forall\varphi\in C_{c}^{\infty}((t_{1},t_{2})\times\mathbb{S}^{1}),\
\varphi\geq 0.$

Letting $\varphi$ approximate the characteristic function of
$(t_{1},t_{2})\times\mathbb{S}^{1}$, we obtain

$0\leq\frac{d}{dt}\Big{|}_{t=t_{2}}\left(\int_{\\{t\\}\times\mathbb{S}^{1}}\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}d\psi\right)-\frac{d}{dt}\Big{|}_{t=t_{1}}\left(\int_{\\{t\\}\times\mathbb{S}^{1}}\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}d\psi\right).$

In other words,

$F(t):=\int_{\\{t\\}\times\mathbb{S}^{1}}\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\
d\psi,\ \ \ t\in(0,\infty)$

is a convex function. By Lemma 3.6, $\int_{0}^{\infty}F(t)dt<\infty$. Since
$F(t)$ is convex and integrable, $F(t)$ is a decreasing. Thus, we obtain

(3.12) $tF(t)\leq 2\int_{\frac{t}{2}}^{t}F(\tau)\ d\tau\leq
2\int_{\frac{t}{2}}^{\infty}F(\tau)\ d\tau.$

With $\mathcal{B}_{1}(t_{0},\psi_{0})$ denoting the unit disk centered at
$(t_{0},\psi_{0})\in\mathcal{C}$, the mean value inequality implies

$\displaystyle
t_{0}\left(\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)(t_{0},\psi_{0})$
$\displaystyle\leq$
$\displaystyle\frac{t_{0}}{\pi}\int_{\mathcal{B}_{1}(t_{0},\psi_{0})}\left(\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)dtd\psi$
$\displaystyle\leq$
$\displaystyle\frac{t_{0}}{\pi}\int_{(t_{0}-1,t_{0}+1)\times\mathbb{S}^{1}}\left(\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)dtd\psi$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\frac{t_{0}}{t_{0}-1}\int_{(t_{0}-1,t_{0}+1)\times\mathbb{S}^{1}}t\left(\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}-\frac{\mathsf{E}_{\rho}}{2\pi}\right)dtd\psi$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\frac{t_{0}}{t_{0}-1}\int_{t_{0}-1}^{t_{0}+1}tF(t)dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\frac{t_{0}}{t_{0}-1}\int_{t_{0}-1}^{t_{0}+1}2\int_{\frac{t}{2}}^{\infty}F(\tau)\
d\tau dt\ \ \ \mbox{(by (\ref{bbbb}))}.$

Since $F(\tau)\geq 0$,

$\int_{\frac{t}{2}}^{\infty}F(\tau)\
d\tau\leq\int_{\frac{t_{0}-1}{2}}^{\infty}F(\tau)\ d\tau,\ \ \
t\in(t_{0}-1,t_{0}+1).$

Combining the above two inequalities and assuming $t_{0}\geq 2$ (and thus
$\frac{t_{0}}{t_{0}-1}\leq 2$),

(3.13) $\displaystyle
t_{0}\left(\left|\frac{\partial(u\circ\Phi)}{\partial\psi}\right|^{2}(t_{0},\psi_{0})-\frac{\mathsf{E}_{\rho}}{2\pi}\right)$
$\displaystyle\leq$
$\displaystyle\frac{4}{\pi}\int_{\frac{t_{0}-1}{2}}^{\infty}F(\tau)\ d\tau.$

Estimates (3.7) and (3.8) both follow from (3.13) by the chain rule and the
fact that $F(\tau)$ is integrable on $(0,\infty)$ with integral bound given by
Lemma 3.6.

Similar argument for

$G(t):=\int_{\\{t\\}\times\mathbb{S}^{1}}\left|\frac{\partial(u\circ\Phi)}{\partial
t}\right|^{2}d\psi,\ \ \ t\in(0,\infty)$

proves (3.9) and (3.10). ∎

######  Lemma 3.8.

Let
$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
be a harmonic section with logarithmic energy growth. For any $\epsilon>0$,
there exists $\rho_{0}>0$ such that

$d^{2}(u(r_{1}e^{i\theta}),u(r_{0}e^{i\theta}))\leq-\epsilon\log r_{1},\ \ \
\forall 0<r_{1}<r_{0}\leq\rho_{0},\,0<\theta<2\pi.$

###### Proof.

Fix $\epsilon>0$. The convergence of $(-\log r)r^{2}\left|\frac{\partial
u}{\partial r}\right|^{2}(r,\theta)$ to 0 as $r\rightarrow 0$ (cf. (3.10))
implies that there exists $\rho_{0}>0$ such that

$\left|\frac{\partial u}{\partial r}\right|\leq\frac{\sqrt{\epsilon}}{2r(-\log
r)^{\frac{1}{2}}},\ \ r\in(0,\rho_{0}].$

Thus, we have for $0<r_{1}<r_{0}\leq\rho_{0}$, and noting $\rho$-equivariance
of $\tilde{u}$, we have

$\displaystyle d^{2}(u(r_{1}e^{i\theta}),u(r_{0}e^{i\theta}))$
$\displaystyle\leq$
$\displaystyle\left(\int_{r_{0}}^{r_{1}}\frac{d}{dr}d(u(re^{i\theta}),u(r_{0}e^{i\theta}))dr\right)^{2}$
$\displaystyle\leq$
$\displaystyle\left(\int_{r_{1}}^{r_{0}}\left|\frac{\partial u}{\partial
r}\right|(re^{i\theta})dr\right)^{2}$ $\displaystyle\leq$
$\displaystyle\frac{\epsilon}{4}\left(\int_{r_{1}}^{r_{0}}\frac{dr}{r(-\log
r)^{\frac{1}{2}}}\right)^{2}$ $\displaystyle\leq$
$\displaystyle\epsilon\left((-\log r_{1})^{\frac{1}{2}}-(-\log
r_{0})^{\frac{1}{2}}\right)^{2}$ $\displaystyle\leq$
$\displaystyle-\epsilon\log r_{1}.$

∎

######  Lemma 3.9 (Property (iii) of Theorem 3.1).

If
$u:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
is a harmonic section with logarithmic energy growth, then $u$ has sub-
logarithmic growth.

###### Proof.

By Lemma 3.8, for $\epsilon>0$, there exists $r_{0}>0$ sufficiently small such
that

$d^{2}(u(re^{i\theta}),u(r_{0}e^{i\theta}))\leq-\frac{\epsilon}{4}\log r,\ \ \
r\in(0,r_{0}).$

Set $P_{0}=\tilde{u}(r_{0}e^{i\theta_{0}})\in\tilde{X}$ and define
$f_{0}(p)=[(\tilde{p},P_{0})]$ as in Definition 2.4. For $z=re^{i\theta}$,

$\displaystyle d^{2}(u(z),f_{0}(z))+\epsilon\log|z|$ $\displaystyle=$
$\displaystyle d^{2}(u(re^{i\theta}),u(r_{0}e^{i\theta_{0}}))+\epsilon\log r$
$\displaystyle\leq$ $\displaystyle
2d^{2}(u(re^{\i\theta}),u(r_{0}e^{i\theta}))+2d^{2}(u(r_{0}e^{i\theta}),u(r_{0}e^{i\theta_{0}}))+\epsilon\log
r$ $\displaystyle\leq$ $\displaystyle\frac{\epsilon}{2}\log
r+d^{2}(u(r_{0}e^{i\theta}),u(r_{0}e^{i\theta_{0}})).$

Now let $r\rightarrow 0$. ∎

######  Lemma 3.10 (Uniqueness for sub-logarithmic growth maps).

If
$u,v:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
are harmonic sections with sub-logarithmic growth with $u=v$ on
$\partial\mathbb{D}$, then $u=v$ on $\mathbb{D}^{*}$.

###### Proof.

The function $d^{2}(u,v)$ (as defined by (2.2)) is a continuous subharmonic
function (cf. [KS1, Remark 2.4.3]). Furthermore, $d^{2}(u,v)\equiv 0$ on
$\partial\mathbb{D}$. Since $u$, $v$ both have sub-logarithmic growth, the
triangle inequality implies

$\lim_{|z|\rightarrow 0}d^{2}(u(z),v(z))+\epsilon\log|z|=-\infty.$

Thus, we can apply Lemma 3.2 to conclude $d^{2}(u(z),v(z))\equiv 0$ on
$\mathbb{D}^{*}$ which implies $u\equiv v$. ∎

######  Corollary 3.11.

Any harmonic section
$v:\bar{\mathbb{D}}^{*}\rightarrow\widetilde{\bar{\mathbb{D}}^{*}}\times_{\rho}\tilde{X}$
with sub-logarithmic growth satisfies properties (i), (ii) and (iii) of
Theorem 3.1.

###### Proof.

By the uniqueness assertion of Lemma 3.10, $v$ must be the harmonic section
$u$ constructed in Theorem 3.1. ∎

##  4\. Existence of infinite energy harmonic maps

In this section, we prove existence of equivariant harmonic maps from the
punctured Riemann surface $\mathcal{R}$. We use the following notation:

  * •

$\mathbb{D}^{j*}=\mathbb{D}^{j}\backslash\\{0\\}$

  * •

$\mathbb{D}^{j}_{r}=\\{z\in\mathbb{D}^{j}:|z|<r\\}$

  * •

$\mathbb{D}^{j}_{r,r_{0}}=\mathbb{D}^{j}_{r_{0}}\backslash\mathbb{D}^{j}_{r}$

  * •

$\mathcal{R}_{r}=\mathcal{R}\backslash\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r}$

  * •

$2\pi\mathbb{Z}\simeq\langle\lambda^{j}\rangle$ is the free group generated by
a loop around the puncture $p^{j}$

  * •

$\rho^{j}:\langle\lambda^{j}\rangle\rightarrow\mathsf{Isom}(\tilde{X})$ be the
restriction of $\rho:\pi_{1}(\mathcal{R})\rightarrow\mathsf{Isom}(\tilde{X})$

  * •

$k:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ is a
locally Lipschitz section (cf. [KS1, Proposition 2.6.1])

  * •

$k^{j}:=k|_{\bar{\mathbb{D}}^{j*}}$, restriction to the conformal disk around
the puncture $p^{j}$

Applying Lemma 3.4 with $\rho=\rho^{j}$, $I=I^{j}$ and $k=k^{j}$ yields a
prototype section

$v^{j}:\mathbb{D}^{*}\rightarrow\widetilde{\mathbb{D}^{*}}\times_{\rho^{j}}\tilde{X}\subset\widetilde{{\mathcal{R}}}\times_{\rho^{j}}\tilde{X}.$

The composition of $v^{j}$ and the quotient map
$\widetilde{{\mathcal{R}}}\times_{\rho^{j}}\tilde{X}\rightarrow\widetilde{{\mathcal{R}}}\times_{\rho}\tilde{X}$
defines a section of
$\tilde{\mathcal{R}}\times_{\rho}\tilde{X}\rightarrow\mathcal{R}$ over
${\mathbb{D}}^{j*}$ which we call again $v^{j}$. We extend these local
sections
$v^{j}:\mathbb{D}^{j*}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$
for $j=1,\dots,n$ to define a smooth section
$v:{\mathcal{R}}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$. By the
construction and Lemma 3.4,

(4.1) $\sum_{j=1}^{n}\mathsf{E}_{j}\log\frac{1}{r}\leq
E^{v}[\mathcal{R}_{r}]\leq\sum_{j=1}^{n}\mathsf{E}_{j}\log\frac{1}{r}+C,\ \ \
0<r\leq 1.$

where $\mathsf{E}_{j}:=\frac{\Delta_{j}^{2}}{2\pi}$.

######  Definition 4.1.

The locally Lipschitz section

(4.2) $v:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$

constructed above is called the prototype section of the fiber bundle
$\tilde{\mathcal{R}}\times_{\rho}\tilde{X}\rightarrow\mathcal{R}$. The
associated $\rho$-equivariant map
$\tilde{v}:\tilde{\mathcal{R}}\rightarrow\tilde{X}$ is called the prototype
map.

Define

$u_{\mathcal{R}_{r}}:\mathcal{R}_{r}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$

to be the unique harmonic section with boundary values equal to that of
$v|_{\partial\mathcal{R}_{r}}$ (cf. [KS1, Theorem 2.7.2]). Fix
$r_{0}\in(0,\frac{1}{2}]$ and let $r\in(0,r_{0})$. By Lemma 3.3,

$\mathsf{E}_{j}\log\frac{r_{0}}{r}\leq\int_{\mathbb{D}_{r,r_{0}}}\frac{1}{r^{2}}\left|\frac{\partial
u_{\mathcal{R}_{r}}}{\partial\theta}\right|^{2}rdr\wedge d\theta\leq
E^{u_{\mathcal{R}_{r}}}[\mathbb{D}^{j}_{r,r_{0}}].$

Thus, by (4.1),

$E^{v}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}]\leq
E^{u_{\mathcal{R}_{r}}}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}]+C$

which implies

$\displaystyle
E^{u_{\mathcal{R}_{r}}}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}]+E^{u_{\mathcal{R}_{r}}}[\mathcal{R}_{r_{0}}]$
$\displaystyle=$ $\displaystyle E^{u_{\mathcal{R}_{r}}}[\mathcal{R}_{r}]$
$\displaystyle\leq$ $\displaystyle E^{v}[\mathcal{R}_{r}]$ $\displaystyle=$
$\displaystyle
E^{v}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}]+E^{v}[\mathcal{R}_{r_{0}}]$
$\displaystyle\leq$ $\displaystyle
E^{u_{\mathcal{R}_{r}}}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}]+C+E^{v}[\mathcal{R}_{r_{0}}].$

In other words,

(4.3) $E^{u_{\mathcal{R}_{r}}}[\mathcal{R}_{r_{0}}]\leq
C+E^{v}[\mathcal{R}_{r_{0}}],\ \ \ \forall r\in(0,r_{0}).$

The right hand side of the inequality (4.3) is independent of the parameter
$r$; i.e. once we fix $r_{0}$, the quantity
$E^{u_{\mathcal{R}_{r}}}[\mathcal{R}_{r_{0}}]$ is uniformly bounded for all
$r\in(0,r_{0})$. This implies a uniform Lipschitz bound, say $L$, of
$u_{\mathcal{R}_{r}}$ for $r\in(0,r_{0})$ in $\mathcal{R}_{2r_{0}}$ (cf. [KS1,
Theorem 2.4.6]).

Let $\\{\mu_{1},\dots,\mu_{N}\\}$ be a set of generators of
$\pi_{1}(\mathcal{R})$ and $\tilde{u}_{\mathcal{R}_{r}}$, $\tilde{v}$ be the
$\rho$-equivariant maps associated to sections $u_{\mathcal{R}_{r}}$, $v$
respectively. Thus,

$d(\tilde{u}_{\mathcal{R}_{r}}(\mu_{i}p),\tilde{u}_{\mathcal{R}_{r}}(p))\leq
Ld_{\tilde{\mathcal{R}}}(\mu_{i}p,p),\ \ \forall
p\in\tilde{\mathcal{R}}_{2r_{0}},\ i=1,\dots,N\ \ r\in(0,r_{0}).$

If we let

$c=L\sup\\{d_{\tilde{\mathcal{R}}}(\mu_{i}p,p):i=1,\dots,N,\
p\in\tilde{\mathcal{R}}_{2r_{0}}\\},$

then by equivariance

$d(\rho(\mu_{i})\tilde{u}_{\mathcal{R}_{r}}(p),\tilde{u}_{\mathcal{R}_{r}}(p))\leq
c,\ \ \ p\in\tilde{\mathcal{R}}_{2r_{0}},\ i=1,\dots,N,\ \ r\in(0,r_{0}).$

In other words, $\delta(\tilde{u}_{\mathcal{R}_{r}}(p))\leq c$ for all
$p\in\tilde{\mathcal{R}}_{2r_{0}}$ and $r\in(0,r_{0})$. By the properness of
$\rho$, there exists $P_{0}\in\tilde{X}$ and $R_{0}>0$ such that

$\\{\tilde{u}_{\mathcal{R}_{r}}(p):\ p\in\tilde{\mathcal{R}}_{2r_{0}},\
r\in(0,r_{0})\\}\subset B_{R_{0}}(P_{0}).$

Thus, by taking a compact exhaustion and applying [KS2, Theorem 2.1.3], we
conclude that there exists a sequence $r_{i}\rightarrow 0$ and a
$\rho$-equivariant harmonic map
$\tilde{u}:\tilde{\mathcal{R}}\rightarrow\tilde{X}$ such that
$u_{\mathcal{R}_{r_{i}}}$ converges to $u$ in $L^{2}$ (i.e.
$d^{2}(u_{\mathcal{R}_{r_{i}}},u)\rightarrow 0$) on every compact subsets of
$\mathcal{R}$. Let
$u:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ be the
associated harmonic section. The lower semicontinuity of energy (cf. [KS1,
Theorem 1.6.1]) and (4.3) imply

$E^{u}[\mathcal{R}_{r_{0}}]\leq\sum_{j=1}^{n}\mathsf{E}_{j}\log\frac{1}{r_{0}}+C,\
\ \forall r_{0}\in(0,\frac{1}{2}]$

which, along with Lemma 3.3, proves that $u$ has logarithmic energy growth
(cf. Definition 2.4). The fact that $u$ has sub-logarithmic growth follows
from Lemma 3.9. This completes the proof of Theorem 1.1.

##  5\. Uniqueness of infinite energy harmonic maps

In this section, we prove uniqueness of harmonic maps from punctured Riemann
surfaces $\mathcal{R}$ (cf. Theorem 1.2). First, let

$E^{f}(r)=E^{f}[\mathcal{R}_{r}]-\sum_{j=1}^{n}\mathsf{E}_{j}\log\frac{1}{r}.$

By Lemma 3.3, for $0<r<r_{0}<1$ and any section
$f:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$,

(5.1) $\sum_{j=1}^{n}\mathsf{E}_{j}\log\frac{r_{0}}{r}\leq
E^{f}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r,r_{0}}].$

Thus, $r\mapsto E^{f}(r)$ is an increasing function.

######  Definition 5.1.

The modified energy of $f$ is

$E^{f}(0)=\lim_{r\rightarrow 0}E^{f}(r).$

Let $\mathcal{L}_{\rho}$ be the set of all sections
$\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ such that
$E^{f}(0)<\infty$. Let $\mathcal{H}_{\rho}\subset\mathcal{L}_{\rho}$ denote
the set of all harmonic sections
$u:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ of sub-
logarithmic growth such that the associated $\rho$-equivariant map $u$ that is
not identically constant or does not map into a geodesic.

The goal is to prove that $\mathcal{H}_{\rho}$ contains only one element. We
first prove the following series of preliminary lemmas.

######  Lemma 5.2.

If $u\in{\mathcal{H}}_{\rho}$ and $\mathbb{D}^{j}\subset\bar{\mathcal{R}}$ is
the fixed conformal disk at the puncture $p^{j}$ (cf. (2.3)), then the
restriction map $u|_{\mathbb{D}^{j*}}$ satisfies the properties (i), (ii) and
(iii) of Theorem 3.1.

###### Proof.

This follows immediately from Corollary 3.11. ∎

######  Lemma 5.3.

If $u_{0},u_{1}\in{\mathcal{H}}_{\rho}$, then $d^{2}(u_{0},u_{1})=c$ for some
constant $c$.

###### Proof.

We prove Lemma 5.3 by showing that $d^{2}(u_{0},u_{1})$ extends as a
subharmonic function to $\bar{\mathcal{R}}$. Since $\bar{\mathcal{R}}$ is
compact, this implies $d^{2}(u_{0},u_{1})$ is constant. Indeed, let
$\mathbb{D}\subset\mathcal{R}$ be a holomorphic disk. By [KS1, Lemma 2.4.2 and
Remark 2.4.3], the function $d^{2}(u_{0},u_{1})$ is subharmonic in
$\mathbb{D}$. Since this statement is true for any holomorphic disk
$\mathbb{D}\subset\mathcal{R}$, we conclude that $d^{2}(u_{0},u_{1})$ is
subharmonic in $\mathcal{R}$.

Next, consider the fixed conformal disk
$\mathbb{D}^{j}\subset\bar{\mathcal{R}}$ centered at $p^{j}\in{\mathcal{P}}$
(cf. (2.3)). Since both $u_{0}$ and $u_{1}$ have sub-logarithmic growth,
$d(u_{0}(z),u_{1}(z))+\epsilon\log|z|\rightarrow-\infty$ as $z\rightarrow 0$
in $\mathbb{D}^{j}$ by the triangle inequality. Thus, by Lemma 3.2,
$d^{2}(u_{0},u_{1})$ is bounded in $\mathbb{D}^{j*}$ and extends as a
subharmonic function on $\mathbb{D}^{j}$. Hence, we conclude that
$d^{2}(u_{0},u_{1})$ extends as a subharmonic function on $\bar{\mathcal{R}}$,
thereby proving Lemma 5.3. ∎

For $u_{0},u_{1}\in{\mathcal{H}}_{\rho}$, let $\tilde{u}_{0}$, $\tilde{u}_{1}$
be the associated $\rho$-equivariant maps. For $s\in[0,1]$, define
$u_{s}:\mathcal{R}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$ to be
the associated section of the $\rho$-equivariant map

(5.2) $\tilde{u}_{s}:\tilde{\mathcal{R}}\rightarrow\tilde{X},\ \ \
\tilde{u}_{s}(z)=(1-s)\tilde{u}_{0}(z)+s\tilde{u}_{1}(z)$

where the sum on the right hand side above denotes geodesic interpolation (cf.
Notation 2.2). Since $\tilde{u}_{0}$ and $\tilde{u}_{1}$ are
$\rho$-equivariant, $\tilde{u}_{s}$ is also $\rho$-equivariant.

From Lemma 5.2, Lemma 5.3, the convexity of the distance function and the
convexity of energy (cf. [KS1, (2.2vi)]), it follows that
$u_{s}|_{\mathbb{D}^{j*}}$ also satisfies the properties $(i)$, $(ii)$ and
$(iii)$ of Theorem 3.1 for all $s\in[0,1]$; i.e.

  * •

$\displaystyle{E^{u_{s}}[\mathbb{D}^{j}_{r,r_{0}}]\leq
C+\mathsf{E}_{j}\log\frac{r_{0}}{r}}$

  * •

$\displaystyle{\lim_{r\rightarrow
0}E^{u_{s}|_{\partial\mathbb{D}^{j}_{r}}}[\mathbb{S}^{1}]=\mathsf{E}_{j},}\vspace{0.5mm}$

  * •

$u_{s}$ has sub-logarithmic growth.

######  Lemma 5.4.

Let $u_{0},u_{1}\in{\mathcal{H}}_{\rho}$ and $\epsilon>0$. For any
$\rho_{0}\in(0,1)$, there exists $r_{0}\in(0,\rho_{0})$ such that

$\frac{1}{-\log
r_{0}}\sum_{j=1}^{n}\int_{0}^{2\pi}d^{2}(u_{0}|_{\partial\mathbb{D}^{j}_{r_{0}}},u_{s}|_{\partial\mathbb{D}^{j}_{r_{0}^{2}}})d\theta<\epsilon,\
\ \ \forall s\in[0,1].$

###### Proof.

It suffices to prove Lemma 5.4 for $s=1$. Let $\rho_{0}>0$ be given. First,
Lemma 3.8 asserts that there exists $\rho_{1}\in(0,\rho_{0})$ such that

$\frac{1}{-\log
r_{0}}\sum_{j=1}^{n}\int_{0}^{2\pi}d^{2}(u_{1}|_{\partial\mathbb{D}^{j}_{r_{0}}},u_{1}|_{\partial\mathbb{D}^{j}_{r_{0}^{2}}})d\theta<\frac{\epsilon}{4},\
\ \ \ \forall r_{0}\in(0,\rho_{1}).$

For $c>0$ as in Lemma 5.3, choose $r_{0}\in(0,\rho_{1})$ such that $\frac{2\pi
nc}{-\log r_{0}}<\frac{\epsilon}{4}.$ Then

$\frac{1}{-\log
r_{0}}\sum_{j=1}^{n}\int_{0}^{2\pi}d^{2}(u_{0}|_{\partial\mathbb{D}^{j}_{r_{0}}},u_{1}|_{\partial\mathbb{D}^{j}_{r_{0}}})d\theta=\frac{2\pi
nc}{-\log r_{0}}<\frac{\epsilon}{4}.$

The inequality of Lemma 5.4 for $s=1$ follows from the above two inequalities
and the triangle inequality. ∎

######  Lemma 5.5.

Let $u_{0},u_{1}\in{\mathcal{H}}_{\rho}$ and $s\in[0,1]$. For $\epsilon>0$,
there exists $\rho_{0}>0$ sufficiently small such that

$-\log
r_{0}\sum_{j=1}^{n}\left(E^{u_{0}|_{\partial\mathbb{D}^{j}_{r_{0}}}}[\mathbb{S}^{1}]+E^{u_{s}|_{\partial\mathbb{D}^{j}_{r_{0}^{2}}}}[\mathbb{S}^{1}]\right)<-2\log
r_{0}\sum_{j=1}^{n}\mathsf{E}_{j}+\epsilon,\ \ \ 0<r_{0}<\rho_{0}.$

###### Proof.

For $s=1$, Lemma 5.5 follows from Lemma 3.7. The general case of $s\in[0,1]$
follows immediately by convexity of energy (cf. [KS1, (2.2vi)]). ∎

######  Lemma 5.6.

For $u_{0},u_{1}\in{\mathcal{H}}_{\rho}$ and $s\in[0,1]$, we have
$E^{u_{0}}(0)=E^{u_{s}}(0).$

###### Proof.

It suffices to prove Lemma 5.6 for $s=1$. Assume on the contrary that
$E^{u_{0}}(0)\neq E^{u_{1}}(0)$. By changing the role of $u_{0}$ and $u_{1}$
if necessary, we can assume $E^{u_{0}}(0)<E^{u_{1}}(0)$. Let $\rho_{0}>0$
smaller of the $\rho_{0}$ in Lemma 5.4 and Lemma 5.5. Choose $\epsilon>0$ and
$r_{0}\in(0,\rho_{0}]$ such that

$E^{u_{0}}(r_{0})<E^{u_{1}}(r_{0})-2\epsilon$

which implies (cf. Definition 5.1)

(5.3)
$E^{u_{0}}[\mathcal{R}_{r_{0}}]<E^{u_{1}}[\mathcal{R}_{r_{0}}]-2\epsilon.$

Next fix $r_{1}\in(0,r_{0})$. Let $\tilde{u}_{0}$,
$\tilde{u}_{1}:\tilde{\mathcal{R}}\rightarrow\tilde{X}$ be the
$\rho$-equivariant maps associated to sections $u_{0}$, $u_{1}$. For the fixed
conformal disk $\mathbb{D}^{j}\subset\bar{\mathcal{R}}$ centered at the
puncture $p^{j}$ (cf. (2.3)), let
$\tilde{\mathbb{D}}^{j}_{r_{1},r_{0}}\subset\tilde{\mathcal{R}}$ the lift of
$\mathbb{D}^{j}_{r_{1},r_{0}}\subset\mathcal{R}$. We define a “bridge” between
map $\tilde{u}_{0}$ and $\tilde{u}_{1}$ by setting

$\tilde{b}:\bigcup_{j=1}^{n}\tilde{\mathbb{D}}^{j}_{r_{1},r_{0}}\rightarrow\tilde{X}$

to be the geodesic interpolation $(1-t)U_{0}(\theta)+tU_{1}(\theta)$ where

$t=\frac{\log|z|-\log r_{0}}{\log r_{1}-\log r_{0}},\ \
U_{0}(\theta)=\tilde{u}_{0}(r_{0},\theta)\ \text{ and }\
U_{1}(\theta)=\tilde{u}_{1}(r_{1},\theta).$

In other words,

$\tilde{b}(r,\theta)=\frac{\log r_{1}-\log r}{\log r_{1}-\log
r_{0}}\tilde{u}_{0}(r_{0},\theta)+\frac{\log r-\log r_{0}}{\log r_{1}-\log
r_{0}}\tilde{u}_{1}(r_{1},\theta)$

for $(r,\theta)\in\tilde{\mathbb{D}}^{j}_{r_{1},r_{0}}$, $j=1,\dots,n$. Let

$b:\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{1},r_{0}}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$

be the local section associated with $\tilde{b}$.

The CAT(0) condition implies (by an argument analogous to the proof the bridge
lemma [KS2, Lemma 3.12])

$\displaystyle E^{b}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{1},r_{0}}]$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\log\frac{r_{0}}{r_{1}}\sum_{j=1}^{n}\left(E^{u_{0}|_{\partial\mathbb{D}^{j}_{r_{0}}}}[\mathbb{S}^{1}]+E^{u_{1}|_{\partial\mathbb{D}^{j}_{r_{1}}}}[\mathbb{S}^{1}]\right)$
$\displaystyle\
+\frac{1}{\log\frac{r_{0}}{r_{1}}}\sum_{j=1}^{n}\int_{0}^{2\pi}d^{2}(u_{0}|_{\partial\mathbb{D}^{j}_{r_{0}}},u_{1}|_{\partial\mathbb{D}^{j}_{r_{1}}})d\theta.$

Choose $r_{1}=r_{0}^{2}$ (cf. Lemma 5.4 and Lemma 5.5) to obtain

(5.4) $E^{b}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{0}^{2},r_{0}}]<-\log
r_{0}\sum_{j=1}^{n}\mathsf{E}_{j}+2\epsilon.$

Since the section

$h:\mathcal{R}_{r_{0}^{2}}\rightarrow\tilde{\mathcal{R}}\times_{\rho}\tilde{X}$

defined by setting

$h=\left\\{\begin{array}[]{ll}u_{0}&\mbox{ in }\mathcal{R}_{r_{0}}\\\ b&\mbox{
in
}\displaystyle{\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{0}^{2},r_{0}}}\end{array}\right.$

is a competitor for $u_{1}$, we have

$\displaystyle E^{u_{1}}[\mathcal{R}_{r_{0}^{2}}]$ $\displaystyle\leq$
$\displaystyle E^{h}[\mathcal{R}_{r_{0}^{2}}]$ $\displaystyle=$ $\displaystyle
E^{u_{0}}[\mathcal{R}_{r_{0}}]+E^{b}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{0}^{2},r_{0}}]$
$\displaystyle<$ $\displaystyle
E^{u_{1}}[\mathcal{R}_{r_{0}}]-2\epsilon+E^{b}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{0}^{2},r_{0}}]\
\ \ \ \ \ \mbox{(by (\ref{ococ}))}$ $\displaystyle<$ $\displaystyle
E^{u_{1}}[\mathcal{R}_{r_{0}}]-\log
r_{0}\sum_{j=1}^{n}\frac{\Delta_{I^{j}}^{2}}{2\pi}\ \ \ \ \ \ \mbox{(by
(\ref{acac}))}.$

Thus,

$E^{u_{1}}[\bigcup_{j=1}^{n}\mathbb{D}^{j}_{r_{0}^{2},r_{0}}]<-\log
r_{0}\sum_{j=1}^{n}\mathsf{E}_{j}.$

This contradicts (5.1) and proves Lemma 5.6. ∎

######  Lemma 5.7.

For $u_{0},u_{1}\in\mathcal{H}_{\rho}$, there exists a constant $c$ such that

(5.5) $\displaystyle d(u_{s}(p),u_{1}(p))$ $\displaystyle\equiv$
$\displaystyle cs,\forall p\in\mathcal{R}$ (5.6)
$\displaystyle|(u_{s})_{*}(V)|^{2}(p)$ $\displaystyle=$
$\displaystyle|(u_{0})_{*}(V)|^{2}(p),\ \ \mbox{for a.e.~{}}s\in[0,1],\
p\in\mathcal{R},V\in T_{p}\tilde{\mathcal{R}}.$

###### Proof.

By the convexity of energy (cf. [KS1, (2.2vi)]),

$E^{u_{s}}[\mathcal{R}_{r}]\leq(1-s)E^{u_{0}}[\mathcal{R}_{r}]+sE^{u_{1}}[\mathcal{R}_{r}]-s(1-s)\int_{\mathcal{R}_{r}}|\nabla
d(u_{0},u_{1})|^{2}d\mbox{vol}_{M}$

for any $r>0$.

$E^{u_{s}}(r)\leq(1-s)E^{u_{0}}(r)+sE^{u_{1}}(r)-s(1-s)\int_{\mathcal{R}_{r}}|\nabla
d(u_{0},u_{1})|^{2}d\mbox{vol}_{M}.$

Letting $r\rightarrow 0$ and applying Lemma 5.6, we conclude

(5.7) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{R}}|\nabla d(u_{0},u_{1})|^{2}d\mbox{vol}_{M}$
(5.8) $\displaystyle E^{u_{s}}$ $\displaystyle=$ $\displaystyle E^{u_{0}},\ \
\forall s\in[0,1]$

First, (5.7) implies that $\nabla d(u_{0},u_{1})=0$ a.e. in $M$ which in turn
implies $d(u_{0},u_{1})\equiv c$ for some $c$. By the definition of the map
$u_{s}$, (5.5) follows immediately.

Next, for $\\{P,Q,R,S\\}\subset\tilde{X}$, the quadrilateral comparison for
CAT(0) spaces (cf. [Re]) implies

$d^{2}(P_{t},Q_{t})\leq(1-t)d^{2}(P,Q)+td^{2}(R,S)$

where $P_{t}=(1-t)P+tS$ and $Q_{t}=(1-t)Q+tR$. Applying the above inequality
with $P=u_{0}(p)$, $Q=u_{1}(p)$, $S=u_{1}(\exp_{p}(tV))$ and
$Q=u_{0}(\exp_{t}(tV))$ where $t>0$ and $V\in T_{p}\tilde{M}$, dividing by $y$
and letting $t\rightarrow 0$, we obtain (cf. [KS1, Theorem 1.9.6])

$|(u_{s})_{*}(V)|^{2}(p)\leq(1-s)|(u_{0})_{*}(V)|^{2}(p)+s|(u_{1})_{*}(V)|^{2}(p),\
\mbox{a.e.~{}$p\in\tilde{M},V\in T_{p}\tilde{M}$.}$

Integrating the above over all unit vectors $V\in T_{p}\tilde{M}$ and then
over $p\in F$, we obtain

$E^{u_{s}}\leq(1-s)E^{u_{0}}+sE^{u_{1}}.$

Combining this with (5.8) implies (5.6). ∎

Proof of Theorem 1.2. We assume there exist $u_{0},u_{1}\in\mathcal{H}_{\rho}$
such that $u_{0}\not\equiv u_{1}$ and treat the different cases of Theorem 1.2
separately.

  * •

$\tilde{X}$ is a negatively curved space: For $p\in\tilde{\mathcal{R}}$,
choose $\kappa>0$ and $R>0$ such that $B_{R}(u_{0}(p))$ is a CAT(-$\kappa$)
space. Let $\mathcal{U}$ be a sufficiently small neighborhood of $p$ and
$s_{0}\in[0,1]$ sufficiently small such that
$\tilde{u}_{s}(\mathcal{U})\subset B_{R}(u(p))$ for $s\in[0,s_{0}]$. Thus, if
$\tilde{u}_{s_{0}}(p)\neq\tilde{u}_{0}(p)$, then applying [Me, Section 5]
implies that the image under $u_{0}$ of a sufficiently small neighborhood of
$p$ is contained in an image $\sigma(\mathbb{R})$ of a geodesic line. (If
$\tilde{X}$ is a smooth manifold, this follows by Hartman [Ha]). If
$\tilde{u}_{0}\not\equiv\tilde{u}_{1}$, then
$\tilde{u}_{s}(p)\neq\tilde{u}_{0}(p)$ for all $p\in\tilde{\mathcal{R}}$ and
$s\in(0,s_{0}]$ by (5.5). Thus, we conclude that
$\tilde{u}_{0}(\tilde{\mathcal{R}})$ is contained in $\sigma(\mathbb{R})$.
Consequently, $\rho(\pi_{1}(\tilde{\mathcal{R}}))$ fixes $\sigma(\mathbb{R})$
which contradicts the fact that $\rho(\pi_{1}(\mathcal{R}))$ is satisfies
assumption (B).

  * •

$\tilde{X}$ is an irreducible symmetric space of non-compact type or locally
finite Euclidean buildings: For these two target spaces, the conclusion of
Lemma 5.7 is the same as that of [DM1, Lemma 3.1]. Thus, we can apply the
arguments of [DM1, Section 3.2 and Section 3.3].

$\Box$

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