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

We extend Siu's and Sampson's celebrated rigidity results to  non-compact domains. More precisely,
let $M$ be a smooth quasi-projective variety with universal cover $\tilde M$ and let $\tilde X$ be a symmetric space of non-compact type, a locally finite Euclidean building or the Weil-Petersson completion of the Teichmüller space of a surface of genus $g$ and $p$ punctures with $3g-3+p>0$.      Under suitable assumptions on a homomorphism $\rho: \pi_1(M) \rightarrow  \mathsf{Isom}(\tilde X)$,  we show that there exists a $\rho$-equivariant pluriharmonic map
$\tilde u: \tilde M \rightarrow \tilde X$ of possibly infinite energy. In the case when the target is  Kähler  and  $\mathsf{rank}(d \tilde u) \geq 3$ at some point,  $\tilde u$  is  holomorphic or conjugate holomorphic. This  builds on previous important work by [24] and [34]. We also extend these results to the case when the target is a Riemannian manifold with sectional curvature bounded from above by a negative constant.

In his influential paper [43], Y.-T. Siu proved   that harmonic maps between Kähler manifolds of complex dimension $\geq 2$ are pluriharmonic, provided that the domain manifold is compact and  the target  non-positively curved in a strong sense.  This result partially settled a conjecture by Yau made a few years earlier.  Siu’s ideas included an extension of the Bochner's formula and were further developed by others.  Notably, Sampson [41] showed that a harmonic map from  a compact Kähler manifold to a Riemannian manifold of  nonpositive Hermitian sectional curvature is pluriharmonic.  The compactness  of the domain  can be replaced by  completeness and finite volume, provided that there exists a finite energy map (cf. [44] and [3]).   However, in many situations,  it may be difficult or even impossible to prove that a finite energy  maps exists.  The goal of this manuscript is  to extend Siu's and Sampson's work  by   proving  the  existence of possibly infinite energy pluriharmonic or even holomorphic  maps between Kähler manifolds.

Infinite energy  harmonic maps between  manifolds  previously appeared the work of Lohkamp and Wolf.  Lohcamp [32]  proved  the existence of a harmonic map in a given homotopy class of maps between two non-compact manifolds, provided that  a certain simplicity condition is satisfied. The most important case  is when the domain  is metrically a product  near infinity.  Wolf [48] studied  harmonic maps of infinite energy  when the domain is a nodal Riemann surface and applied this study to describe degenerations of surfaces in the Riemann moduli space.

A few years later, Jost and Zuo (cf. [24], [49]) sketched a proof of the existence of  infinite energy pluriharmonic maps from  non-compact Kähler manifolds to symmetric spaces and  buildings.
In a remarkable paper, Mochizuki  [34] rigorously constructed pluriharmonic  maps into the symmetric space    $GL(r, \C)/ U(r)$. More precisely, Mochizuki constructed pluriharmonic metrics on flat vector bundles.  These metrics correspond to harmonic maps  by the Donaldson-Corlette theorem (cf. [13], [3]).
Mochizuki  brought into the picture a new Bochner formula (cf.  <cit.>) for harmonic bundles that can be used to bypass the potential obstruction coming from the non-triviality of the normal bundle of the divisor at infinity.

In this paper, we prove the existence of pluriharmonic maps without assuming the existence of a finite energy maps nor the triviality assumption.  One of the contributions of this paper is to broaden Mochizuki's Bochner formula for  harmonic maps.   Applying these Bochner formulas, we prove the following theorems:

Let $M$, $\tilde X$ and $\rho:\pi_1(M) \rightarrow \tilde X$ satisfy the following:

* $M$ is a smooth quasi-projective variety with universal cover $\tilde M$

* $\tilde X$ is a symmetric space of non-compact type

* $\rho: \pi_1(M) \rightarrow  \mathsf{Isom}(\tilde X)$  is a proper homomorphism that  does not fix an unbounded closed convex strict subset of $\tilde X$.

Then  there exists a unique  $\rho$-equivariant pluriharmonic map $\tilde u: \tilde M \rightarrow \tilde X$ of sub-logarithmic growth.

Here, sub-logarithmic growth means that the map grows slower than a logarithmic function as it approaches the divisor, cf. Definition <ref> and Definition <ref>.

In Theorem <ref>, if we additionally assume that  $\tilde X$ is a  Kähler manifold and the $\rho$-equivariant harmonic map $\tilde u:\tilde M \rightarrow \tilde X$ has $\mathsf{rank}_{\R}(d \tilde u) \geq 3$ at some point, then $\tilde u$ is holomorphic or conjugate holomorphic.

The conclusions of Theorem <ref>  and Theorem <ref> hold if we replace $\tilde X$ by  a complete simply connected   Kähler manifold with strong nonpositive  curvature in the sense of Siu and sectional curvature bounded from above by $\kappa<0$.
the conclusion of Theorem <ref>   holds if we replace $\tilde X$ by a Riemannian manifold $\tilde X$ of  Hermitian-negative curvature and sectional curvature bounded from above by $\kappa<0$.

The conclusion of Theorem <ref>  holds if we place $\tilde X$ by a locally finite Euclidean building without a Euclidean factor.

The conclusion of Theorem <ref>  holds if we replace $\tilde X$ by the Weil-Petersson completion of Teichmüller space and assume only that $\rho$ is a proper homomorphism (i.e. we do not need to assume   that it fixes an unbounded closed convex strict subset of $\tilde X$).

Theorems <ref>, <ref> and <ref>  were first stated in [24] and [49] where some of the ideas originate.
One of the contributions of this paper is to provide complete proofs.  However, our main motivation of this paper is to develop techniques
to study rigidity questions associated to the mapping class group of surfaces.   Indeed, in a sequel to this paper, we will use Theorem <ref> to provide a proof of the following generalization of the  holomorphic rigidity theorem of [9]:

Let $ \Gamma$ and $\mathcal T$ denote the mapping class group and the Teichmüller space of an oriented  surface  $S$ of genus $g$ and $p$ punctures where $3g-3+p>0$.   Assume that $ \Gamma$ acts as a discrete automorphism group on a contractible Kähler manifold $\tilde{M}$ and there exists a finite cover  $M'$ of $\tilde M/\Gamma$ that is a smooth quasi-projective variety.
Then $\tilde{M}$ is equivariantly biholomorphic or conjugate biholomorphic to  the Teichmüller space $\mathcal T$ of $S$.

The above statement strengthens  <cit.>. In  [9], we made the assumption that $M$ admits a compactification $\overline M$ as an algebraic variety such that   the codimension  of $\overline M \backslash M$ is $\geq 3$. This condition originates in <cit.>, where they use this to prove the existence of an equivariant homotopy equivalence of finite energy between $\tilde M$ and $\mathcal T$. This is also the starting point  of <cit.>. With the theory developed in this paper, we will show that we can remove this unnecessary assumption.

Acknowledgements. The authors would like to thank T. Mochizuki and Y. Siu for illuminating discussions.

Outline of the paper

In Chapter <ref>, we record the relevant background material needed in this paper.  Of particular importance is the characterization of certain class of  isometries in a space of non-positive curvature $\tilde X$.  To give a rough description of these isometries, first note that a semi-simple (i.e. elliptic or hyperbolic) isometry $I$ with translation length $\Delta_I$ is characterized by the property that there exists a point $P \in \tilde X$ such that

\[
d(P, I(P))=\Delta_I.
\]

We want to allow for isometries that are more general than semi-simple ones.  This motivates the following:  An isometry $I:\tilde X \rightarrow \tilde X$ is said to have an exponential decay to its translation length  if  there exists a  geodesic ray $c:[0,\infty) \rightarrow \tilde X$ such that

\[
d(c(t),I(c(t))) = \Delta_I(1 +be^{-at})
\]

for some constants $a,b>0$ (cf. <cit.>).
We show that all isometries of $\tilde X$ considered  in  Theorem <ref>, Theorem <ref>, and Theorem <ref> satisfy this property.  We use this to obtain the energy growth estimates for the harmonic map  we construct later.

In Chapter <ref>, we present the key tools of this paper:  the variations on the Siu's and Sampson's Bochner formulas.  If  the domain Kähler manifold $M$ is compact, then Siu's or Sampson's Bochner formulas can be used to prove that harmonicity implies pluriharmonicity.    With $M$ as in Theorem <ref>, neither Siu's nor Sampson's formulas are sufficient to overcome  the complication coming from the non-compactness of the domain.  Indeed,   they need to be rewritten in a different form  to get the  cancellations of the  residues  near the divisor.
The new formulas, contained in
Theorem <ref> and Theorem <ref>, are  inspired by Mochizuki's work on harmonic bundles  <cit.>.

Chapter <ref>, Chapter <ref>, and Chapter <ref> comprise  the proofs  of Theorem <ref>, Theorem <ref> and Theorem <ref>.  We follow the proof of  [34] with adjustments made  on the account of  the  more general target spaces.
A brief sketch of the proof is given below.

Let $M$,
$\tilde X$ and $\rho:\pi_1(M) \rightarrow \tilde X$ be as in Theorem <ref> with $\dim_\C M=n$.  We construct a pluriharmonic map by an induction on $n$ (cf. Chapter <ref>).
We can assume that  $M=\bar M \backslash \Sigma$ where $\bar M$ is a  smooth projective variety of complex dimension $\geq 2$ and  $\Sigma$ is a divisor with normal crossings.
The inductive hypothesis implies that if

* $\bar Y \subset \bar M$ is a sufficiently ample divisor such that $\bar Y \cap \Sigma$ is a normal crossing,

* $Y=\bar Y \backslash \Sigma$,

* $\iota: Y \hookrightarrow M$ is the inclusion map, and

* $\rho_Y: \pi_1(Y) \rightarrow \tilde X$ is the composition $\rho_Y=\rho \circ \iota_*$,

then there exists a unique pluriharmonic $\rho_Y$-equivariant map $\tilde u_Y: \tilde Y \rightarrow \tilde X$.
Letting $\{\bar Y_s\}$ be a collection of  such subvarieties whose union is $\bar M$,  we show that the map   $\tilde u:\tilde M \rightarrow \tilde X$ given by

\begin{equation} \label{x983}
\tilde u=\tilde u_{Y_s} \mbox{ on $Y_s:=\bar Y_s \backslash \Sigma$}
\end{equation}

is a well-defined, thereby proving the existence result for $\dim_\C M=n$.  To get this induction started, we first need  to show existence and uniqueness of   pluriharmonic maps in  dimensions $n=1$ and $n=2$.

First assume  $\dim_\C M=1$.  In other words, we construct harmonic maps from punctured Riemann surfaces of possibly infinite energy (cf. Chapter <ref>).  For the purpose of explaining this case, we assume that $\tilde X$ is a universal cover of a compact manifold $X$ and let  $\gamma:\Sp^1 \rightarrow X$ 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 defined on the  infinite cylinder $[0,\infty) \times \Sp^1$ and is   equal to $\gamma$ on slices $\{t\} \times \Sp^1$ for  all sufficiently large $t$.  More precisely, given any Lipschitz map $k: \Sp^1 \rightarrow  X$, let

\[
v: [1, \infty)  \times \Sp^1 \rightarrow X
\]

be a  Lipschitz continuous map with

\begin{eqnarray*}
v(1,\theta)=k(1,\theta), & & \forall \theta \in \Sp^1,
\\
v(t,\theta)=\gamma(\theta), & & \forall t \in [1,\infty), \, \theta \in \Sp^1.
\end{eqnarray*}

The energy of $v$ on the finite cylinder $[0,T] \times \Sp^1$ is bounded from above by $T \cdot E^\gamma +C$ where $C$ is a constant independent of $T$.  Let

\[
u_T:[0,T] \times \Sp^1 \rightarrow X
\]

be the Dirichlet solution on the finite cylinder $[0,T] \times \Sp^1$  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 an energy minimizing map, the energy of $u_T$ in any finite cylinder $[0,t] \times \Sp^1$ 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 $[t_1,t_2] \times \Sp^1$ 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 $\{u_T\}_{T\geq 1}$ have a uniform Lipschitz bound on any compact subset of $[0,\infty) \times \Sp^1$.  By the Arzela-Ascoli theorem, there exists a sequence $T_i \rightarrow \infty$ such that  $u_{T_i}$ converges locally uniformly to a harmonic map $u:[0,\infty) \times \Sp^1 \rightarrow  X$.  We extend this  idea  to prove the existence of an equivariant  harmonic map from any punctured Riemann surface $\Sigma$.  In doing so, we replace the existence of a minimizing geodesic $\gamma$ with the assumption that the the isometry $\rho([\gamma])$ has  exponential decay to its translation length for any $[\gamma] \in \pi_1(M)$  representing a loop around a puncture.  We  note that the idea of the prototype map already appears in [32].  The main contribution of Chapter <ref> to the theory of infinite energy harmonic maps  is the extension of the existence result to general NPC target spaces (i.e. not necessarily manifolds) and, more importantly,  the  uniqueness result (cf. Theorem <ref> and Theorem <ref> respectively). The uniqueness is a crucial part of the inductive argument.  Indeed, it is used to show that the the pluriharmonic map in dimension $n$ (<ref>)    is well-defined.

Next, assume $\dim_\C M=2$.   In other words, we consider harmonic maps from a complex surface (cf. Chapter <ref>). The idea, as in the case of $\dim_\C M=1$, is to first construct a prototype map and then use it to construct a harmonic map (cf. <cit.>).  By the assumption that the divisor $\Sigma \subset \bar M$ is a normal crossing divisor,  $\Sigma$ consists of a finite set of  irreducible smooth components $\Sigma_j$ intersecting pairwise at most isolated points. We  choose a canonical section $\sigma_j$ of the line bundle $\mathcal O(\Sigma_j)$ vanishing along $\Sigma_j$ and let $h_j$ be a Hermitian metric on $\mathcal O(\Sigma_j)$.  We endow the quasi-projective variety $M=\bar M \backslash \Sigma$ with a  Poincare type metric $g$ (cf. Definition <ref> and Cornabla and Griffiths [5]) making $M$ into a Kähler manifold.
Following [34], we construct a fiber-wise harmonic map along the divisor at infinity.  In other words, we construct a prototype map $v$ as follows:
on any punctured disk $\D^*$ where $\D$ is a disk transverse to $\Sigma_j$, let $v$ be  such that the restriction to $\D^*$  is a harmonic map.  Much like in  the construction of the harmonic map from the infinite cylinder $[0,\infty) \times \Sp^1$ (which is conformally equivalent to the punctured disk), we use the prototype map $v$ to construct a sequence of    Dirichlet solutions defined on compact sets that exhaust $M$.  This sequence contains a subsequence that  converges locally uniformly to a $\rho$-equivariant harmonic map $u: M \rightarrow \tilde X$.  By the Bochner method, we then  show that $u$ is in fact pluriharmonic (cf. Theorem <ref>).  Technically, the most challenging part is the  analysis near the crossings of the divisor and dealing with the non-triviality of the normal bundle of the divisor.  In fact, even when the divisor is smooth,  an application of the Siu's or Sampson's Bochner formula does not immediately yield pluriharmonicity.   We need the variations of the Siu's and Sampson Bochner formulas provided in Chapter <ref>.  Integrating these formulas and applying integration by parts,  a miraculous cancellation (originally discovered by Mochizuki <cit.> in terms of flat vector bundles)  allows us to prove pluriharmonicity and even holomorphicity of $u$.   We can thus start the induction process to prove the existence in higher dimensions.

In Chapter <ref>, we combine the harmonic map theory to Euclidean buildings developed in [17] and to the Weil-Petersson completion of Teichmüller space developed   in [6], [7], [8] and [9]  with  ideas in the previous chapters to prove Theorem <ref>.
The main difference in the proof of these theorems is that we must  deal with the singularities of harmonic maps that apriori may exist because of the singular nature of the target spaces.
We also use the important fact that every element of the  isometry group for a Euclidean building (resp. the mapping class group)  acts as an elliptic or hyperbolic element.  For the isometry group of a Euclidean building (resp. the mapping class group), this follows from <cit.> (resp. Thurston's classification of mapping classes and [12]).

Table of Contents

Chapter <ref>  Preliminaries

<ref> Definitions  p. part1prelim

<ref> Exponential decay of an isometry
p. sec:Expdecay

<ref> Exponential decay of a commuting pair of isometries
p. sec:Expdecay2

Chapter <ref>  Bochner Formulae

<ref> Between Kähler manifolds
p. sec:KtoK

<ref> From Kähler manifolds to Riemannian manifolds
p. sec:KtoR

<ref> Appendix to Chapter <ref>
p. boch-appendix

Chapter <ref>  Harmonic maps from Riemann surfaces

<ref> Infinite energy harmonic maps from punctured disk
p. disc

<ref> Proof of existence, Theorem <ref>
p. sec:proofexistence

<ref> Proof of uniqueness, Theorem <ref>
p. sec:proofuniqueness

Chapter <ref> Harmonic maps from Kähler surfaces

<ref> Neighborhoods near the divisor p. sec:neardivisor

<ref> Poincaré-type metric and its estimates p. sec:metric

<ref> Prototype section p. prototype

<ref> Energy estimates of the prototype section p. sec:ptm

<ref> Harmonic maps of possibly infinite energy p. sec:existence

<ref> Pluriharmonicity p. sec:pluriharmonicity

<ref> Appendix to Chapter <ref> p. appendix:calculus

Chapter <ref> Harmonic maps in higher dimensions

<ref> Preliminary set-up
p. premsetup

Proof of Theorem <ref>, Theorem <ref>, and Theorem <ref>
p. proofpluriharmonic

Chapter <ref> Euclidean buildings and Teichmüller spaces

<ref> Singular sets of harmonic maps
p. sec:singularsets

<ref> Proof of  Theorem <ref> and Theorem <ref> p. proofofthm4

References p. references

CHAPTER: PRELIMINARIES

§ DEFINITIONS

§.§ NPC spaces

We refer to [2] for more details than provided here.

A curve $c:[a,b] \rightarrow \tilde X$ into a metric space is called a geodesic if $\mathsf{length}(l|_{[\alpha,\beta]})=d(c(\alpha),c(\beta))$ for any subinterval $[\alpha,\beta] \subset [a,b]$.  (Note that a identically constant map from an interval is a geodesic.). A metric space $\tilde X$ is a geodesic space if there exists a geodesic connecting every pair of points in $\tilde X$.

An NPC space $\tilde X$ 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 $Q_{t}=c(tl)$.

It follows immediately from Definition <ref> 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

\[
\]

Let $\tilde X$ be an NPC space.  We say that two geodesics rays $c,c':[0,\infty) \rightarrow X$ are equivalent if there exists a constant $K$ such that $d(c(t), c'(t)) \leq K$ for all $t \in [0,\infty)$.  Denote the equivalence class of a geodesic ray $c$ by $[c]$.  The set $\partial \tilde X$ of boundary points of $\tilde X$ is the set of equivalence classes of geodesic rays.

For any $\kappa>0$, we say an NPC space $\tilde X$ is CAT$(-\kappa)$ if for any  $P,Q,R$ and $Q_t$ as in Definition <ref>,

\begin{equation} \label{cat}
\cosh d(P,Q_t) \leq \frac{\sinh (1-t)\kappa d(P,Q)}{\sinh \kappa d(P,Q))} \cosh d(P,Q)+ \frac{\sinh t\kappa d(P,Q)}{\sinh \kappa d(P,Q))} \cosh d(P,R).
\end{equation}

We say that an NPC space $\tilde X$ is negatively curved if, for any point $P \in \tilde X$, there exists $r>0$ and $\kappa>0$  such that every $P,Q,R \in B_r(P)$ satisfies (<ref>).

§.§ Maps into NPC spaces

In this paper, we consider harmonic maps into  NPC spaces.  In many cases, the target space $\tilde X$ is a smooth Riemannian manifold of non-positive sectional curvature.  In this case, the energy of a smooth map $u: \Omega \rightarrow \tilde X$ is

\[
E^u =\int_\Omega |df|^2 d\mbox{vol}_g
\]

where $(\Omega,g)$ is a Riemannian domain and $d\mbox{vol}_g$ is the volume form of $\Omega$.
We say $u:\tilde M \rightarrow \tilde X$ is harmonic if it is locally energy minimizing; i.e. for any $p \in \tilde M$, there exists $r>0$ such that the restriction  $u|_{B_r(p)}$ minimizes energy amongst all maps $v:B_r(p) \rightarrow \tilde X$ with the same boundary values as $u|_{B_r(p)}$.

In other cases, we consider the target $\tilde X$ to be an NPC space, not necessarily smooth.  Below, we will review harmonic maps in this context, and refer to [26] for more details than provided here.

Let $(\Omega, g)$ be a bounded Lipschitz Riemannian domain.   Let $\Omega_{\epsilon}$  be the set of points in $\Omega$ at a distance least $\epsilon$ from $\partial \Omega$.  Let $B_{\epsilon}(x)$ be a geodesic ball centered at $x$  and $S_{\epsilon}(x)=\partial B_{\epsilon}(x)$.   We say $f: \Omega \rightarrow X$ is  an $L^2$-map (or that $f \in L^2(\Omega,X)$) if

\[
\int_{\Omega} d^2(f,P) d\mbox{vol}_g< \infty.
\]

For $f \in L^2(\Omega,X)$, define

\[e_{\epsilon}:\Omega \rightarrow {\bf R}, \ \ \
e_{\epsilon}(x) =
\left\{
\begin{array}{ll}
\displaystyle{\int_{y \in S_{\epsilon}(x)}
\frac{d^2(f(x),f(y))}{\epsilon^2}
\frac{d\sigma_{x,\epsilon}}{\epsilon} }& x \in \Omega_{\epsilon}\\
0 & \mbox{otherwise}
\end{array}
\right.
\]

where $\sigma_{x,\epsilon}$ is the induced measure on  $S_{\epsilon}(x)$. We define a
family of  functionals

\[
E^f_{\epsilon}:C_c(X) \rightarrow
{\bf R}, \ \ \ \
E^f_{\epsilon}(\varphi)=\int_{\Omega} \varphi e_{\epsilon} d\mbox{vol}_g.
\]

We say $f$ has finite energy (or that $f \in W^{1,2}(\Omega,X)$)

\[ E^f := \sup_{\varphi \in C_c(\Omega ), 0 \leq \varphi \leq 1}
\limsup_{\epsilon \rightarrow 0} E^f_{\epsilon}(\varphi) < \infty.
\]
It is shown in [26] that if $f$ has finite energy, the measures $
e_{\epsilon}(x)d\mbox{vol}_g$ converge weakly to a measure which is
absolutely continuous with respect to the Lebesgue measure.
Therefore, there exists a function $e(x)$, which we call the
energy density, so that $e_{\epsilon}(x)d\mbox{vol}_g \rightharpoonup
e(x)d\mbox{vol}_g$. In analogy to the case of smooth targets, we
write $|\nabla f|^2(x)$ in place of $e(x)$. In particular, the (Korevaar-Schoen) energy of $f$ in $\Omega$ is

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

We say a continuous map $u: \Omega \rightarrow \tilde X$ from a Lipschitz domain $\Omega$ is harmonic if it is locally energy minimizing;   more precisely, at each $p \in \Omega$, 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

\[
\pi_f(V,W)=\Gamma \Omega \times \Gamma \Omega \rightarrow
L^1(\Omega, {\bf R})
\]

\[
\pi_f(V,W)=\frac{1}{2}|f_*(V+W)|^2-\frac{1}{2}|f_*(V-W)|^2.
\]

We refer to [26] for more

Let  $(x^1, \dots, x^n)$ be local coordinates of $(\Omega,g)$ and $g=(g_{ij})$, $g^{-1}=(g^{ij})$ be the local metric expressions.  Then energy density function of
$f$ can be written (cf. <cit.>)

\[
|\nabla f|^2 =  g^{ij} \pi_f(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j})
\]

Next assume  $(\Omega, g)$ is a Hermitian domain and let $(z^1=x^1+ix^2, \dots, z^n=x^{2n-1}+ix^{2n})$ be local complex coordinates.  We extend $\pi_f$ linearly cover $\C$ and  denote

\[
\frac{\partial f}{\partial z^i} \cdot  \frac{\partial f}{\partial \bar z^j}=\pi_f(\frac{\partial}{\partial z^i},  \frac{\partial}{\partial \bar z^j})
\]

\[
\left| \frac{\partial f}{\partial z^i}\right|^2= \pi_f(\frac{\partial}{\partial z^i},  \frac{\partial}{\partial \bar z^i}).
\]

\[
\frac{1}{4}|\nabla f|^2= g^{i\bar j} \frac{\partial f}{\partial z^i} \cdot
\frac{\partial f}{\partial \bar z^j}.
\]

§.§ Isometries of an NPC space

Throughout this paper, we denote the group of isometries of an NPC space $\tilde X$ by $\mathsf{Isom}(\tilde X)$.  Isometries of an NPC space are classified as follows.

For $I\in \mathsf{Isom}(\tilde X)$,

\[
\Delta_I: =\inf_{P \in \tilde X} d(I(P), P)
\]

denote  its translation length and define

\[
\mathsf{Min}(I):=\{ P \in \tilde X:  d(I(P),P)=\Delta_I\}.
\]

The isometry $I$ is  elliptic   if $\Delta_I=0$ and $\mathsf{Min}(I) \neq \emptyset$.     It is hyperbolic  if $\Delta_I>0$ and $\mathsf{Min}(I) \neq \emptyset$.   If $I$ is elliptic or hyperbolic, then we  say $I$ is semisimple.  Otherwise, $I$ is said to be parabolic.

If  $\tilde X$ is locally compact and $I \in \mathsf{Isom}(\tilde X)$ is a parabolic isometry, then there exists a arclength parameterized geodesic ray

\begin{equation} \label{quasi}
c:[0,\infty) \rightarrow \tilde X \ \mbox{ such that  }\
\lim_{s \rightarrow \infty} d(I(c(s)), c(s))=\Delta_I.
\end{equation}

Let  $P_i \in \tilde X$ be such that

\[
\lim_{i \rightarrow \infty} d(I(P_i), P_i) = \Delta_I.
\]

Fix $P_0 \in \tilde X$ and let $T_i=d(P_i, P_0)$.  Since  $\mathsf{Min}(I)=\emptyset$, we have that $T_i \rightarrow \infty$.
Let $c_i:[0,T_i] \rightarrow \tilde X$ be the arclength parameterized geodesic from $P_0$ to $P_i$.  Since $\tilde X$ is assumed to be locally compact, by taking a subsequence if necessary, the sequence  $c_i(s)$ converges uniformly on every compact set to an arclength parameterized geodesic ray $c(s)$.

Let $\Gamma$ be a finitely generated group, $\Lambda$ be a finite set of generators of $\Gamma$, $\tilde{X}$ be an NPC space and $\rho:  \Gamma \rightarrow \mathsf{Isom}(\tilde{X})$ be a homomorphism. Define $\delta: \tilde{X} \rightarrow [0,\infty)$ to be the function

\[
\delta(P)=\max \{d(\rho(\lambda)P,P): \lambda \in \Lambda\}.
\]
We say $\rho$ is proper if the sublevel sets of the function $\delta$ is bounded in $\tilde X$; i.e. given $c>0$, there exists $P_0 \in X$ and $R_0>0$ such that

\[
\{P \in \tilde X:  \delta(P)\leq c\} \subset B_{R_0}(P_0).
\]

§.§ Equivariant maps and section of the flat $\tilde X$-bundle

Following Donaldson  [13],  we will replace equivariant   maps with  sections of an associated fiber bundle.
Assume we have the following:

* a complete Riemannian manifold $(M,g)$ with universal covering $\Pi:\tilde M \rightarrow M$

* NPC space $\tilde X$

* action of $\pi_1(M)$ on  $\tilde \domain$  by deck transformations

* homomorphism $\rho:\pi_1(M) \rightarrow \mathsf{Isom}(\tilde X)$

A map $\tilde f:\tilde \domain \rightarrow  \tilde X$ is said to be $\rho$-equivariant if

\[
\tilde f(\gamma p) = \rho(\gamma) \tilde f(p), \ \ \forall \gamma \in \pi_1(M), \ p \in \tilde \domain.
\]

Assume  $\rho:\pi_1(M) \rightarrow \mathsf{Isom}(\tilde X)$ is a proper homomorphism (cf. Definition <ref>). If there exists a finite energy $\rho$-equivariant map $f:\tilde M \rightarrow \tilde X$, then there exists a Lipschitz harmonic map $u:\tilde M \rightarrow \tilde X$ (cf. <cit.>).    In this paper, we are trying to establish the existence of a harmonic map without the assuming that there exists a finite energy map to start with.

The quotient space 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

\[
|\nabla f|^2(p):=|\nabla \tilde f|^2(\tilde p).
\]

(In literature, this quantity is sometimes referred  as the vertical part the derivative of $f$, e.g. [13].)
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 <ref>.
For $U \subset M$,
the (vertical) energy of a section $f$ is

\begin{equation} \label{vertical}
E^f[U] = \int_U |\nabla f|^2 d\mbox{vol}_g.
\end{equation}

for   sections $f_1$, $f_2$, we define

\begin{equation} \label{defdis}
d(f_1(p),f_2(p)):=d(\tilde f_1(\tilde p), \tilde f_2(\tilde p))
\end{equation}

where  $\tilde f_1$,  $\tilde f_2$ are  the associated  $\rho$-equivariant maps  to sections $f_1$, $f_2$ respectively.

§ EXPONENTIAL DECAY OF AN ISOMETRY

The distinguishing characteristic of a semisimple  isometry $I:\tilde X \rightarrow \tilde X$  is that there is a point $P_* \in \tilde X$ such that  $d(P_*,I(P_*))$ is equal to its translation length $\Delta_I$.  No such point $P_*$ exists if $I$ is a parabolic isometry.
This motivates the following:

We say that $I \in \mathsf{Isom}(\tilde X)$ has an exponential decay to its translation length (or simply exponential decay for short)  if one of the following conditions hold:

* $I$ is semisimple, or

* $I$ is parabolic and there exist an arclength parameterized geodesic ray $c:[0,\infty) \rightarrow \tilde X$ and constants $a,b>0$ such that

\[
d^2(I(c(t)), c(t)) \leq  \Delta_I^2(1+be^{-at}).
\]

If $I$ is semisimple, we can simply let $c(t)$ be identically equal to the point $P_*$ as above, and thus $I$ has an exponential decay to its translation length.  On the other hand, for a parabolic isometry  in a general NPC space, we may not be able to find a geodesic ray $c(t)$ such that that    $d(c(t),I(c(t)))$ converges to $\Delta_I$ exponentially fast.

In this section, we  prove that isometries of  important examples of NPC spaces (i.e. symmetric spaces of non-compact type, CAT($-1$) spaces, and Euclidean buildings) have an exponential decay to their translation length.

§.§ Symmetric spaces of non-compact type

In this subsection, we prove that every isometry of a symmetric space  of non-compact type has exponential decay.  To this end, we introduce the following notations:

* $\Mn$ is the algebra of $(n \times n)$-matrices with real coefficients.

* $\Sn$ is the vector subspace of $\Mn$ consisting of symmetric matrices.

* $\Pn$ is the open cone of positive definite symmetric matrices.

* $\Gl$ is the general linear group, or the  invertible $(n \times n)$-matrices.

* $\On$ is the orthogonal group, i.e. the elements of $\Gl$ such that its transpose is its inverse.

* $\An$ is the subgroup of diagonal matices with positive diagonals.

* $\Nn$ is the subgroup of upper triangular matrices with diagonal entries 1.

The group $\Gl$ acts on $\Pn$ by the standard left action; i.e.

\[
G.p:=G p G^T,  \ \ \  \forall G \in \Gl, \, p \in \Pn
\]

where the left hand side consists of matrix multiplication.
Denoting the identity matrix by $e$, we have

\[
\]

The manifold $\Pn$ along with the Riemannian metric given by

\[
\langle V,W \rangle_p = \mbox{Trace}(p^{-1} V p^{-1}W), \ \ \ p \in \Pn, \ V,W \in T_p\Pn
\]

defines a  NPC symmetric space with isometry group $\mathsf{Isom}(\Pn)=\Gl$ (cf. <cit.>).

Let $e$ be the identity element of $\Pn$.
A geodesic line

\[
c:\R \rightarrow \Pn \mbox{ \  with $c(0)=e$ and $c'(0)=V \in T_e\Pn=\Sn$}
\]

is given by $c(t)=\exp(tV)$.  If $p =\exp(V)$, then $d(e,p)=|V|$ (cf. <cit.>).

For a geodesic ray $c:[0,\infty) \rightarrow \Pn$, let $\xi=[c] \in \partial \Pn$ be the equivalence class of geodesic rays containing $c$.   The parabolic subgroup associated to $\xi=[c]$ is

\[
\Gsf:=\{G \in \Pn:  G.\xi=\xi\}.
\]

The horospherical subgroup associated to $\xi=[c]$ is

\[
\Nsf=\{G \in G_\xi:  \exp(-tV)G\exp(tV) \rightarrow I \mbox{ as } t \rightarrow \infty\}.
\]

We refer to  <cit.> for details regarding these subgroups.

Following an argument of [34],  we will first prove that all isometries of $\Pn$ have exponential decay.

For $G \in \Gl$, define

\[
\rho(G):=\left( \sum_{i=1}^n (\log |b_i|^2)^2 \right)^{\frac{1}{2}}
\]

where $b_1, \dots, b_n$ are the eigenvalues of $G$.

The translation length of $G \in \Gl$ is at least $\rho(G)$; i.e.
\rho(G) \leq \Delta_G.$

We need to show

\begin{equation} \label{tl}
\rho(G) \leq d(p,G.p), \ \ \forall p \in \Pn, \forall G \in \Gl.
\end{equation}

Before we proceed, note the following:

* It is sufficient to prove the inequality of (<ref>) when $p$ is the identity element $e$ in $\Pn$.
We prove this claim in two steps.

* Let $q$ be a diagonal matrix; i.e. $q=\mathsf{diag}(\lambda_1, \dots, \lambda_n)$   and  $q^{\frac{1}{2}}=\mathsf{diag}(\lambda_1^{\frac{1}{2}}, \dots, \lambda_n^{\frac{1}{2}})$.  Thus,  $q=q^{\frac{1}{2}} e q^{\frac{1}{2}}$ and hence
for any $G_0 \in \Gl$ and $G_1:=q^{-\frac{1}{2}} G_0 q^{\frac{1}{2}}$,

\begin{eqnarray*}
d(G_0.q, q) & = & d(G_0 q G_0^t, q)
\ = \
d(G_0 q^{\frac{1}{2}} e q^{\frac{1}{2}} G_0^t, \,q^{\frac{1}{2}}  e q^{\frac{1}{2}})
\\
& = &
d(q^{-\frac{1}{2}} G_0 q^{\frac{1}{2}} e q^{\frac{1}{2}} G_0^t q^{-\frac{1}{2}}, e)
\ = \
\end{eqnarray*}

* Let  $p \in \Pn$.   Choose $O \in \On$ be such that $p=O q O^t=O.q$ where $q$ is a diagonal matrix.   If  $G_0:=OGO^t$ and $G_1:=q^{-\frac{1}{2}} G_0 q^{\frac{1}{2}}$ and, then

\begin{eqnarray*}
d(G.p,p) & = &  d(GO.q, \,O.q)
\ = \
d(O^tGO.q, q)
\\
& = &
d(G_0.q, q)
\ = \
\end{eqnarray*}

and $\rho(G)=\rho(G_1)$.

* It is sufficient to prove (<ref>) assuming the eigenvalues of $G$ are distinct.  Indeed, if $G$ does not have distinct eigenvalues, we can take a sequence $G^{(j)}$ of matrices with distinct eigenvalues such that $\rho(G^{(j)}) \rightarrow \rho(G)$ and $d(G^{(j)}.e,e) \rightarrow d(G.e,e)$.

By virtue of the above two bullet points, we assume that  $G \in \Gl$ has distinct eigenvalues $b_1, b_2, \dots, b_n$ and show (<ref>) with $p=e$.

Let   $V \in T_e\Pn=\Sn$ be a diagonal matrix with diagonal entries

\[
\frac{\log |b_1|^2}{\rho(G)}, \ \frac{\log |b_2|^2}{\rho(G)}, \ \dots, \frac{\log |b_n|^2}{\rho(G)}.
\]

Conjugating by elemen
Let  $c(t)=\exp(tV)$ and $\xi=[c]$. We apply the Iwasawa decomposition to write $G=O A_0 N$  where  $O \in \On$, $A_0 \in \An$ and $N \in \Nn$.
Since $A'=A_0 N$ is an upper triangular matrix, we can decompose
A'=N' A_0
where $N' \in \Nn$.  In summary,

\[
G=O  A_0  N = O  A'=O  N'  A_0.
\]

The diagonal matrix $A_0$ is the hyperbolic isometry fixing the geodesic $c(t)=\exp (tV)$ and the upper triangular matrix  $N$ with 1's on the diagonals is an  element of the horospherical subgroup $\Nsf$ associated to $\xi$ (cf. <cit.>).
By construction,

\[
c(\rho(G))=\exp(\rho(G) V)=A_0A_0^T=A_0.e.
\]

Since the geodesic line $c(t)$ intersects the horospheres centered at $\xi=[c]$ orthogonally,   the point $A_0.e$ is the closest point  to $e$ amongst all points on the  horosphere containing $A_0.e$ (cf. <cit.>).  Furthermore, any orbit of $N'$ is contained in a single horosphere (cf. <cit.>), and hence

\[
d(A_0.e,e) \leq d(N'A_0.e, \, e)=d(G.e,e).
\]

This in turn implies

\[
\rho(G)=\rho(A_0) = d(A_0.e,e)  \leq d(G.e,e).
\]

For a parabolic element $G\in \Pn$, let

* $\xi \in \partial \Pn$ be such that $G \in \Gsf$ (i.e. $G$ is an element of the parabolic subgroup associated to $\xi$),

* $c:[0,\infty) \rightarrow \Pn$ be the geodesic ray such that $\xi=[c]$ and $c(0)=e$ (cf. <cit.>).

Then there exist constants $a,b>0$ such that

\[
\lim_{t \rightarrow \infty} d(c(t),G.c(t)) =  \triangle_G+be^{-at}.
\]

We will prove this by an inductive argument on $n$.

\[
V \in T_e(\Pn)=\Sn \mbox{ such that }c(t)=\exp(tV).
\]

By conjugating by an element of $\So$, we may assume that $V$ is a diagonal matrix

\[
V = \mathsf{diag}(\lambda_1, \dots, \lambda_n)
\mbox{
with $\lambda_1 \geq \dots \geq \lambda_n$}.
\]

Let $r_1, \dots, r_k$ be the multiplicities of these eigenvalues.
By <cit.>,

\[
\]

* $A$ is a block diagonal matrix,

\begin{equation} \label{eq:A}
\begin{pmatrix}
A_{11} & 0 & \dots & 0\\
0 & A_{22} & \dots & 0\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & A_{kk}
\end{pmatrix}
\end{equation}

* $N \in \Nsf$, the horospherical subgroup associated to $\xi \in [c_0]$, and $N$ is a block upper triangular matrix

\begin{equation} \label{eqN}
\begin{pmatrix}
I_{r_1} & A_{12} & \dots & A_{1k}\\
0 & I_{r_2} & \dots & A_{2k}\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & I_{r_k}
\end{pmatrix}
\end{equation}

where $A_{ij}$ is an $(r_j \times r_j)$-matrix and $I_{ij}$ is the  $(r_j \times r_j)$-identity matrix.

Let $F(c)$ be the union of all geodesic lines parallel to $c$.  We have that (cf. <cit.>)

\begin{equation} \label{eq:F}
\begin{pmatrix}
\mathsf{P}(r_1,\R) & 0 & \dots & 0\\
0 & \mathsf{P}(r_2,\R) & \dots & 0\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & \mathsf{P}(r_k,\R)
\end{pmatrix}.
\end{equation}

We consider two cases:

Case 1. The eigenvalues $\lambda_1, \dots, \lambda_n$ of $V$ are distinct; i.e. $r_i=1$ for all $i$.

The assumption implies that $c(t)=\exp (tV)$ is a regular geodesic (cf. <cit.>).  Moreover,  $F(c)$  is the unique maximal flat containing $c(t)$, $A$ is a diagonal matrix $\mathsf{diag}(
\alpha_1, \dots, \alpha_n)$  and acts on  $F(c)$ by translation (cf. <cit.>).  In particular,   $t \mapsto d(c(t),A.c(t))$ is a constant function in $t$.  Thus,

\begin{equation} \label{anotanot}
d(c(t), A.c(t))  = d(c(0), A.c(0)) = d(e,A.e)=\rho(A)=\rho(G).
\end{equation}

Furthermore, letting $\hat c(t):=A.c(t)$, we have

\begin{eqnarray*}
d(\hat c(t),N.\hat c(t)) & = & d(\hat c(t), \, N  \hat c(t) N^T)  \\
& = & d(\hat c(t)^{\frac{1}{2}} \hat c(t)^{\frac{1}{2}} , \  N  \hat c(t)^{\frac{1}{2}} \hat c(t)^{\frac{1}{2}}  N^T)  \\
& = & d(e, \, (\hat c(t)^{-\frac{1}{2}}N \hat c(t)^{\frac{1}{2}})  e(\hat c(t)^{-\frac{1}{2}}  N \hat c(t)^{\frac{1}{2}})^T)
\\
& = & d(e, \, (\hat c(t)^{-\frac{1}{2}}N \hat c(t)^{\frac{1}{2}}).  e).
\end{eqnarray*}

Since $\hat c(t)$ is a diagonal matrix with diagonal entries $b_1^2 e^{t\lambda_1},  b_2^2 e^{t\lambda_2},  \dots,  b_n^2 e^{t \lambda_n}$ and $N$ is an upper triangular matrix with  diagonal entries equal to 1,  there exists $C_1, C_2>0$ such that

\[
\| e - \hat c(t)^{-\frac{1}{2}}  N  \hat c(t)^{\frac{1}{2}}\| \leq C_1e^{-C_2t}
\]

Note that the choice of $V$ having distinct diagonal entries is used in the estimate above.  Indeed,  the constant $C_2$ is chosen less than $\min \{\lambda_i-\lambda_{i+1}: i=1, \dots, n-1 \}>0$.
We thus conclude

\begin{equation} \label{eq:N}
d(\hat c(t),N.\hat c(t)) \leq be^{-at}
\end{equation}

for some $a, b>0$.
Combining this with (<ref>), the triangle inequality implies that

\begin{equation} \label{c}
d(c(t), G. c(t)) \leq d(c(t), \hat c(t))+ d(\hat c(t), N.\hat c(t)) \leq
\rho(G)+be^{-at}.
\end{equation}

Case 2:  The eigenvalues of $V$  are not distinct.

The assumption implies that $c(t)=\exp (tV)$ is a singular geodesic (cf. <cit.>).  We write

\[
\begin{pmatrix}
c_1(t) & 0 & \dots & 0\\
0 & c_2(t) & \dots & 0\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & c_k(t)
\end{pmatrix}.
\]

where $c_i(t)$ is a geodesic ray in  $\mathsf{P}(r_i,\R)$, the block diagonal entry that appear in (<ref>).     If $c_i(t)$ is not constant, then the block diagonal $A_{ii}$ of (<ref>) is an element of  parabolic subgroup associated to $\xi_i=[c_i] \in \partial \mathsf{P}(r_i,\R)$ by <cit.>.  By the induction hypothesis, there exists a geodesic ray $c_i:[0,\infty)  \rightarrow  \partial \mathsf{P}(r_i,\R)$ such that

\[
d(c_i(t), A_{ii}.c_i(t)) \leq
\rho(A_{ii})+b_ie^{-a_it}.
\]

If $c(t)$ is a constant, then $c(t)=e$ is a fixed point of $A_{ii}$.
there exist $\alpha, \beta>0$ such that

\[
d(c(t), A.c(t)) \leq
\rho(A)+\beta e^{-\alpha t}.
\]

By an analogous proof as in (<ref>), there exist constants  $\bar \alpha, \bar \beta>0$ such that

\[
d(A.c(t), (NA).c(t)) \leq
\bar \beta e^{-\bar \alpha t}.
\]

Let $a=\min\{\alpha, \bar \alpha\}$,  $b=\beta+\bar \beta$.  By the triangle inequality,

\[
d(c(t),G.c(t)) \leq d(c(t), A.c(t)) + d(A.c(t), G.c(t)) \leq
\rho(A)+b e^{-a t}.
\]

Let $\tilde X$ be a  irreducible symmetric space of non-compact type.  For $I \in \mathsf{Isom}(\tilde X)$, there exist a geodesic line $c:\R \rightarrow \tilde X$ and constants $a,b>0$ such that

\[
d(c(t),I(c(t))) \leq \triangle_{I} + be^{-at}.
\]

After scaling, we can assume that  $\tilde X$  is a  totally geodesic submanifold  of $\Pn$ (cf. <cit.> or <cit.>)   and $\mathsf{Isom}(\tilde X)=\{I=G|_{\tilde X}:  G \in \Gl, G.\tilde X=\tilde X\}$.
Let $I=G|_{\tilde X} \in \mathsf{Isom}(\tilde X)$.  Then $I$ semisimple if and only if $G$ is semisimple in $\Gl$ (cf. <cit.>).  Thus, the assertion follows from Theorem <ref>.

We have therefore shown the following.

If  $\tilde X$ is an irreducible symmetric space of non-compact type, then every isometry $I \in \mathsf{Isom}(\tilde X)$ has exponential decay.

§.§ CAT($-1$) spaces

In the following important cases, every isometry has exponential decay.

If  $\tilde X$ is a CAT$(-1)$ space, then every isometry $I:\tilde X \rightarrow \tilde X$ has exponential decay.

Let  $I$ be a parabolic isometry and $c_0:[0,\infty) \rightarrow \tilde X$ be a geodesic ray such that

\[
\lim_{t \rightarrow \infty} d(c_0(t), I(c_0(t))) = \Delta_{I}=\inf_{t \in [0,\infty)} d(c_0(t), I(c_0(t))).
\]

Let $c_1(t)=I(c_0(t))$.  For each $R \in [0,\infty)$, let $p=c_0(0)$, $q=c_0(R)$, $r=I(c_0(R))$, $s=c_1(0)$ and $\tilde p$, $\tilde q$, $\tilde r$, $\tilde s$ be the comparison quadrilateral in $\Hyp^2$ to the quadrilateral $(p,q,r,s)$ in $\tilde X$ (cf. [40] or <cit.>).
Let $\tilde c_0^R$ ($\tilde c_1^R$ resp.) be the ray geodesic emanating from $\tilde p$ ($\tilde s$ resp.)and passing through $\tilde q$ ($\tilde r$ resp.). Let $\tilde \gamma_0$ ($\tilde \gamma_1$ resp.) be the geodesic from $\tilde p$ to $\tilde s$ ($\tilde q$ to $\tilde r$ resp.).
By the quadrilateral comparison (cf. [40] or <cit.>),

\[
d(c_0(t), c_1(t)) \leq d(\tilde c_0^R(t), \tilde c_1^R(t)), \ \ \forall t \in [0,R].
\]

Without  loss of generality, we may assume $\tilde c_0^R(0) =\tilde c_0^{R'}(0)$ and $\tilde c_1^R(0) =\tilde c_1^{R'}(0)$ for all $R,R' \in [0,\infty)$.  Arzela-Ascoli then implies that  there exists $R_j \rightarrow \infty$ and geodesic rays $\tilde c_0$, $\tilde c_1=I. \tilde c_0$ such that $\tilde c_0^R \rightarrow \tilde c_0$ and $\tilde c_1^R \rightarrow \tilde c_1$ on every compact subsets of $[0,\infty)$.  Thus, by Theorem <ref> for $M=\Hyp^2$,

\[
d(c_0(t), c_1(t)) \leq d(\tilde c_0(t), \tilde c_1(t)) \leq \Delta_{I}+be^{-at}.
\]

§ EXPONENTIAL DECAY OF A COMMUTING PAIR OF ISOMETRIES

In this subsection, we consider the notion of exponential decay for a commuting pair of isometries $I_m: \tilde X \rightarrow \tilde X$, $ m=1,2$.

We say a commuting pair $I_1, I_2 \in \mathsf{Isom}(\tilde X)$ has exponential decay to their translation length (or exponential decay for short) if one of the following conditions hold:

* $I_1$ and $I_2$ are semisimple, or

* $I_1$ and $I_2$ are parabolic and there exist an arclength parameterized geodesic ray $c:[0,\infty) \rightarrow \tilde X$ and constants  $a,b>0$ such that

\[
d^2(I_m(c(t)), c(t)) \leq  \Delta_{I_m}^2(1+be^{-at}), \ \ m=1,2.
\]

The following is due to [16].
If  $\tilde X$ is  locally compact, $I_1, I_2 \in \mathsf{Isom}(\tilde X)$ commute and $I_1$ is parabolic, then $I_2$ is also parabolic  and fix the same point at infinity.
Thus, commuting isometries  are   always both semisimple or both parabolic.   In the latter case, there exists a geodesic ray $c:[0,\infty) \rightarrow \infty$ such that

\[
\lim_{t \rightarrow \infty} d(I_m(c(t)),c(t)) =\Delta_{I_m}, \ \ m=1,2.
\]

If the subgroup $\langle I_1, I_2 \rangle \subset \mathsf{Isom}(\tilde X)$  is generated by commuting semisimple isometries $I_1, I_2$ and

\[
\rho:2\pi\Z \times 2\pi \Z \rightarrow \Gamma
\]

is a homomorphism defined by $(2\pi,0), (0,2\pi) \in 2\pi\Z \times 2\pi \Z$ mapping to $I_1, I_2$ respectively, then
there exists a totally geodesic $\rho$-equivariant map

\[
h: \R \times \R \rightarrow \tilde X
\]

such that

\[
\left| \frac{\partial h}{\partial x}\right|^2=\frac{\Delta_{I_1}^2}{4\pi^2}, \ \ \ \left| \frac{\partial h}{\partial y} \right|=\frac{\Delta_{I_2}^2}{4\pi^2}
\]

where $(x,y)$ are the standard coordinates of $\R \times \R$.

By <cit.>, we can choose $P_* \in \mathsf{Min}(I_1) \cap \mathsf{Min}(I_2)$. We consider the following cases:

* Assume  $I_1$ and $I_2$ are both elliptic.   Then let $h(x,y)$ be identically equal to $P_*$.

* Assume $I_1$ is hyperbolic and $I_2$ is elliptic. Then $P_*$ is a fixed point of $I_2$.  Let $c_1:\R \rightarrow \tilde X$ be an arclength parameterized geodesic such that   $c_1(0)=P_*$ and  $c_1(\R)$ is an axis of $I_1$.
By commutativity,

\[
I_2 \circ I_1^k(P_*) = I_1^k \circ I_2 (P_*)=I_1^k(P_*), \ \ \forall k \in \Z.
\]

Thus,    $c_1(k\Delta_{I_1})=I_1^k(P_*)$ is a fixed point of $I_2$ for any $k \in \Z$ which implies that  every point of  $c_1(\R)$ is a fixed point of $I_2$.  Define $h(x,y)=c_1(\frac{\Delta_I}{2\pi}x)$.

* Assume $I_1$ and $I_2$ are both hyperbolic.  Then $\langle I_1, I_2 \rangle$ is properly discontinuous, and we can  apply the flat torus theorem  (cf. <cit.>) to assert the existence of a $\rho$-equivariant totally geodesic map from $\R \times \R$ to $\tilde X$.  After an appropriate reparameterization, we obtain a map $h$ with the desired derivative estimates.

If the subgroup $\langle I_1, I_2 \rangle \subset \mathsf{Isom}(\tilde X)$  is generated by commuting parabolic isometries $I_1, I_2$ with exponential decay and

\[
\rho:2\pi\Z \times 2\pi \Z \rightarrow \Gamma
\]

is a homomorphism defined by $(2\pi,0), (0,2\pi) \in 2\pi\Z \times 2\pi \Z$ mapping to $I_1, I_2$ respectively, then we have the following:   there exist  $a,b >0$ and a

\[
\tilde h: [0,\infty) \times \R \times \R \rightarrow \tilde X
\]

such that for each $t \in [0,\infty)$, $\tilde h_t(x,y):=\tilde h(t,x,y)$ is a $\rho$-equivariant map and

\[
\left| \frac{\partial \tilde h}{\partial t}\right|^2\leq 1, \ \ \ \left| \frac{\partial \tilde h}{\partial x}\right|^2\leq \frac{\Delta_{I_1}^2}{4\pi^2}(1+be^{-at}), \ \ \ \left| \frac{\partial \tilde h}{\partial y} \right|\leq \frac{\Delta_{I_2}^2}{4\pi^2}(1+be^{-at}).
\]

Let $c(t)$ be as in Definition <ref>.  Let
\begin{eqnarray*}
\tilde h(t,0,0)=c(t), & & \tilde h(t,2\pi,0)= I_1(c(t)),\\
\tilde h(t,0,2\pi)=I_2(t) & & \tilde h(t,2\pi, 2\pi)=I_1 \circ I_2(c(t))=I_2 \circ I_1(c(t)).
\end{eqnarray*}

For each $t \in [0,\infty)$, let

\begin{eqnarray*}
x  \in [0,2\pi]  & \mapsto & \tilde h(t, x,0) \\
x \in [0,2\pi] & \mapsto & \tilde h(t,x,2\pi)
\end{eqnarray*}

be  geodesics.
For each $x \in [0,2\pi]$, let

\[
y \mapsto \tilde h(t,x,y)
\]

be a geodesic.
Finally, we extend $\tilde h_t(x,y)$ equivariantly from $[0,2\pi] \times [0,2\pi]$ to $\R \times \R$.

The assumption that  the commuting pair $I_1$ and $I_2$ has exponential decay along with  multiple applications of the quadrilateral comparison property of NPC spaces imply the derivative estimates.

§.§ Symmetric spaces of non-compact type

In this section, we prove that every pair of commuting isometries of a symmetric space  of non-compact type has exponential decay.

* $G_1, G_2 \in \Gsf$, and

* $c:[0,\infty) \rightarrow \Pn$ be the geodesic ray such that $\xi=[c]$ and $c(0)=e$.

Then there exist constants $a,b>0$ such that

\begin{eqnarray*}
\lim_{t \rightarrow \infty} d(c(t),G_m.c(t)) & = &  \triangle_{G_m}+be^{-at}, \ \ m=1,2.
\end{eqnarray*}

We will prove this by an inductive argument on $n$.

\[
V \in T_e(\Pn)=\Sn \mbox{ such that }c(t)=\exp(tV).
\]

By conjugating by an element of $\So$, we may assume that $V$ is a diagonal matrix

\[
V = \mathsf{diag}(\lambda_1, \dots, \lambda_n)
\mbox{
with $\lambda_1 \geq \dots \geq \lambda_n$}.
\]

Let $r_1, \dots, r_k$ be the multiplicities of these eigenvalues.
By <cit.>,

\[
G_1=N_1A_1 \ \mbox{ and } \  G_2=N_2A_2
\]

where for $m=1,2$:

* $A_m,$ is a block diagonal matrix as in (<ref>) which we write as

\begin{equation} \label{eq:A2}
\begin{pmatrix}
A_{m,11} & 0 & \dots & 0\\
0 & A_{m,22} & \dots & 0\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & A_{m,kk}
\end{pmatrix}
\end{equation}

* $N \in \Nsf$, the horospherical subgroup associated to $\xi \in [c_0]$, and $N$ is a block upper triangular matrix as in (<ref>).

Let $F(c)$  be the union of all geodesic lines parallel to $c$ as in (<ref>).
We consider two cases:

Case 1. The eigenvalues $\lambda_1, \dots, \lambda_n$ of $V$ are distinct; i.e. $r_i=1$ for all $i$.

The assumption implies that $c(t)=\exp (tV)$ is a regular geodesic (cf. <cit.>).  Moreover,  $F(c)$  is the unique maximal flat containing $c(t)$ and $A_m$ acts on  $F(c)$ by translation (cf. <cit.>).  In particular,   $t \mapsto d(c(t),A_m.c(t))$ is a constant function in $t$.  Thus,

\begin{equation} \label{anotanot}
d(c(t), A_m.c(t))  = d(c(0), A_m.c(0)) = d(e,A_m.e)=\rho(A_m)=\rho(G_m).
\end{equation}

Furthermore, letting $\hat c(t):=A_m.c(t)$, we have

\begin{eqnarray*}
d(\hat c(t),N_m.\hat c(t)) & = & d(\hat c(t), \, N_m  \hat c(t) N_m^T)  \\
& = & d(\hat c(t)^{\frac{1}{2}} \hat c(t)^{\frac{1}{2}} , \  N_m  \hat c(t)^{\frac{1}{2}} \hat c(t)^{\frac{1}{2}}  N_m^T)  \\
& = & d(e, \, (\hat c(t)^{-\frac{1}{2}}N_m \hat c(t)^{\frac{1}{2}})  e(\hat c(t)^{-\frac{1}{2}}  N_m \hat c(t)^{\frac{1}{2}})^T)
\\
& = & d(e, \, (\hat c(t)^{-\frac{1}{2}}N_m \hat c(t)^{\frac{1}{2}}).  e).
\end{eqnarray*}

Since $\hat c(t)$ is a diagonal matrix with diagonal entries $b_1^2 e^{t\lambda_1},  b_2^2 e^{t\lambda_2},  \dots,  b_n^2 e^{t \lambda_n}$ and $N_m$ is an upper triangular matrix with  diagonal entries equal to 1,  there exists $C_1, C_2>0$ such that

\[
\| e - \hat c(t)^{-\frac{1}{2}}  N_m  \hat c(t)^{\frac{1}{2}}\| \leq C_1e^{-C_2t}
\]

Note that the choice of $V$ having distinct diagonal entries is used in the estimate above.  Indeed,  the constant $C_2$ is chosen less than $\min \{\lambda_i-\lambda_{i+1}: i=1, \dots, n-1 \}>0$.
We thus conclude

\begin{equation} \label{eq:N2}
d(\hat c(t),N_m.\hat c(t)) \leq b_me^{-a_mt}
\end{equation}

for some $a_m, b_m>0$.
Combining this with (<ref>), the triangle inequality implies that

\begin{equation} \label{c}
d(c(t), G_m. c(t)) \leq d(c(t), \hat c(t))+ d(\hat c(t), N_m.\hat c(t)) \leq
\rho(G_m)+b_me^{-a_mt}.
\end{equation}

Choose $a=\min\{a_1,a_2\}$ and $b=\max\{b_1,b_2\}$.

Case 2:  The eigenvalues of $V$  are not distinct.

The assumption implies that $c(t)=\exp (tV)$ is a singular geodesic (cf. <cit.>).  We write

\[
\begin{pmatrix}
c_1(t) & 0 & \dots & 0\\
0 & c_2(t) & \dots & 0\\
\dots & \dots & \dots & \dots
\\
0 & 0 & \dots & c_k(t)
\end{pmatrix}.
\]

where $c_i(t)$ is a geodesic ray in  $\mathsf{P}(r_i,\R)$, the block diagonal entry that appear in (<ref>).     If $c_i(t)$ is not constant, then the block diagonal $A_{m,ii}$ of (<ref>) is an element of  parabolic subgroup associated to $\xi_i=[c_i] \in \partial \mathsf{P}(r_i,\R)$ by <cit.>.  By the induction hypothesis, there exists a geodesic ray $c_i:[0,\infty)  \rightarrow  \partial \mathsf{P}(r_i,\R)$ such that

\[
d(c_i(t), A_{m,ii}.c_i(t)) \leq
\rho(A_{m,ii})+b_ie^{-a_it}
\]

If $c(t)$ is a constant, then $c(t)=e$ is a fixed point of $A_{m,ii}$.
there exists $\alpha_m, \beta_m>0$ such that

\[
d(c(t), A_m.c(t)) \leq
\rho(A_m)+\beta_m e^{-\alpha_m t}.
\]

By an analogous proof as in (<ref>), there exist constants  $\bar \alpha_m, \bar \beta_m>0$ such that

\[
d(A_m.c(t), (N_mA_m).c(t)) \leq
\bar \beta_m e^{-\bar \alpha_m t}.
\]

Let $a_m=\min\{\alpha_m, \bar \alpha_m\}$,  $b_m=\beta+\bar \beta$.  By triangle inequality,

\[
d(c(t),G_m.c(t)) \leq d(c(t), A_m.c(t)) + d(A_m.c(t), G_m.c(t)) \leq
\rho(A_m)+b_m e^{-a_m t}.
\]

Choose $a=\min\{a_1,a_2\}$ and $b=\max\{b_1,b_2\}$

If $\tilde X$ is a  irreducible symmetric space of non-compact type, then every pair of commuting isometries has exponential decay.

The proof is similar to the proof of Theorem <ref>.

§.§ CAT($-1$) spaces

If $\tilde X$ is a CAT($-1$), then every pair of commuting isometries  has exponential decay.

Analogously to the proof of Theorem <ref>, the assertion follows from comparing $\tilde X$ to $\mathbb{H}^2$.

CHAPTER: BOCHNER FORMULAE

The goal of this chapter is to prove variations of the Siu's and Sampson's Bochner formulas.  These formulas are contained in  Theorem <ref> and Theorem <ref> and are inspired by Mochizuki's formula in <cit.>.

§ BETWEEN KÄHLER MANIFOLDS

In [43], Siu introduced the following notion of negative curvature.

For local holomorphic coordinates $(w^i)$   of a  Kähler manifold $X$, the curvature tensor component is

\[
R_{i\bar j l \bar k} =\frac{\partial^2 h_{l \bar k}}{\partial w^i \partial \bar w^j}+h^{p\bar q} \frac{\partial h_{l\bar q}}{\partial w^i} \frac{\partial h_{p \bar k}}{\partial \bar w^j}.
\]

For $V=(V^i)$ and $W=(W^i)$, let

\[
R(V,W)=-\frac{1}{2} R_{i\bar j l \bar k}(V^i \overline{W^j}-W^i \overline{V^j})\overline{(V^l \overline{W^k}-W^l \overline{V^k})}
\]

We say $X$ has strongly negative (resp. strongly non-positive)  curvature in the sense of Siu [43] if
$R(V,W)<0$ (resp. $R(V,W) \leq 0$)
for any nonzero $n \times n$ complex matrix $(V^i \overline{W^j}-W^i \overline{V^j})$.

It is well known that a Kähler  symmetric space of non-compact type has strongly non-positive curvature in the sense of Siu (cf. [43]).

Let $X$ be a Kähler manifold with Kähler form $\omega$ and strongly non-positive curvature in the sense of Siu.  Let $M$ be another Kähler manifold and  $u : M \rightarrow X $ be a smooth map.
If the function $Q:M \rightarrow \R$ is defined by setting

\begin{eqnarray*}
Q \frac{ \omega^n}{n!} =-R_{i \bar j l \bar k}  \partial u^l \wedge \bar \partial \bar{u}^k \wedge \bar \partial u^i \wedge \partial \bar{u}^j \wedge \frac{ \omega^{n-2}}{(n-2)!},
\end{eqnarray*}

then $Q \geq 0$.

Follows from  <cit.> .

Let $(M,g)$ and $(X,h)$ be Kähler manifolds.
For a smooth map $u:M \rightarrow X$,

\begin{equation} \label{E}
E:=u^{-1}(TX \otimes \C).
\end{equation}

\[
E=E' \oplus E'' \ \mbox{ where } \ E':=u^{-1} (T^{(1,0)}X) , \ \ E'':= u^{-1} (T^{(0,1)}X).
\]

Denote by $\Omega^{p,q}(E)$, $\Omega^{p,q}(E')$ and $\Omega^{p,q}(E'')$  the space of   $E$-, $E'$- and $E''$-valued $(p,q)$-forms respectively.
If $(w^i)$ are local holomorphic coordinates in $X$, then
$\{\frac{\partial}{\partial u^i}  :=  \frac{\partial}{\partial w^i} \circ u, \
\frac{\partial}{\partial \bar u^i} := \frac{\partial}{\partial \bar w^i} \circ u \}$
is a local  frame  of $E$.
Following [43], denote the  natural projections and  inclusions by

\begin{eqnarray*}
\Pi_{(1,0)}: TX \otimes \C \rightarrow  T^{(1,0)}X, & &
\Pi_{(0,1)}: TX \otimes \C \rightarrow  T^{(0,1)}X,\\
\mbox{I}_{(1,0)}: T^{(1,0)}M \hookrightarrow TM \otimes \C, & &
\mbox{I}_{(0,1)}: T^{(0,1)}M \hookrightarrow TM \otimes \C.
\end{eqnarray*}

The differential $du:TM \rightarrow TX$ gives rise to a map

\[
du \otimes \C:  TM \otimes \C \rightarrow TX \otimes \C.
\]

\begin{eqnarray*}
\partial u = \Pi_{(1,0)} \circ du \otimes \C  \circ \mbox{I}_{(1,0)} \in \Omega^{(1,0)}(E')& &
\bar \partial u = \Pi_{(1,0)} \circ du \otimes \C  \circ \mbox{I}_{(0,1)} \in \Omega^{(0,1)}(E')\\
\partial \bar u = \Pi_{(0,1)} \circ du \otimes \C  \circ \mbox{I}_{(1,0)} \in \Omega^{(1,0)}(E'')& &
\bar \partial \bar u = \Pi_{(0,1)} \circ du \otimes \C  \circ \mbox{I}_{(0,1)} \in \Omega^{(0,1)}(E'').
\end{eqnarray*}

In local holomorphic coordinates of $M$ and $X$,

\begin{eqnarray*}
\partial u=\partial u^i \frac{\partial}{\partial u^i}
& &
\bar \partial u=\bar \partial u^i\frac{\partial}{\partial u^i}
\\
\partial \bar u= \partial\bar u^i \frac{\partial}{\partial \bar u^i}
& &
\bar \partial \bar u=\bar \partial \bar u^i \frac{\partial}{\partial \bar u^i}\\
\overline{\partial u} = \bar \partial \bar u \  & &  \overline{\bar \partial u}=\partial \bar u.
\end{eqnarray*}

Let $\partial_E$, $\bar \partial_E$ denote the exterior derivatives on $E$ induced by the Riemannian  connection  $\nabla$ on $X$.
Since $\nabla$ preserves  $T^{(1,0)}X$ and   $T^{(0,1)}X$, they induce exterior derivatives on $E'$ and $E''$ denoted by $\partial_{E'}$, $\bar\partial_{E'}$, $\partial_{E''}$, $\bar\partial_{E''}$.

The curvature operators of $E'$ and $E''$ are $R_{E'}=(\partial_{E'} + \bar \partial_{E'} )^2$ and $R_{E''}=(\partial_{E''} + \bar \partial_{E''} )^2$ respectively.

Let $\{e^i\}$ be a  local frame of $E$.  For

\begin{eqnarray*}
\psi  =  \psi_i e^i \in \Omega^{p,q}(E) \ \mbox{and }  \
\xi  =  \xi_ie^i \in \Omega^{p',q'}(E),
\end{eqnarray*}

we set

\[
\{\psi, \xi \}= \langle e^i, e^j \rangle
\psi_i \wedge \bar \xi_j
\in \Omega^{p+q',q+p'}
\]

where $ \langle \cdot, \cdot \rangle$ is  the Hermitian metric on $E$ induced by identification $TX  \simeq T^{(1,0)}X$.

With $\{ \cdot, \cdot \}$ given as in Definition <ref>,

\[
\{ \partial u, \partial u \} = - \{ \bar \partial \bar u, \bar \partial \bar u \} , \ \  \{ \partial \bar u, \partial \bar u \} = - \{ \bar \partial u, \bar \partial u \}.
\]

for any smooth map $u : M \rightarrow X $ between  Kähler manifolds,

In local  holomorphic coordinates $(z^\alpha)$ on $M$ and  normal coordinates $(w^i)$ on $X$ at $u(p)$,

\begin{eqnarray*}
\{ \partial u, \partial u\}
\sum_i
\frac{\partial{u^i}}{ {\partial{ z^\beta}}}  \overline{\frac{\partial{u^i}}{\partial{ z^\alpha}}} dz^\beta \wedge d\bar z^\alpha
\frac{\partial{\bar u^i}}{\partial{ \bar z^\alpha}}
\overline{ \frac{\partial{ \bar u^i}}{ {\partial{ \bar z^\beta}}}
d\bar z^\alpha \wedge dz^\beta
= -\{\bar \partial \bar u, \bar \partial \bar u\}
\end{eqnarray*}

\begin{eqnarray*}
\{ \partial \bar u, \partial \bar u\}
\sum_i
\frac{\partial \bar u^i}{ {\partial{ z^\beta}}}  \overline{\frac{\partial \bar u^i}{\partial{ z^\alpha}}} dz^\beta \wedge d\bar z^\alpha
- \sum_i
\frac{\partial  u^i}{\partial{ \bar z^\alpha}}
\overline{
\frac{\partial  u^i}{ {\partial{\bar  z^\beta}}}
d\bar z^\alpha \wedge dz^\beta =-\{\bar \partial u, \bar \partial u\}.
\end{eqnarray*}

For any smooth map $u : M \rightarrow X $ between  Kähler manifolds, we have

\[
\partial_{E'} \bar \partial u = - \bar \partial_{E'} \partial u, \ \ \partial_{E''} \bar \partial \bar u = - \bar \partial_{E''} \partial \bar u
\]

\[
\partial_{E'} \partial u = 0,  \  \bar \partial_{E'} \bar \partial u = 0, \  \partial_{E''} \partial \bar u = 0,  \  \bar \partial_{E''} \bar \partial \bar u = 0.
\]

We will prove the first equality,  all the other being  similar.  In local  holomorphic coordinates $(z^\alpha)$ on $M$ and  normal coordinates $(w^i)$ on $X$ at $u(p)$,

\begin{eqnarray*}
\partial_{E'} \bar \partial u =
\frac{\partial^2 u^i}{\partial z^{\alpha} \partial \bar z^{\beta}}  dz^\alpha \wedge d\bar z^{\beta} \otimes \frac{\partial}{\partial{ u^i}}  =
-\frac{\partial^2 u^i}{\partial \bar z^{\beta} \partial z^{\alpha}}  d\bar z^\beta \wedge d z^\alpha \otimes \frac{\partial}{\partial{ u^i}}
= -\bar \partial_{E'}  \partial u.
\end{eqnarray*}

We now state the well-known Siu-Bochner formula <cit.>.  For the sake of completeness,
we provide a proof in  the appendix to this chapter (cf. Section <ref>).

For a harmonic map $u: M \rightarrow X$ between Kähler manifolds,

\begin{eqnarray*}
\partial  \bar \partial\{\bar \partial u,\bar \partial u \}\wedge \frac{ \omega^{n-2}}{(n-2)!} \nonumber
& = &    \left(4\left|\partial_{E'}\bar \partial u \right|^2 +Q\right) \frac{ \omega^n}{n!} \ = \    \partial  \bar \partial\{\bar \partial \bar u,\bar \partial \bar u \}\wedge \frac{ \omega^{n-2}}{(n-2)!}.
\end{eqnarray*}

The following is the variation of the Siu's Bochner Formula, inspired by the work of  Mochizuki (cf. <cit.>) for harmonic metrics.

For a harmonic map $u: M \rightarrow X$ between Kähler manifolds,

\begin{eqnarray*}
d \{\bar \partial_{E'}  \partial u,  \bar \partial u -  \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
&= &  \left(8\left|{\partial}_{E'} \bar \partial u\right|^2+2Q   \right)  \wedge \frac{\omega^n}{n!}.
\end{eqnarray*}

They key observation is that, since  $\partial \{ \partial_{E'}  \bar\partial u, \bar \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}
$ is an $(n+1,n-1)$-form and $\bar \partial \{\partial_{E'} \bar \partial u, \partial u\}\wedge \frac{ \omega^{n-2}}{(n-2)!} $ is an $(n-1,n+1)$-form,  these two forms  are both   identically equal to zero.  Thus, by applying Lemma <ref>  and Lemma <ref>,

\begin{eqnarray}
\partial \{  \partial_{E'}  \bar \partial u, \partial u -\bar  \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}
\nonumber
& = &  \partial \{\partial_{E'} \bar \partial u,  \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}
\nonumber \\
& = &-  \partial \{\bar \partial_{E'} \partial u,  \partial u \} \wedge\frac{ \omega^{n-2}}{(n-2)!} \nonumber \\
& = &
-  \partial \bar \partial \{ \partial u, \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!} \nonumber
\nonumber \\
& = & \partial \bar \partial\{\bar \partial \bar u, \bar \partial \bar u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\label{sb1},
\\
\bar\partial \{\partial_{E'}\bar \partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!} & = & - \bar\partial \{\partial_{E'}\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\nonumber \\
& = &  -  \bar\partial\partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}  \nonumber  \\
& = & \partial \bar\partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}.
\label{sb2}
\end{eqnarray}

Thus, we obtain

\begin{eqnarray*} \label{mochizukitrick}
\lefteqn{d \{\bar \partial_{E'}  \partial u,  \bar \partial u -  \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!} }\\
& = &  d \{\partial_{E'} \bar\partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\ \ \mbox{(by Lemma~\ref{easycommute})}
\\
& = &  (\partial+\bar \partial) \{\partial_{E'} \bar\partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\\
& = &  \left( \partial \bar\partial\{\bar \partial u, \bar \partial u\}+ \partial \bar \partial\{\bar \partial \bar u, \bar \partial \bar u\}  \right)\wedge \frac{ \omega^{n-2}}{(n-2)!}  \ \ \mbox{(by (\ref{sb1}) and (\ref{sb2}))}.
\end{eqnarray*}

Thus, the asserted identity follows from
Theorem <ref>.

§ FROM A KÄHLER MANIFOLD TO A RIEMANNIAN MANIFOLD

In [41], Sampson introduced the following notion of negative curvature.

Let $X$ be a Riemannian manifold and  $R_{ijkl}$ be its Riemann curvature tensor.
Then $X$ is said to have Hermitian-negative curvature if

\begin{equation} \label{Hnc}
R_{ijkl} A^{i\bar l} A^{j\bar k} \leq 0
\end{equation}

for any Hermitian semi-positive matrix $A = (A^{i\bar l})$.

It is well known that symmetric spaces of non-compact type have Hermitian-negative curvature.  For an exposition, see <cit.>.

Let $(M,g)$ be Kähler manifold, $(X,h)$ be a Riemannian manifold and $u:M \rightarrow X$ be a smooth map.
Again consider the pullback vector bundle

\begin{equation} \label{ERiemannian}
E:=u^{-1}(TX \otimes \C)
\end{equation}

and denote by $\Omega^{p,q}(E)$  the space of   $E$-valued $(p,q)$-forms respectively.
If $(x^i)$ are local  coordinates in $X$, then
$\{\frac{\partial}{\partial u^i}  :=  \frac{\partial}{\partial x^i} \circ u \}$
is a local  frame  of $E$.

\begin{eqnarray*}
\mbox{I}_{(1,0)}: T^{(1,0)}M \hookrightarrow TM \otimes \C, & &
\mbox{I}_{(0,1)}: T^{(0,1)}M \hookrightarrow TM \otimes \C.
\end{eqnarray*}

be the inclusions and define

\begin{eqnarray*}
\partial u = du  \circ \mbox{I}_{(1,0)} \in \Omega^{(1,0)}(E),& &
\bar \partial u = du  \circ \mbox{I}_{(0,1)} \in \Omega^{(0,1)}(E).
\end{eqnarray*}

In local holomorphic coordinates of $M$,

\begin{eqnarray*}
\partial u=\partial u^i   \frac{\partial}{\partial u^i}
& &
\bar \partial u=\bar \partial u^i\frac{\partial}{\partial u^i}.
\end{eqnarray*}

Let $\partial_E$, $\bar \partial_E$ denote the exterior derivatives on $E$ induced by the Riemannian  connection  $\nabla$ on $X$.

The curvature operator of $E$  is $R_E=(\partial_E + \bar \partial_E )^2$.

Let $\{e^i\}$ be a  local frame of $E$.  For

\begin{eqnarray*}
\psi  =  \psi_i e^i \in \Omega^{p,q}(E) \ \mbox{and }  \
\xi  =  \xi_ie^i \in \Omega^{p',q'}(E),
\end{eqnarray*}

we set

\[
\{\psi, \xi \}= \langle e^i, e^j \rangle
\psi_i \wedge \bar \xi_j
\in \Omega^{p+q',q+p'}
\]

where  $ \langle \cdot, \cdot \rangle$ is  the pullback to $E$ of the Hermitian metric defined by extending the Riemannian metric $h$ of  $X$.

With $\{ \cdot, \cdot \}$ given as in Definition <ref>,

\[
\{ \partial u, \partial u \} = - \{ \bar \partial  u, \bar \partial u \}.
\]

for any smooth map $u : M \rightarrow X $.

In local  holomorphic coordinates $(z^\alpha)$ on $M$ and  normal coordinates $(x^i)$ on $X$ at $u(p)$,

\begin{eqnarray*}
\{ \partial u, \partial u\}
\sum_i
\frac{\partial{u^i}}{ {\partial{ z^\beta}}}  \overline{\frac{\partial{u^i}}{\partial{ z^\alpha}}} dz^\beta \wedge d\bar z^\alpha
\frac{\partial{u^i}}{\partial{ \bar z^\alpha}}
\overline{ \frac{\partial{u^i}}{ {\partial{ \bar z^\beta}}}
d\bar z^\alpha \wedge dz^\beta
= -\{\bar \partial u, \bar \partial u\}
\end{eqnarray*}

For any smooth map $u : M \rightarrow X $, we have

\[
\partial_{E} \bar \partial u = - \bar \partial_{E} \partial u, \ \ \partial_{E} \partial u = 0,  \  \bar \partial_{E} \bar \partial u = 0.
\]

We will prove the first equality,  all the other being  similar.  In local  holomorphic coordinates $(z^\alpha)$ on $M$ and  normal coordinates $(x^i)$ on $X$ at $u(p)$,

\begin{eqnarray*}
\partial_{E'} \bar \partial u =
\frac{\partial^2 u^i}{\partial z^{\alpha} \partial \bar z^{\beta}}  dz^\alpha \wedge d\bar z^{\beta} \otimes \frac{\partial}{\partial{ u^i}}  =
-\frac{\partial^2 u^i}{\partial \bar z^{\beta} \partial z^{\alpha}}  d\bar z^\beta \wedge d z^\alpha \otimes \frac{\partial}{\partial{ u^i}}
= -\bar \partial_{E'}  \partial u.
\end{eqnarray*}

The following version of Sampson's Bochner formula is contained in <cit.>.

For a harmonic map $u: (M,g) \rightarrow (X,h)$ from a  Kähler manifold to a Riemannian manifold,

\begin{eqnarray}
\partial  \bar \partial\{\bar \partial u,\bar \partial u \}\wedge \frac{ \omega^{n-2}}{(n-2)!}
& = &    \left(4\left|\partial_E\bar \partial u \right|^2 +4Q_0\right) \frac{ \omega^n}{n!}  \label{SampsonFormula}
\end{eqnarray}

\[
Q_0=-2g^{\alpha \bar \delta} g^{\gamma \bar \beta} R_{ijkl}
\frac{\partial u^i}{\partial z^\alpha} \frac{\partial u^k}{\partial \bar z^\beta} \frac{\partial u^j}{\partial z^\gamma} \frac{\partial u^l}{\partial \bar z^\delta}\]

in local coordinates $(z^\alpha)$ of $M$ and $(x^i)$ of $X$.

By applying a similar proof as Theorem <ref>, we obtain a generalization the following.

For a harmonic map $u: M \rightarrow X$ from a Kähler manifold to a Riemannian manifold,

\begin{eqnarray*}
d \{\bar \partial_E  \partial u,  \bar \partial u -  \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
&= &  \left(8\left|{\partial}_E \bar \partial u\right|^2+8Q_0   \right)  \wedge \frac{\omega^n}{n!}.
\end{eqnarray*}

We apply a similar argument as in the proof of Theorem <ref>.  Again, we use the vanishing $\partial \{ \partial_{E'}  \bar\partial u, \bar \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}=0=\bar \partial \{\partial_{E'} \bar \partial u, \partial u\}\wedge \frac{ \omega^{n-2}}{(n-2)!}$ to show
\begin{eqnarray}
\partial \{  \partial_{E'}  \bar \partial u, \partial u -\bar  \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}
& = &  \partial \{\partial_{E'} \bar \partial u,  \partial u\} \wedge\frac{ \omega^{n-2}}{(n-2)!}
\nonumber \\
& = &-  \partial \{\bar \partial_{E'} \partial u,  \partial u \} \wedge\frac{ \omega^{n-2}}{(n-2)!}\ \ \ \mbox{(by Lemma~\ref{easycommute'})}.\nonumber \\
& = &
-  \partial \bar \partial \{ \partial u, \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\nonumber
\nonumber \\
& = & \partial \bar \partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\ \ \ \mbox{(by Lemma~\ref{form4'})}, \label{sb1'}
\\
\bar\partial \{\partial_{E'}\bar \partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!} & = & - \bar\partial \{\partial_{E'}\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\nonumber \\  & = &  -  \bar\partial\partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!} \nonumber \\
& = & \partial \bar\partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}.
\label{sb2'}
\end{eqnarray}

\begin{eqnarray*} \label{mochizukitrick}
d \{\bar \partial_{E'}  \partial u,  \bar \partial u -  \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}   & = &  d \{\partial_{E'} \bar\partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\ \ \mbox{(by Lemma~\ref{easycommute'})}
\\
& = &  (\partial+\bar \partial) \{\partial_{E'} \bar\partial u, \partial u - \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}
\\
& = & 2\partial \bar\partial\{\bar \partial u, \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}  \ \ \mbox{(by (\ref{sb1'}) and (\ref{sb2'}))}.
\end{eqnarray*}

Thus, the asserted identity follows from
Theorem <ref>.

§ APPENDIX TO CHAPTER <REF>

In this section, we provide a proof of Siu's Bochner formula for the sake of completeness.
First, by a simple computation (cf. [31]), for $\psi   \in \Omega^{p,q}(E)$ and $\xi   \in \Omega^{p',q'}(E)$,

\begin{eqnarray}\label{bracket}
\partial\{\psi, \xi \}&=&\{\partial_E\psi, \xi \}+ (-1)^{p+q}\{\psi, \bar \partial_E\xi \} \nonumber \\
\bar \partial\{\psi, \xi \}&=&\{\bar\partial_E\psi, \xi \}+(-1)^{p+q}\{\psi,  \partial_E\xi \} \\
\partial \bar\partial\{\psi, \xi \}&=&\{\partial_E\bar\partial_E\psi, \xi \}+ (-1)^{p+q+1}\{\bar\partial_E\psi, \bar \partial_E\xi \}  \nonumber \\
& & \ + (-1)^{p+q}\{\partial_E\psi,  \partial_E\xi \}+\{\psi, \bar \partial_E \partial_E\xi \} \nonumber.
\end{eqnarray}

For any smooth map $u : M \rightarrow X $ between  Kähler manifolds, we have

\[
\partial  \bar \partial\{\bar \partial u,\bar \partial u \}\wedge \frac{ \omega^{n-2}}{(n-2)!}=\left(-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \}+ \{\bar \partial u, R_{E'}( \bar \partial u) \}\right)\wedge \frac{ \omega^{n-2}}{(n-2)!}
\]

\[
\partial  \bar \partial\{\bar \partial \bar{u},\bar \partial \bar{u} \}\wedge \frac{ \omega^{n-2}}{(n-2)!}
= \left(-\{\partial_{E''}\bar \partial \bar{u},  \partial_{E''}\bar \partial \bar{u} \}+ \{\bar \partial \bar{u}, R_{E''}( \bar \partial \bar{u}) \}\right)\wedge \frac{ \omega^{n-2}}{(n-2)!}.
\]

By setting $\psi=\xi= \bar \partial u \in \Omega^{0,1}(E')$ in (<ref>) and repeatedly using the fact that $\bar \partial_{E'} \bar \partial u=0$ (cf. Lemma <ref>),

\begin{eqnarray*}
\partial  \bar \partial\{\bar \partial u,\bar \partial u \}
& =&-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \}+ \{\bar \partial u, \bar \partial_{E'} \partial_{E'}\bar \partial u \}
\\
& =&-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \}+ \{\bar \partial u, (\bar \partial_{E'} \partial_{E'} +   \partial_{E'} \bar \partial_{E'}+ \bar \partial_{E'}^2) \bar \partial u  \}.
\end{eqnarray*}

Since $\{\bar \partial u, \partial_{E'}^2 \bar \partial u\} \wedge \frac{ \omega^{n-2}}{(n-2)!}$ is an
$(n-1,n+1)$-form and hence zero  for dimensional reasons, we can complete the square to obtain

\begin{eqnarray}
\partial  \bar \partial\{\bar \partial u,\bar \partial u \}\wedge \frac{ \omega^{n-2}}{(n-2)!}
& =&\left(-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \}+ \{\bar \partial u, (\partial_{E'} + \bar \partial_{E'} )^2 \bar \partial u\}\right) \wedge \frac{ \omega^{n-2}}{(n-2)!} \nonumber\\
& =&\left(-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \}+ \{\bar \partial u, R_{E'}( \bar \partial u) \} \right)\wedge \frac{ \omega^{n-2}}{(n-2)!} \nonumber
\end{eqnarray}

which proves the first equation.
The second equation follows by setting $\psi=\xi= \bar \partial \bar{u} \in \Omega^{0,1}(E'')$ in (<ref>) and following exactly the same computation.

For any harmonic map $u : M \rightarrow X $ between  Kähler manifolds, we have

\begin{eqnarray*}
-\{\partial_{E'}\bar \partial u,  \partial_{E'}\bar \partial u \} \wedge \frac{ \omega^{n-2}}{(n-2)!} & = &
4\left|\partial_{E'}\bar \partial u \right|^2 \frac{ \omega^n}{n!} \\
-\{\partial_{E''}\bar \partial \bar{u},  \partial_{E''}\bar \partial \bar{u} \} \wedge \frac{ \omega^{n-2}}{(n-2)!} & = &
4\left|\partial_{E''}\bar \partial \bar u \right|^2 \frac{ \omega^n}{n!}.
\end{eqnarray*}

From a straightforward computation (cf. <cit.>),

\[
-\{\phi, \phi\}\wedge \frac{ \omega^{n-2}}{(n-2)!}=4(|\phi|^2-|\mbox{Tr}_\omega\phi|^2)\frac{ \omega^n}{n!}, \ \ \ \forall \phi \in \Omega^{1,1}(E).
\]

We apply this identity with $\phi=\partial_{E'}\bar \partial u$ (resp. $\phi=\partial_{E''}\bar \partial \bar u$).   Since $u$ is harmonic, $\mbox{Tr}_{\omega} \partial_{E'}\bar \partial u=0$ and $\mbox{Tr}_{\omega} \partial_{E''}\bar \partial \bar u=0$ by  <cit.> and the Kähler identities.

For any smooth map $u : M \rightarrow X $ between  Kähler manifolds, we have

\[
\{\bar \partial  u, R_{E'}  (\bar \partial  u) \}  = -R_{i \bar j l \bar k}  \partial u^l \wedge \bar \partial \bar{u}^k \wedge \bar \partial u^i \wedge \partial \bar{u}^j=\{\bar \partial \bar u, R_{E''}  (\bar \partial \bar u) \}.
\]

We compute

\begin{eqnarray}\label{dminor}
\{\bar \partial  u, R_{E'}  (\bar \partial  u) \}  &=&
\{\bar \partial u^i\frac{\partial}{\partial u^i}, R_{E'}(\bar \partial u^j\frac{\partial}{\partial u^j})\} \nonumber \\
&=&\{\bar \partial u^i\frac{\partial}{\partial u^i}, \bar \partial u^j \wedge R_{E'}(\frac{\partial}{\partial u^j})\} \nonumber \\
&=&\{\bar \partial u^i\frac{\partial}{\partial u^i}, \bar \partial u^j \wedge R^s_{  j k \bar l}\partial u^k \wedge \overline{\partial u^l} \frac{\partial}{\partial u^s}\} \nonumber\\
&=&\bar \partial u^i \wedge \partial \bar{u}^j \wedge G_{i \bar s} \overline{R^s_{  j k \bar l}\partial u^k \wedge \overline{\partial u^l}} \\
&=&\bar \partial u^i \wedge \partial \bar{u}^j \wedge \overline{ G_{s \bar i} R^s_{  j k \bar l}\partial u^k \wedge \overline{\partial u^l}} \nonumber\\
&=&\bar \partial u^i \wedge \partial \bar{u}^j \wedge \overline{  R_{  j \bar i k \bar l}\partial u^k \wedge \overline{\partial u^l}} \ \mbox{(by [25], (1.7.10))} \nonumber\\
&=& -R_{i \bar j l \bar k}  \partial u^l \wedge \bar \partial \bar{u}^k \wedge \bar \partial u^i \wedge \partial \bar{u}^j \ \mbox{(by [25], (1.7.21))}
\nonumber
\end{eqnarray}

which proves the first equality.
Similarly as in (<ref>),

\begin{eqnarray}\label{emajor}
\{\bar \partial  \bar{u}, R_{E''}  (\bar \partial  \bar{u}) \}  &=&
\{\bar \partial \bar{u}^i\frac{\partial}{\partial \bar{u}^i}, R_{E''}(\bar \partial \bar{u}^j\frac{\partial}{\partial \bar{u}^j})\} \nonumber \\ &=&
\{\bar \partial \bar{u}^i\frac{\partial}{\partial \bar{u}^i}, \bar \partial \bar{u}^j \wedge R_{E''}(\frac{\partial}{\partial \bar{u}^j})\} \nonumber \\
&=&\{\bar \partial \bar{u}^i\frac{\partial}{\partial \bar{u}^i}, \bar \partial \bar{u}^j \wedge \overline{R^s_{  j k \bar l}}\partial \bar{u}^k \wedge \bar \partial u^l  \frac{\partial}{\partial \bar{u}^s} \} \ \ \ \ \mbox{(by \cite[(1.5.10)]{koba})}\\
&=&\bar \partial \bar{u}^i \wedge \partial u^j \wedge G_{s \bar i} R^s_{  j k \bar l}\bar \partial u^k \wedge \partial \bar{u}^l  \nonumber \\
&=&\bar \partial \bar{u}^i \wedge \partial u^j \wedge \  R_{  j \bar i k \bar l}\bar \partial u^k  \wedge \partial \bar{u}^l  \ \mbox{(by \cite[(1.7.10)]{koba})} \nonumber\\
&=& -R_{  j \bar i k \bar l}  \partial u^j \wedge \bar \partial \bar{u}^i \wedge \bar \partial u^k \wedge \partial \bar{u}^l
\nonumber \\
&=& -R_{ k \bar l j \bar i }  \partial u^j \wedge \bar \partial \bar{u}^i \wedge \bar \partial u^k \wedge \partial \bar{u}^l \ \mbox{(by \cite[(1.7.22)]{koba})}
\nonumber \\
&=& -R_{i \bar j l \bar k}  \partial u^l \wedge \bar \partial \bar{u}^k \wedge \bar \partial u^i \wedge \partial \bar{u}^j.
\nonumber
\end{eqnarray}

This proves the second equality.

For any harmonic map $u : M \rightarrow X $ between  Kähler manifolds, we have

\[
\left|\partial_{E''}\bar \partial \bar u \right|^2=\left|\bar  \partial_{E''}\partial \bar u \right|^2=\left|\partial_{E'}\bar \partial u \right|^2.
\]

The left equality follows from Lemma <ref> and the right from the fact that conjugation is an isometry.

Combine Lemma <ref>, Lemma <ref>, Lemma <ref>, Lemma <ref> and Lemma <ref>.

CHAPTER: HARMONIC MAPS FROM RIEMANN SURFACES

In this chapter,  we present some results about harmonic section with possibly infinite energy from punctured Riemann surfaces.   The target space  $\tilde{X}$ is an NPC space.  For smooth targets, the existence results are due to [32] and [48].
We will also prove new uniqueness theorems for these harmonic maps.   In the case when the harmonic map is of finite energy,  uniqueness of  harmonic maps  follows from previous work (cf. [33], [10]) of the authors. Additionally, we give precise estimates  of these harmonic maps near the punctures.  These results are used to find pluriharmonic maps from smooth quasi-projective varieties  of arbitrary dimensions in Chapter <ref>.

Throughout this chapter, we use the following notation:

* $\bar \RR$ is a compact Riemann surface

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

* $\Pi:\tilde \RR \rightarrow \RR$ is the universal covering map

* $\rho:\pi_1(\RR) \rightarrow \mathsf{Isom}(\tilde X)$ is a homomorphism.

For each $j=1, \dots, n$,

* $\lambda^j \in \pi_1(\RR)$ is the element associated to the loop around the puncture $p^j$

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

The main results of this chapter are the following two theorems.

Assume that $\tilde X$ is an NPC space.

(A) the action of $\rho(\pi_1(\RR))$ is proper (cf. Definition <ref>), and

(B) $I^j$ has exponential decay (cf. Definition <ref>) for $j=1,\dots, n$,
then there exists a harmonic section $u:  \RR \rightarrow \tilde \RR \times_\rho \tilde X$  with logarithmic energy growth (cf. Definition <ref> below).

The section $u$ of Theorem <ref> is the unique harmonic section with logarithmic energy growth in the following cases:

(i) $\tilde X$ is a negatively curved space as in Definition <ref>

(ii) $\tilde X$ is an irreducible symmetric space of non-compact type or
is an irreducible locally finite Euclidean building such that the action of $\rho(\pi_1(\RR))$ does not fix an unbounded closed convex strict subset  of $\tilde X$.

We explain some of the terminology contained Theorem <ref>.  Loosely speaking,  sub-logarithmic growth means that, near  the punctures of the Riemann surface,  the section grows at most logarithmically, and  logarithmic energy growth means that the energy of the section grows like a logarithmic function.

In order to  give a precise definition, consider the universal cover $\R \rightarrow \Sp^1$  with the free group  $2\pi \Z \simeq \pi_1(\Sp^1)$  acting on $\R$ by translation.

For a homomorphism $\rho:  2\pi \Z \simeq \pi_1(\Sp^1) \rightarrow \tilde X$,
let  $\R \times_\rho \tilde X \rightarrow \Sp^1$ be the flat $\tilde X$-fiber bundle defined by $\rho$ (cf. Subsection <ref>).
Define $\mathsf{E}_{\rho}$ to be  the infimum of the (vertical) energies of sections  $\Sp^1 \rightarrow \R \times_{\rho} \tilde X$ (cf. (<ref>)).

If  $\Delta_I$ is the translation length of $I=\rho([\Sp^1])$, then

\[
\mathsf{E}_{\rho}= \frac{\Delta^2_{I}}{2\pi}.
\]

Next, we will

\begin{equation}  \label{fixeddisk}
\mbox{fix a conformal disk $\D^j \subset \bar \RR$ centered at each puncture $p^j$}
\end{equation}

such that $\D^i \cap \D^j=\emptyset$ for $i\neq j$.  Furthermore, we use the following notation:

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

* $\D^j_r=\{z \in \D^j: |z|<r\}$

* $\RR_r=\RR \backslash \bigcup_{j=1}^n \D^j_r$

* $2\pi \Z \simeq \langle \lambda^j  \rangle$ is  the free group generated by $\lambda^j$

* $\rho^j:\langle \lambda^j \rangle \rightarrow \mathsf{Isom}(\tilde X)$ be the restriction of $\rho: \pi_1(\RR) \rightarrow \mathsf{Isom}(\tilde X)$.

Fix $P_0 \in \tilde X$ and a fundamental domain $F$ of $\tilde \RR$.  Let  $f_0$ be the section of the fiber bundle $\tilde \RR \times_\rho \tilde X \rightarrow \RR$  such that, for any $p \in \RR \cap \Pi(F)$,   $f_0(p)=[(\tilde p,P_0)]$ where $\tilde p = \Pi^{-1}(p) \cap F$.  (Note that  $\Pi(F)$ is of full measure in $\RR$.) Let  $p^j \in \RR$ be a puncture and define

\begin{equation} \label{tildeyes}
\delta_j:\D^{*j} \rightarrow [0,\infty), \ \ \ \ \delta_j(z) = \essinf_{\{  z \in \D^{*j}\}} d( f(z),f_0(z)).
\end{equation}

We say  a section $f:\RR \rightarrow \tilde \RR \times_\rho \tilde X$  has sub-logarithmic growth if for any $j=1, \dots, n$

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

By the triangle inequality, this definition is independent of the choice of  $P_0 \in \tilde X$.
We say $f$ has logarithmic energy growth if there exists $\rho_0>0$ such that

\begin{equation} \label{Cbddef}
\sum_{j=1}^n
\mathsf{E}_{\rho^j} \log \frac{1}{r} \leq E^{f}[\RR_r] \leq \sum_{j=1}^n\mathsf{E}_{\rho^j} \log \frac{1}{r}+C, \ \ \  0<r<r_0<\rho_0
\end{equation}

for some constant $C>0$.

§ INFINITE ENERGY 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 $\bar \D^*$ to an NPC space $\tilde X$.  Identify the boundary of the disk with the circle; i.e.

\[
\partial \D=\Sp^1,
\]

This induces a natural identification

\[
2\pi \Z \simeq \pi_1(\Sp^1) \simeq  \pi_1( \bar \D^*).
\]

The main result of this section is the following.

Assume the following:

* $\rho:2\pi \Z \simeq \pi_1(\Sp^1) \simeq  \pi_1( \bar \D^*) \rightarrow \mathsf{Isom}(\tilde X)$
is a homomorphism

* $k:\DDDD$ is a locally Lipschitz section

* $I:=\rho([\Sp^1])$ has exponential decay to its translation length

Then there  exists a  harmonic section

\[
u:\DDDD \mbox{  with  }u|_{\Sp^1} =k|_{\Sp^1}.
\]

there exists a constant $C>0$   that depends only on  $\mathsf{E}_\rho$ of Definition <ref>,  $a$, $b$ from Definition <ref> and the section $k$ satisfying the following properties:

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

(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   $\D_{\frac{1}{2}}^*$

(iii)  $u$ has sub-logarithmic growth.

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

Combine Lemma <ref>, Lemma <ref>, Lemma <ref> and Lemma <ref> below.

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

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

\begin{equation} \label{slowblowup}
\lim_{z \rightarrow 0} \  \nu(z) + \epsilon \log |z| =-\infty, \ \ \forall \epsilon>0,
\end{equation}

\[
\sup_{z \in \D_r^*} \nu(z)  \leq \max_{\zeta \in \partial \D_r} \nu(\zeta).
\]

In particular, $\nu$ extends to a subharmonic function on $\D_r$ (cf. [19]).

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

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

is subharmonic on $\D_r^*$ with
f_{\epsilon}|_{\partial \D_r} \equiv 0$ on $\partial \D_r$.
Furthermore, since $h$ is a bounded function and $\nu$ satisfies (<ref>),

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

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

\[
\sup_{z \in \D_r^*} \nu(z)  \leq \sup_{z \in \D_r}h(z) \leq    \max_{\zeta \in \partial \D_r} h(\zeta)=\max_{\zeta \in \partial \D_r} \nu(\zeta).
\]

If  $\rho$ and $I$ are as in Theorem <ref>, then for any section $f: \D_{r,r_0} \rightarrow \DDD \times_{\rho} \tilde X$,

\[
\mathsf{E}_\rho \log \frac{r_0}{r}  \leq
\int_{\D_{r,r_0}}  \frac{1}{r^2} \left| \frac{\partial f}{\partial \theta} \right|^2 rdr \wedge d\theta \leq E^{f}[\D_{r,r_0}], \ \ 0<r<r_0<1.
\]

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

\[
\mathsf{E}_\rho
\leq   \int_0^{2\pi} \left| \frac{\partial c}{\partial \theta} \right|^2 d\theta.
\]

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

\begin{eqnarray*}\label{ggeo}
\mathsf{E}_\rho \log \frac{r_0}{r} & = & \mathsf{E}_\rho  \int_r^{r_0} \frac{1}{r^2}  rdr \wedge d\theta \nonumber\\
&  \leq &   \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 \\
& \leq & E^f[\D_{r,r_0}] \nonumber.
\end{eqnarray*}

If $\rho$, $I$ and $k:\DDDD$ are as in  Theorem <ref>, then there exists a constant $C>0$ and a  Lipschitz section $v:\DDDD$  with $v|_{\partial \D}=k|_{\partial \D}$ such that

\begin{equation}   \label{glog}
\mathsf{E}_\rho \log \frac{r_0}{r} \leq  E^v[\D_{r,r_0}]  \leq \mathsf{E}_\rho \log \frac{r_0}{r}+C, \ \ \ 0<r<r_0\leq \frac{1}{2}.
\end{equation}

Moreover, the Lipschitz constant of $v$ and $C$ are dependent only on $\mathsf{E}_\rho$, $a$, $b$ from Definition <ref> and $k$.

We will  first construct an one-parameter family of  sections   $\gamma_s:\Sp^1 \mapsto \R \times_{\rho} \tilde X$ as follows:

* If $I$ is semisimple (i.e. elliptic or hyperbolic), let $P_{\star} \in \tilde X$ be a point where the infimum $\Delta_I$ is attained.  For any $s \in [0,\infty)$, define
$\tilde \gamma_s:\R \rightarrow \tilde X$ 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 $\gamma_s:\Sp^1 \rightarrow \tilde \Sp^1 \times_{\rho} \tilde X$ be the associated section.

* If $I$ is parabolic, then
$c:[0,\infty) \rightarrow \tilde X$ be a geodesic ray defined in Definition <ref>.  Define the $\rho$-equivariant curve
$\tilde \gamma_s:\R  \rightarrow \tilde X$ such that $\tilde \gamma_s|_{[0,2\pi]}$ is  the geodesic from $c(s)$ and $I(c(s))$ and  let $\gamma_s:\Sp^1 \rightarrow \tilde \Sp^1 \times_{\rho} \tilde X$ be the associated section.

We note that by the quadrilateral comparison of NPC spaces, for $\theta
\in [0,2\pi]$,

\[
d(\tilde \gamma_{s_1}(\theta), \tilde \gamma_{s_2}(\theta)) \leq (1-\frac{\theta}{2\pi}) d(c(s_1), c(s_2))+\frac{\theta}{2\pi}d(I \circ c(s_1)), I \circ c(s_2)))=|s_1-s_2|.
\]

\begin{equation} \label{npcquad}
\left| \frac{d}{ds} (\tilde \gamma_s(\theta)) \right| \leq 1, \ \ \ \forall  s \in (0,\infty), \  \theta \in \Sp^1.
\end{equation}

Since  $I \in \mathsf{Isom}(\tilde X)$ has exponential decay of Definition <ref>, we have

\begin{equation} \label{gammatbd}
\int_{\Sp^1} \left| \frac{\partial \gamma_s}{\partial \theta} \right|^2 d\tau \leq \mathsf{E}_\rho(1+2\pi be^{-as}).
\end{equation}

Define a $\rho$-equivariant map $\tilde v:  \DDD \rightarrow \tilde X$ as follows:  First, for $(r,t) \in  \DDD$,  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 $t \in \R$, 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 $\tilde k(\frac{1}{2},t)$.  Finally, let
$v:\DDDD$ be the section associated to the $\rho$-equivariant map $\tilde v$.

For $0<r<r_0<\frac{1}{2}$, we have

\begin{eqnarray*}
\mathsf{E}_\rho \log \frac{r_0}{r} & \leq  &  \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})} \nonumber \\
& \leq  &  \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
\\
& =  &  \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 \nonumber \\
& \leq  &  \int_{-\log r_0}^{-\log r}( \mathsf{E}_\rho+be^{-a(t-\log 2)^{\frac{1}{3}}})
dt \ \ \ \ \ \ \ \ \mbox{(by (\ref{gammatbd}))}\nonumber \\
& = &
\mathsf{E}_\rho \log \frac{r_0}{r}+C.
\end{eqnarray*}

By (<ref>),

\begin{eqnarray*}
\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}}},
\end{eqnarray*}

and thus

\begin{eqnarray*}
\int_0^{2\pi} \int_r^{r_0} \left| \frac{\partial v}{\partial r} \right|^2 rdr \wedge d\theta  & = &  \int_0^{2\pi} \int_r^{r_0}  \frac{1}{r(-\log r-\log 2)^{\frac{4}{3}}} dr \wedge d\theta  \leq C.
\end{eqnarray*}

If $\tilde X$, $\rho$,  $\mathsf{E}_\rho$ and $k$ as in  Theorem <ref>,
then there  exists a  harmonic section  $u:\D^* \rightarrow \widetilde{\D^*} \times_\rho \tilde X$  with  $u|_{\partial \D} =k|_{\partial \D}$ and a constant $C>0$ that   depends only on $\mathsf{E}_\rho$, $a$, $b$ from Definition <ref> and the section $k$ such that

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

Let  $v$ be as in Lemma <ref>. By <cit.>, there exists a  unique harmonic section

\[
u_r:   \D_{r,1} \rightarrow  \widetilde{ \D_{r,1}}\times_{\rho} \tilde X
\mbox{ with } u_r|_{\partial \D_{r,1}}=v|_{\partial \D_{r,1}}.
\]

We have that

\begin{eqnarray}
\mathsf{E}_\rho \log \frac{r_0}{r} +E^{u_r}[\D_{r_0,1}] & \leq & E^{u_r}[\D_{r,r_0}] +E^{u_r}[\D_{r_0,1}]  \ \ \ \ \  \mbox{(by Lemma~\ref{lowerbd})} \nonumber
\\
& = & E^{u_r}[\D_{r,1}]\nonumber  \\
& \leq  &  E^{v}[\D_{r,1}] \ \ (\mbox{since $u_r$ is minimizing}) \nonumber  \\
& = &  E^{v}[\D_{r,r_0}]+ E^{v}[\D_{r_0,1}]  \nonumber  \\
& \leq &
\mathsf{E}_\rho \log \frac{r_0}{r} +C +E^{v}[\D_{r_0,1}]
\ \ \ \ \  \mbox{(by (\ref{glog}))}.  \nonumber
\end{eqnarray}

\begin{equation} \label{bdvr}
E^{u_r}[\D_{r_0,1}] \leq C+ E^{v}[\D_{r_0,1}].
\end{equation}

Note that the right hand side of the above is independent of $r \in (0,\frac{1}{2}]$.
Thus, $\{u_r|_{\D_{2r_0,1}}\}$ is an equicontinuous family of sections (cf. <cit.>). Consequently, there exists a sequence  $r_i \rightarrow 0$ and a harmonic section

\begin{equation} \label{hmpd}
u:  \DDDD \mbox{ with }  u|_{\partial \D} =k|_{\partial \D}.
\end{equation}

such that the sequence $\{u_{r_i}\}$ converges uniformly on every on compact subset of $\D^*$ to  $u$.

To prove property (i),  we note that for $0<r<r_1<1$,
\begin{eqnarray*}
E^{u_r}[\D_{r,r_1}]  +E^{u}[\D_{r_1,1}] & = & E^{u_r}[\D_{r,1}] \\
& \leq &   E^{v}[\D_{r,1}]
\\
& = & E^{v}[\D_{r,r_1}]+E^{v}[\D_{r_1,1}] \end{eqnarray*}

which, combined with Lemma <ref> and (<ref>), imply that

\[
E^{u_r}[\D_{r_1,1}]  \leq E^v[\D_{r_1,1}]+C.
\]

Letting $r \rightarrow 0$ and applying the lower semicontinuity of energy <cit.>,  we obtain

\[
E^{u}[\D_{r_1,1}]  \leq E^v[\D_{r_1,1}]+C.
\]

Applying Lemma <ref> and (<ref>), we obtain

\[
\mathsf{E}_\rho \log \frac{1}{r_1}  \leq E^{u}[\D_{r_1,1}]  \leq E^v[\D_{r_1,1}]+C \leq \mathsf{E}_\rho \log \frac{1}{r_1}+C.
\]

If    $u:\DDDD$ is a harmonic section  with logarithmic energy growth,

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

where $C$ that depends $C_0$ from (<ref>).

Follows immediately from the inequality (<ref>) in the Definition <ref>.

If    $u:\DDDD$ is a harmonic section with logarithmic energy growth,

\begin{eqnarray} \label{mika1}
\label{mika2}
\left(\left| \frac{\partial u}{\partial \theta}\right|^2 - \frac{\mathsf{E}_\rho}{2\pi}\right) & \leq &  \frac{C}{(-\log r)}
\ \text{ in } \,
\D_{\frac{1}{4}}^*
\\
\lim_{r \rightarrow 0}  (-\log r)\left(  \left| \frac{\partial u}{\partial \theta} \right|^2(r,\theta) -     \frac{\mathsf{E}_\rho}{2\pi} \right)
&  = &  0   \label{limitisF}
\\
\left| \frac{\partial u}{\partial r}\right| & \leq &  \frac{C}{r^2 (-\log r)}
\ \mbox{ in } \,
\D_{\frac{1}{4}}^*
\label{mika1}
\\
\label{limitisFrad}
\lim_{r \rightarrow 0}  (-\log r) r^2  \left| \frac{\partial u}{\partial r} \right|^2 (r,\theta) &  = &  0.
\end{eqnarray}

where  $C$ that depends $C_0$ from (<ref>).

Consider the cylinder

\[
{\mathcal C}
=(0,\infty) \times \Sp^1
\]

and let

\begin{equation} \label{Phi}
\Phi:{\mathcal C} \rightarrow \D^*, \ \ \ \ (t , \psi) = (r=e^{-t}, \theta=\psi).
\end{equation}

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 <cit.>.
$\left| \frac{\partial (u \circ \Phi)}{\partial t} \right|^2$ and $\left| \frac{\partial (u \circ \Phi)}{\partial \psi} \right|^2 - \frac{\mathsf{E}_\rho}{2\pi}$ are integrable in ${\mathcal C}=(0,\infty) \times \Sp^1$ by Lemma <ref> and the chain rule.

The subharmonicity   of the directional energy density functions implies

\[
0 \leq \int_{(t_1,t_2) \times \Sp^1} \triangle \varphi  \left| \frac{\partial (u \circ \Phi)}{\partial \psi} \right|^2 dtd\psi, \ \ \forall \varphi \in C_c^{\infty}((t_1,t_2) \times \Sp^1), \ \varphi \geq 0.
\]

Letting $\varphi$ approximate the characteristic function of $(t_1,t_2) \times \Sp^1$, we obtain

\[
0 \leq  \frac{d}{dt}\Big|_{t=t_2} \left( \int_{\{t\} \times \Sp^1} \left| \frac{\partial (u \circ \Phi)}{\partial \psi} \right|^2 d\psi \right)-\frac{d}{dt}\Big|_{t=t_1} \left( \int_{\{t\} \times \Sp^1} \left| \frac{\partial (u \circ \Phi)}{\partial \psi} \right|^2 d\psi \right).
\]

In other words,

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

is a convex function.    By Lemma <ref>, $\int_0^{\infty} F(t) dt< \infty$.
Since $F(t)$ is convex and  integrable,
$F(t)$ is a  decreasing.  Thus,  we obtain

\begin{equation} \label{bbbb}
tF(t) \leq 2 \int_{\frac{t}{2}}^{t} F(\tau)\ d\tau \leq 2 \int_{\frac{t}{2}}^{\infty} F(\tau)\ d\tau.
\end{equation}

With $\B_1(t_0,\psi_0)$ denoting the unit disk centered at $(t_0,\psi_0) \in \mathcal C$,  the mean value inequality implies

\begin{eqnarray*}
\lefteqn{
t_0 \left( \left| \frac{\partial (u \circ \Phi)}{\partial \psi}\right|^2
\right)(t_0,\psi_0)
\\
& \leq &
\frac{t_0}{\pi} \int_{\B_1(t_0,\psi_0)}\left( \left| \frac{\partial (u \circ \Phi)}{\partial \psi}\right|^2
\right) dtd\psi \\
& \leq &
\frac{t_0}{\pi} \int_{(t_0-1,t_0+1) \times \Sp^1} \left( \left| \frac{\partial (u \circ \Phi)}{\partial \psi}\right|^2
\right) dtd\psi
\\
&  \leq & \frac{1}{\pi}
\frac{t_0}{t_0-1} \int_{(t_0-1,t_0+1) \times \Sp^1} t   \left( \left| \frac{\partial (u \circ \Phi)}{\partial \psi}\right|^2
\right)  dt d\psi
\\
&  = &
\frac{1}{\pi}
\frac{t_0}{t_0-1} \int_{t_0-1}^{t_0+1} t  F(t) dt
\\
&  \leq &
\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}))}.
\end{eqnarray*}

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 $\frac{t_0}{t_0-1}\leq 2$),

\begin{eqnarray}
t_0 \left(  \left| \frac{\partial (u \circ \Phi)}{\partial \psi}\right|^2 (t_0,\psi_0)  -\frac{\mathsf{E}_\rho}{2\pi} \right)
& \leq & \frac{4}{\pi} \int_{\frac{t_0-1}{2}}^{\infty} F(\tau)\ d\tau.
\label{inspector}
\end{eqnarray}

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

Similar argument for

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

proves (<ref>) and (<ref>).

Let  $u:\DDDD$ be a harmonic section  with logarithmic energy growth.   For any $\epsilon>0$, there exists $\rho_0>0$ such that

\[
d^2(u(r_1e^{i\theta}),u(r_0 e^{i\theta}))
\leq
-\epsilon \log r_1, \ \ \ \forall 0<r_1<r_0\leq \rho_0, \, 0<\theta<2\pi.
\]

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. (<ref>)) 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

\begin{eqnarray*}
d^2(u(r_1e^{i\theta}),u(r_0e^{i\theta})) & \leq &
\left(  \int_{r_0}^{r_1}
\frac{d}{dr} d(u(r e^{i\theta}),u(r_0 e^{i\theta}))dr \right)^2
\nonumber \\
& \leq  &
\left(
\int_{r_1}^{r_0}
\left| \frac{\partial u}{\partial r} \right|(re^{i\theta}) dr
\right)^2
\nonumber \\
& \leq  &
\frac{\epsilon}{4}  \left(  \int_{r_1}^{r_0} \frac{dr}{r(-\log r)^{\frac{1}{2}}}   \right)^2
\nonumber \\
& \leq  &
\epsilon \left(  (-\log r_1)^{\frac{1}{2}} - (-\log r_0)^{\frac{1}{2}} \right)^2
\nonumber \\
& \leq &
-\epsilon \log r_1.
\end{eqnarray*}

If    $u:\DDDD$ is a harmonic section with logarithmic energy growth,
then $u$ has sub-logarithmic growth.

By Lemma <ref>, for $\epsilon>0$, there exists
$r_0>0$ sufficiently small such that

\[
d^2(u(re^{i\theta}), u(r_0e^{i\theta})) \leq  - \frac{\epsilon}{4}\log r, \ \ \ r \in (0,r_0).
\]

Set $P_0=\tilde u(r_0e^{i\theta_0}) \in \tilde X$ and define $f_0(p)=[(\tilde p, P_0)]$ as in Definition <ref>.  For $z =re^{i\theta}$,

\begin{eqnarray*}
d^2(u(z),f_0(z))+\epsilon \log |z|& = &
d^2(u(re^{i\theta}),u(r_0e^{i\theta_0}))   +\epsilon \log r \\
& \leq  &
2d^2(u(re^{\i\theta}),u(r_0e^{i\theta}))+2d^2(u(r_0e^{i\theta}),u(r_0e^{i\theta_0}))   +\epsilon \log r
\\
& \leq &  \frac{\epsilon}{2}\log r +d^2(u(r_0e^{i\theta}),u(r_0e^{i\theta_0})). \end{eqnarray*}

Now let  $r \rightarrow 0$.

If   $u, v:\DDDD$ are  harmonic sections  with sub-logarithmic growth  with   $u=v$ on $\partial \D$, then $u=v$ on $\D^*$.

The function $d^2(u,v)$ (as defined by (<ref>))
is a continuous subharmonic function (c.f. <cit.>).  Furthermore,  $d^2(u,v)\equiv 0$ on $\partial \D$.  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 <ref>  to conclude $d^2(u(z),v(z))\equiv 0$ on $ \D^*$ which implies $u \equiv v$.

Any harmonic section  $v:\DDDD$
with sub-logarithmic growth satisfies properties (i), (ii) and (iii)  of Theorem <ref>.

By the uniqueness assertion of Lemma <ref>, $v$ must be the harmonic section $u$ constructed in Theorem <ref>.

§ PROOF OF EXISTENCE, THEOREM <REF>

In this section, we prove existence of an equivariant harmonic map from a punctured Riemann surface  $\RR$.
We  use the following notation:

* $k: \RR \rightarrow  \tilde{\mathcal R} \times_\rho \tilde X$ is a locally Lipschitz section  (cf. <cit.>)

* $k^j:=k|_{\bar \D^{j*}}$

* $\langle \lambda^j \rangle$ is the subgroup of $\pi_1(\RR)$ generated by a loop $\lambda^j$ around the puncture $p^j$

* $\rho^j:  \langle \lambda^j \rangle \rightarrow \mathsf{Isom}(\tilde X)$ is the restriction of $\rho$ to $ \langle \lambda^j \rangle$

* $I^j=\rho(\lambda^j)$

* $\Ej:=\mathsf{E}_{\rho^j}$ as in Definition <ref> where we identify $[\lambda^j]$ with $[\Sp^1]$

Applying   Lemma <ref> with $\rho=\rho^j$, $I=I^j$ and $k=k^j$ yields  a prototype  section

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

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  ${\D}^{*j}$ which we call again $v^j$.
We extend  these local sections $v^j:{\D}^{*j} \rightarrow \tilde {\mathcal R} \times_{\rho} \tilde X$ 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 <ref>,

\begin{equation} \label{ebv}
\sum_{j=1}^n\Ej \log \frac{1}{r} \leq
E^v[\RR_r] \leq
\sum_{j=1}^n \Ej \log \frac{1}{r}+C, \ \ \ 0<r
\leq  1.
\end{equation}

The locally Lipschitz section

\begin{equation} \label{mochizukimap}
v:  \mathcal R \rightarrow \tilde{\mathcal R} \times_{\rho} \tilde{X}
\end{equation}

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 $\tilde{v}:\tilde{M} \rightarrow \tilde{X}$  is called the prototype map.

\[
u_{\RR_r}:   \RR_r \rightarrow \tilde \RR \times_{\rho} \tilde X
\]

to be the unique harmonic section with boundary values equal to that of $v|_{\partial \RR_r}$  (cf. <cit.>).  Fix $r_0 \in (0,\frac{1}{2}]$ and let $r \in (0,r_0)$.
By Lemma <ref>,

\[
\Ej\log \frac{r_0}{r}
\leq    \int_{\D_{r,r_0}} \frac{1}{r^2} \left| \frac{\partial u_{\RR_r}}{\partial \theta} \right|^2 rdr \wedge d\theta\leq  E^{u_{\RR_r}}[\D^j_{r,r_0}].
\]

Thus, by (<ref>),

\[
E^v[\bigcup_{j=1}^n \D^j_{r,r_0}] \leq E^{u_{\RR_r}}[\bigcup_{j=1}^n\D^j_{r,r_0}]+C
\]

which implies

\begin{eqnarray}
E^{u_{\RR_r}}[\bigcup_{j=1}^n \D^j_{r,r_0}] +E^{u_{\RR_r}}[\RR_{r_0}]& = & E^{u_{\RR_r}}[\RR_r]\nonumber  \\
& \leq  &  E^v[\RR_r] \nonumber  \\
& = &
E^v[\bigcup_{j=1}^n \D^j_{r,r_0}] +E^v[\RR_{r_0}]  \nonumber\\
& \leq &
E^{u_{\RR_r}}[\bigcup_{j=1}^n \D^j_{r,r_0}] +C+E^v[\RR_{r_0}].   \nonumber
\end{eqnarray}

In other words,

\begin{equation} \label{bdvr}
E^{u_{\RR_r}}[\RR_{r_0}] \leq C+ E^v[\RR_{r_0}], \ \ \ \forall r \in (0,r_0).
\end{equation}

The right hand side of the inequality (<ref>)  is independent of the parameter $r$; i.e. once we fix $r_0$, the quantity $E^{u_{\RR_r}}[\RR_{r_0}]$ is uniformly bounded for all $r\in (0, r_0)$.  This implies a uniform Lipschitz bound, say $L$,  of $u_{\RR_r}$ for $r \in (0,r_0)$ in $\RR_{2r_0}$ (cf. <cit.>).

Let $\{\mu_1, \dots, \mu_N\}$ be a set of generators of $\pi_1(\RR)$ and  $\tilde u_{\RR_r}$, $\tilde v$ be the $\rho$-equivariant maps associated to  sections $u_{\RR_r}$, $v$ respectively.

\[
d(\tilde u_{\RR_r}(\mu_i p), \tilde u_{\RR_r}(p)) \leq L d_{\tilde M}(\mu_i p,p),
\ \
\forall p \in \tilde \RR_{2r_0}, \ i=1, \dots, N \ \ r \in (0,r_0).
\]

If we let

\[
c=L \sup \{d_{\tilde{M}}(\mu_i p,p): i=1,\dots, N, \ p \in \tilde \RR_{2r_0} \},
\]

then by  equivariance

\[
d(\rho(\mu_i)\tilde u_{\RR_r}(p),\tilde u_{\RR_r}(p))) \leq c, \ \ \
p \in \tilde \RR_{2r_0}, \ i=1, \dots, N, \ \ r \in (0,r_0).
\]

In other words, $\delta(\tilde u_{\RR_r}(p)) \leq c$ for all $p \in \tilde \RR_{2r_0}$ and $r \in (0,r_0)$.
By the properness of $\rho$, there exists $P_0 \in \tilde X$ and $R_0>0$ such that

\[
\{\tilde u_{\RR_r}(p): \  p \in \tilde \RR_{2r_0}, \ r \in (0,r_0)\} \subset B_{R_0}(P_0).
\]

Thus,  by taking a compact exhaustion, applying Arzela-Ascoli and the usual diagonalization argument, there exists a subsequence of  $\tilde u_{\RR_r}$ converges locally uniformly  to a $\rho$-equivariant harmonic map $\tilde u: \tilde \RR \rightarrow
\tilde X$.   Let $u: \RR \rightarrow \tilde \RR \times_{\rho} \tilde{X}$ be the associated  harmonic section.
The lower semicontinuity of energy (cf. <cit.>) and (<ref>) imply

\[
E^{u}[\RR_{r_0}] \leq  \sum_{j=1}^n\Ej  \log \frac{1}{r_0}+C,
\ \ \forall r_0 \in (0,\frac{1}{2}]
\]

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

§ PROOF OF UNIQUENESS, THEOREM <REF>

In this section, we prove the uniqueness of  a harmonic map from a punctured Riemann surface  $\RR$ (cf. Theorem <ref>).
First note that by Lemma <ref>,  for $ 0<r<r_0<1$ and any section $f:\DDDD$,

\begin{equation} \label{lwrbd}
\sum_{j=1}^n\Ej \log \frac{r_0}{r}   \leq  E^f[\bigcup_{j=1}^n \D^j_{r,r_0}].
\end{equation}

\[
E^f(r)=E^f[\RR_r] - \sum_{j=1}^n \Ej \log \frac{1}{r}.
\]

Note that by (<ref>),   $r \mapsto E^f(r)$ is an increasing function.
Define the modified energy of $f$ as

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

Let $\LL_\rho$ be the set of all sections $\RR \rightarrow \tilde \RR \times_\rho \tilde X$ such that  $E^f(0)<\infty$.  Let $\mathcal H_{\rho} \subset \mathcal L_{\rho}$ denote the set of all   harmonic sections $u:  \RR \rightarrow \tilde \RR \times_\rho \tilde X$ 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.

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

This follows immediately from Corollary <ref>.

If $u_0, u_1 \in {\mathcal H}_{\rho}$,  then
for some constant $c$.

We prove  Lemma <ref> by showing that $d^2(u_0,u_1)$ extends as a subharmonic function to $\bar \RR$.  Since $\bar \RR$ is compact,  this implies  $d^2(u_0,u_1)$ is constant.  Indeed,   let   $\D \subset \RR$ be a holomorphic disk.  By  <cit.>,    the function $d^2(u_0,u_1)$ is subharmonic in $\D$.  Since this statement is true for any holomorphic disk $\D \subset \RR$, we conclude that $d^2(u_0,u_1)$ is subharmonic in $\RR$.

Next, consider the fixed conformal disk $\D^j \subset \bar{\RR}$  centered at   $p^j \in {\mathcal P}$ (cf. (<ref>)).
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 $\D^j$ by the triangle inequality.  Thus,  by Lemma <ref>,
$d^2(u_0,u_1)$   is bounded in $\D^{j*}$
and  extends as a subharmonic function on $\D^j$.  Hence, we conclude that $d^2(u_0,u_1)$ extends as a subharmonic function on $\bar \RR$, thereby proving    Lemma <ref>.

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  $u_s:\RR \rightarrow  \tilde \RR \times_{\rho} \tilde X$  to be the associated section of the $\rho$-equivariant map

\begin{equation} \label{us}
\tilde u_s:  \tilde \RR \rightarrow \tilde X, \ \ \ \tilde u_s(z)=(1-s)\tilde u_0(z)+s \tilde u_1(z)
\end{equation}

where the sum on the right hand side above denotes geodesic  interpolation (cf. Notation <ref>).
Since $\tilde u_0$ and $\tilde u_1$ are $\rho$-equivariant, $\tilde u_s$ is also $\rho$-equivariant.

From Lemma <ref>, Lemma <ref>, the convexity of the distance function and the convexity of  energy (cf. <cit.>), it follows that $u_s|_{\D^{j*}}$ also satisfies the properties $(i)$, $(ii)$ and $(iii)$ of Theorem <ref> for all $s \in [0,1]$; i.e.

* $\displaystyle{
E^{u_s}[\D^j_{r,r_0}] \leq  C+ \Ej \log \frac{r_0}{r}

* $\displaystyle{
\lim_{r \rightarrow 0} E^{u_s|_{\partial \D^j_r}}[\Sp^1]  =  \Ej,
\vspace{0.5mm}

* $u_s$ has sub-logarithmic growth.

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

\[
\frac{1}{-\log r_0}  \sum_{j=1}^n \int_0^{2\pi} d^2(u_0|_{\partial \D^j_{r_0}}, u_{s}|_{\partial \D^j_{r_0^2}})d\theta<\epsilon,
\ \  \ \forall s \in [0,1].
\]

It suffices to prove Lemma <ref> for $s=1$. Let $\rho_0>0$ be given. First, Lemma <ref> asserts that there exists $\rho_1 \in (0,\rho_0)$ such that

\[
\frac{1}{-\log r_0}  \sum_{j=1}^n \int_0^{2\pi} d^2(u_1|_{\partial \D^j_{r_0}}, u_1|_{\partial \D^j_{r_0^2}})d\theta
< \frac{\epsilon}{4}, \ \  \ \ \forall r_0 \in (0,\rho_1).
\]

For $c>0$ as in Lemma <ref>, choose $r_0 \in (0,\rho_1)$ such that
\frac{2\pi  n c}{-\log r_0}<\frac{\epsilon}{4}.

\[
\frac{1}{-\log r_0} \sum_{j=1}^n \int_0^{2\pi} d^2(u_0|_{\partial \D^j_{r_0}}, u_1|_{\partial \D^j_{r_0}})d\theta
=\frac{2\pi n c}{-\log r_0} < \frac{\epsilon}{4}.
\]

The inequality of Lemma <ref> for $s=1$ follows from the above two inequalities and the triangle inequality.

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

\[
-\log r_0  \sum_{j=1}^n \left(E^{u_0|_{\partial \D^j_{r_0}}}[\Sp^1] +E^{u_s |_{\partial \D^j_{r_0^2}}}[\Sp^1] \right) <-2 \log r_0 \sum_{j=1}^n \Ej +\epsilon, \ \ \ 0<r_0<\rho_0.
\]

For $s=1$,  Lemma <ref>  follows from Lemma <ref>.  The general case of $s \in [0,1]$ follows immediately by convexity of energy (cf. <cit.>).

For $u_0, u_1 \in {\mathcal H}_{\rho}$ and $s \in [0,1]$,  we have
E^{u_0}(0) = E^{u_s}(0).

It suffices to prove Lemma <ref> 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 <ref> and Lemma <ref>.  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 <ref>)
\begin{equation} \label{ococ}
E^{u_0}[\RR_{r_0}] <  E^{u_1}[\RR_{r_0}] -2\epsilon.
\end{equation}

Next fix $r_1 \in (0,r_0)$.  Let  $\tilde u_0$, $\tilde u_1:  \tilde \RR \rightarrow \tilde X$ be the $\rho$-equivariant maps associated to sections $u_0$, $u_1$.
For the fixed conformal disk $\D^j \subset \bar \RR$ centered at the puncture $p^j$ (cf. (<ref>)), let  $ \tilde \D^j_{r_1,r_0} \subset \tilde \RR$ the lift of   $\D^j_{r_1,r_0} \subset \RR$.
We define a “bridge" between  map $\tilde u_0$ and $\tilde u_1$  by setting

\[
\tilde b: \bigcup_{j=1}^n \tilde \D^j_{r_1,r_0}  \rightarrow \tilde X
\]

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 $(r,\theta)  \in \tilde \D^j_{r_1,r_0}$,  $j=1,\dots, n$.

\[
b:\bigcup_{j=1}^n \D^j_{r_1,r_0} \rightarrow
\tilde \RR \times_{\rho}  \tilde X
\]

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

The NPC condition implies (by an argument analogous to the proof the  bridge lemma  <cit.>)

\begin{eqnarray*}
E^b[\bigcup_{j=1}^n \D^j_{r_1,r_0}]
& \leq &
\frac{1}{2} \log \frac{r_0}{r_1} \sum_{j=1}^n \left(E^{u_0|_{\partial \D^j_{r_0}}}[\Sp^1] +E^{u_1|_{\partial \D^j_{r_1}}}[\Sp^1] \right) \\
& & \  + \frac{1}{\log \frac{r_0}{r_1}} \sum_{j=1}^n \int_0^{2\pi} d^2(u_0|_{\partial \D^j_{r_0}}, u_1|_{\partial \D^j_{r_1}})d\theta.
\end{eqnarray*}

Choose $r_1=r_0^2$ (cf.  Lemma <ref> and Lemma <ref>) to obtain

\begin{equation} \label{acac}
E^b[\bigcup_{j=1}^n \D^j_{r_0^2,r_0}] < -\log r_0
\sum_{j=1}^n \Ej  +2\epsilon.
\end{equation}

Since the section

\[
h:\RR_{r_0^2} \rightarrow \tilde \RR \times_\rho \tilde X
\]

defined by setting

\[
\left\{
\begin{array}{ll}
u_0 & \mbox{ in }\RR_{r_0}\\
b & \mbox{ in } \displaystyle{\bigcup_{j=1}^n \D^j_{r_0^2,r_0}}
\end{array}
\right.
\]
is a competitor for $u_1$, we have

\begin{eqnarray*}
E^{u_1}[\RR_{r_0^2}]  & \leq &  E^h[\RR_{r_0^2}]
\\
& = & E^{u_0}[\RR_{r_0}]+ E^b[\bigcup_{j=1}^n \D^j_{r_0^2,r_0}]\\
&  <&  E^{u_1}[\RR_{r_0}]-2\epsilon + E^b[\bigcup_{j=1}^n \D^j_{r_0^2,r_0}] \ \ \ \ \ \ \mbox{(by (\ref{ococ}))}\\
& < & E^{u_1}[\RR_{r_0}] -\log r_0 \sum_{j=1}^n\frac{\Delta_{I^j}^2}{2\pi}  \ \ \ \ \ \ \mbox{(by (\ref{acac}))}.
\end{eqnarray*}

\[
E^{u_1}[\bigcup_{j=1}^n \D^j_{r_0^2,r_0}] < -\log r_0 \sum_{j=1}^n \Ej.
\]

This contradicts   (<ref>) and proves Lemma <ref>.

For $u_0, u_1 \in \HH_\rho$, there exists a constant $c$ such that

\begin{eqnarray}
d(u_s(p), u_1(p)) & \equiv  & cs, \forall p \in  \RR
\label{d=1}
\\
|(u_s)_*(V)|^2(p) & = & |(u_0)_*(V)|^2(p), \ \ \mbox{for a.e.~}s \in [0,1], \ p \in \RR, V \in T_p\tilde M. \label{pullback}
\end{eqnarray}

By the  convexity of energy (cf. <cit.>),

\[
E^{u_s}[\RR_r] \leq (1-s)E^{u_0}[\RR_r] + s E^{u_1}[\RR_r] -s(1-s) \int_{\RR_r} |\nabla d(u_0, u_1)|^2 d\mbox{vol}_{\domain}
\]

for any $r>0$.

\[
E^{u_s}(r) \leq (1-s)E^{u_0}(r) + s E^{u_1}(r) -s(1-s) \int_{\RR_r} |\nabla d(u_0, u_1)|^2 d\mbox{vol}_{\domain}.
\]

Letting $r \rightarrow 0$ and applying Lemma <ref>,  we conclude

\begin{eqnarray}
0 &= & \int_{\RR} |\nabla d(u_0, u_1)|^2 d\mbox{vol}_{\domain}
\label{graddist}
\\
E^{u_s}& = & E^{u_0}, \ \ \forall s \in [0,1] \label{energyconstant}
\end{eqnarray}

First, (<ref>) implies that $\nabla d(u_0, u_1)=0$ a.e. in $\domain$ which in turn implies $d(u_0,u_1) \equiv c$ for some $c$.  By the definition of the map $u_s$, (<ref>) follows immediately.

Next, for $\{P,Q,R,S\} \subset \tilde X$, the quadrilateral comparison for NPC spaces   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. <cit.>)

\[
|(u_s)_*(V)|^2(p) \leq (1-s) |(u_0)_*(V)|^2(p) + s|(u_1)_*(V)|^2(p),
\
\mbox{a.e.~$p \in \tilde M, V \in T_p\tilde M$.}
\]
Integrating the above over all unit vectors $V \in T_p\tilde M$  and then over $p \in F$, we obtain

\[
E^{u_s} \leq (1-s) E^{u_0}+ s E^{u_1}.
\]

Combining this with  (<ref>) implies (<ref>).

We assume there exists $u_0, u_1 \in \HH_\rho$ such that $u_0 \not \equiv u_1$ and treat the two  cases (i) and (ii) of Theorem <ref> separately.

* $\tilde X$ is a negatively curved space as in Definition <ref>:
For  $p \in \tilde \RR$,  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  <cit.> implies that the image under $u_0$ of a  sufficiently small  neighborhood of $p$ is contained in an image $\sigma(\R)$ of a geodesic line.   (If $\tilde X$ is a smooth manifold, this follows by Hartman [18]).  If $ \tilde u_0 \not \equiv \tilde u_1$, then $\tilde u_s(p) \neq \tilde u_0(p)$ for all $p \in \tilde \RR$ and $s \in (0,s_0]$ by (<ref>).  Thus, we conclude that $\tilde u_0(\tilde \RR)$ is contained in $\sigma(\R)$.  Consequently, $\rho(\pi_1(\tilde \RR))$ fixes $\sigma(\R)$ which contradicts  the fact that $\rho(\pi_1(\RR))$ 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, Lemma <ref> combined with [10]  implies  the uniqueness result.

CHAPTER: HARMONIC MAPS FROM KÄHLER SURFACES

In this chapter, we assume that  the complex dimension of $M$ is 2.
In dimension 2, the intersection of two irreducible components of the normal crossings divisor consists of  isolated points.  Thus, the analysis around the juncture of a normal crossing divisor  is much easier to understand than in higher dimensions.

Throughout this section, we assume:

* $M$ is a smooth quasi-projective variety such that $M=\bar M \backslash \Sigma$ where $\bar M$ is a  smooth projective variety of complex dimension $2$,  $\Sigma$ is a divisor with normal crossings and $M=\bar M \backslash \Sigma$

* $\tilde X$ is an NPC space

* $\rho: \pi_1(M) \rightarrow  \mathsf{Isom}(\tilde X)$  is a proper homomorphism that  does not fix an unbounded closed convex strict subset of $\tilde X$.

* every element (resp. commuting pair) of $\mathsf{Isom}(\tilde X)$ has exponential decay (cf. Definition <ref> and Definition <ref>)

In other words, $M$ and $\rho$ are as in
Theorem <ref> with the additional assumption that

\[
\dim_\C M=2,
\]

but the target space $\tilde X$ is more general  than in Theorem <ref>.  By relaxing the restriction on the target space, we  include for example the case when $\tilde X$ is a Euclidean building   (cf. Theorem <ref>) and the Weil-Petersson completion of Teichmüller space (cf. Theorem <ref>).

The main theorems  of Chapter <ref> are:

With the Poincaré-type Kähler metric $g$  on $M$ (cf. Section <ref>), there exists    a   harmonic section
u: M \rightarrow \tilde M \times_\rho \tilde{X}
of logarithmic energy growth.

If in addition $\tilde X$ is as in Theorem <ref>, the harmonic section $
u: M \rightarrow \tilde M \times_\rho \tilde{X}
$ of Theorem <ref>
is pluriharmonic.

The main idea of the proof  of Theorem <ref> is to   construct a prototype section which almost minimizes energy near infinity.  This idea goes back to [32] and  [24].  On the other hand,   since we want include the case when  the normal bundle of $\Sigma$ is non-trivial and also the case when the divisor consists of more than one irreducible components,  our argument closely follows the construction in Mochizuki [34].  We use the prototype section to construct a harmonic section.  The idea is to consider the Dirichlet problem on a compactly supported domain with boundary values equal to $v$ and then to take a compact exhaustion.

Throughout this Chapter, we let $\bar M$ be a   smooth projective variety of complex dimension $\geq 2$ and  $\Sigma$ be a divisor with normal crossings such that

\[
M=\bar M \backslash \Sigma.
\]
Furthermore, let

\[
\Sigma= \bigcup_{j=1}^L \Sigma_j
\]

where $\{\Sigma_j\}$ is the set of irreducible components of $\Sigma$.

Chapter <ref> consists of the following:

In Section <ref>, we study neighborhoods of the divisor of $\bar M$.  To do so, we consider the following:

* A neighborhood $\Omega \times \D_{\frac{1}{4}}$  of a point in $\Sigma_j$ away from the crossings

* A neighborhood
$\D_{\frac{1}{4}}  \times  \D_{\frac{1}{4}}$ near  the crossing $\Sigma_j \cap \Sigma_i$.

In Section <ref>,  we discuss a  Poincaré-type metric $g$ (cf. Definition <ref>) due to Cornalba and Griffiths [5].  This is a complete metric which puts the divisor $\Sigma$ at infinity.

In Section <ref>, we construct  a $\rho$-equivariant prototype section $v:M \rightarrow \tilde M \times_\rho \tilde X$   with controlled growth near infinity. The crucial tool in this contruction is Theorem <ref> (i.e. the Dirichlet solution on the punctured disk).  This enables us to construct a fiber-wise harmonic map on the normal bundle of the divisor.  This map defined near the divisor is then extended to all of $M$.

In Section <ref>,  we give  precise estimates for energy growth of the prototype section near the divisor at infinity.  These are important because they imply the estimates for the harmonic section.

In Section <ref>, we use the prototype section $v:M \rightarrow \tilde M \times_\rho \tilde X$  in order to construct a harmonic section $u:M \rightarrow \tilde M \times_\rho \tilde X$ (cf. Theorem <ref>). We end with some energy estimates of the harmonic section constructed.

In Section <ref>, we show that the harmonic section is in fact pluriharmonic (cf. Theorem <ref>). This is where we take advantage of the variation of the Bochner formulas proved in Chapter <ref>.

§ NEIGHBORHOODS NEAR THE DIVISOR

We continue to denote by $\D$ the unit disk in the complex plane. For clarity, we will also denote the unit disk by $\D_{z}$ to indicate that $\D$ is being parameterized by the complex variable $z=re^{i\theta}$.    We use analogous notation $\D_{z,r}$, $\D_{z, r_1,r_2}$, etc. that corresponding to the sets defined in Section <ref>. We also use the notation $\Sp^1_{\theta}$ to denote the circle  $\Sp^1$  parameterized by the real variable $\theta$ and identify $\Sp^1_{\theta}$ as the boundary of ${\D}_{z}$ via the map $\theta \mapsto e^{i\theta}$.

Recall that $\bar M$ is a compact Kähler surface with a divisor $\Sigma$ containing only  simple normal crossings,
M=\bar M \backslash \Sigma,
and $\{\Sigma_j : j=1,\cdots, L\}$ are the irreducible components of  $\Sigma$.  Let $\sigma_j$ be the canonical section of ${\mathcal O}(\Sigma_j)$ with zero set $\Sigma_j$.

To study a neighborhood of the juncture, let $P \in \Sigma_i \cap \Sigma_j$ for some $i, j \in \{1, \dots, L\}$ with $i \neq j$, and let $V_P$ be a neighborhood of $P$ containing no other crossings.
Choose holomorphic  trivializations $e_i$ (resp. $e_j$)  of $\mathcal O(\Sigma_i)$ (resp. $\mathcal O(\Sigma_j)$)  on
$V_P$ and define $z^1$ (resp. $z^2$) by setting

\begin{equation} \label{z12}
\sigma_i=z^1e_i, \mbox{ (resp. }\sigma_j=z^2e_j).
\end{equation}

For each $j=1, \dots, L$, let $h_j$ be a Hermitian metric  on $\mathcal\mathcal {\mathcal O}(\Sigma_j)$ such that $|e_j|_{h_j}=1$ in $V_P$ for any crossing $P$.
Let $h$ be a Hermitian metric on $\bar{M}$,
not necessarily Kähler, such that the following holds:

(i) The metric $h$ is the Euclidean metric in a neighborhood $V_P$ of every crossing $P$, i.e.

\begin{equation} \label{hiseuc}
h|_{V_P} =dz^1d\bar z^1+ dz^2d\bar z^2.
\end{equation}

By rescaling $\sigma_1$ and $\sigma_2$ if necessary, we can assume without the loss of generality that

\begin{equation} \label{D12}
\bar{\D}_{z^1} \times \bar{\D}_{z^2} \subset V_P.
\end{equation}

(ii) The metric $h$ induces the orthogonal decomposition $T \bar M|_{\Sigma_j} = T \Sigma_j \oplus N \Sigma_j$ and  under the natural isomorphism

\begin{equation} \label{natiso}
N \Sigma_j\simeq \mathcal\mathcal O(\Sigma_j)|_{\Sigma_j},
\end{equation}

the restriction of $h$ to $N \Sigma_j$ is same as $ h_j$.

For   $r \in (0,1]$, we  set

\begin{eqnarray*} \label{nbhds}
{\mathcal D}_{j,r}  & = &   \{\nu \in  N\Sigma_j: |\nu|_{h_j} < r\},   \nonumber \\
{\mathcal D}_{j,r}^*  & = &   \{\nu \in   N\Sigma_j: 0<|\nu|_{h_j} < r\},    \nonumber \\
\bar{\mathcal D}_{j,r}  & = &   \{\nu  \in N\Sigma_j: |\nu|_{h_j} \leq r\} \nonumber\\
\bar{\mathcal D}_{j,r}^*  & = &    \{\nu \in \subset N\Sigma_j: 0<|\nu|_{h_j} \leq r\},
\end{eqnarray*}

\begin{eqnarray*}
{\mathcal D}_r = \bigcup_{j=1}^L {\mathcal D}_{j,r}, \ \  {\mathcal D}_r^* = \bigcup_{j=1}^L {\mathcal D}_{j,r}^*, \ \
\bar{\mathcal D}_r = \bigcup_{j=1}^L\bar{\mathcal D}_{j,r}, \ \   \bar{\mathcal D}_r^* = \bigcup_{j=1}^L\mathcal D_{j,r}^*
\end{eqnarray*}

\begin{eqnarray} \label{nbhd12'}
{\mathcal D}_{r_1,r_2} = {\mathcal D}_{r_2} \backslash \bar{\mathcal D}_{r_1}
\mbox{ for
0< r_1< r_2 \leq 1.
\end{eqnarray}

There exists  $r>0$ such that the restriction of the exponential map

\[
\exp : N \Sigma_j \subset T \bar M|_{\Sigma_j} \rightarrow \bar M
\]

defines diffeomorphism of
${\mathcal D}_{j,r}$ to a neighborhood of $\Sigma_j$ in $\bar M$. By rescaling $\sigma_j$ if necessary, we may assume $r>1$.
In particular, we identify (<ref>) for sufficiently small $r>0$ as an open subset of $M$ via the exponential map; i.e.

\begin{equation} \label{nbhd12}
{\mathcal D}_{r_1,r_2} := \bigcup_j \{\exp \nu: \nu \in N\Sigma_j, \ r_1<|\nu|_{h_j} <r_2\} \subset M.
\end{equation}

\[
\bar{\mathcal D}_j:=\bar{\mathcal D}_{j,1} \subset N\Sigma_j.
\]

The restriction of the normal bundle $N\Sigma_j \rightarrow \Sigma_j$ to $\bar{\mathcal D}_j$ defines a disk bundle

\begin{equation} \label{diskbundle}
\pi_j : \bar{\mathcal D}_j \rightarrow \Sigma_j.
\end{equation}

We also identity $\bar{\mathcal D}_j$  as a subset of $\bar{M}$; i.e.

\begin{equation}  \label{assubsets}
\bar{\mathcal D}_j \simeq \exp(\bar{\mathcal D}_j) \subset \bar{M}.
\end{equation}

We denote by $J_{\bar M}$  the  holomorphic structure on $\bar{\mathcal D}_j$ defined by pulling back the complex  structure on $\bar M$ via the exponential map.

We now consider a finite collection of sets near the divisor
of the following two types:

* A  set   of type (A) admits a local unitary trivialization

\begin{equation} \label{ref:typeA}
\pi_j^{-1}(\Omega) \simeq  \Omega \times \bar{\D}_{z^2}, \ \
\end{equation}

of $\pi_j:  \bar {\mathcal D}_j \rightarrow \Sigma_j$ where $\Omega \subset \Sigma_j$ is a contractible open subset of  $\Sigma_j$ containing no crossings.  We will use coordinates $(z^1,z^2)$ where  $z^1$  is a holomorphic local coordinate in  $\Omega$ and $z^2$ is the standard coordinate of $\bar \D$.
Although the coordinates $(z^1,z^2)$  are holomorphic with respect to the  product complex structure $J_{prod}$ of $\Omega \times \bar \D_{z^2} $, they  are not holomorphic with respect to the complex structure $J_{\bar M}$.  However, by construction, we have that

\[
J_{\bar M}=J_{prod} \ \ \mbox{on} \ \  T_{(z^1,0)} (\Omega \times \bar \D_{z^2}).
\]

* A set of type (B) is as in (<ref>); i.e.

\begin{equation} \label{ref:typeB}
\bar{\D}_{z^1} \times \bar{\D}_{z^2} \subset V_P
\end{equation}

where $V_P$ be an  open set containing a single crossing $P \in \Sigma_i \cap \Sigma_j$ ($i\neq j$).
By the property (i) of the hermitian metric $h$,
$(z^1,z^2)$ are holomorphic coordinates with respect to $J_{\bar M}$.
Furthermore, with the identification
$\bar\D_{z^1} \simeq \bar\D_{z^1} \times \{0\} \subset \Sigma_1$ (resp. $\bar\D_{z^2} \simeq  \{0\} \times \bar\D_{z^2}\subset \Sigma_2$),

\begin{equation} \label{unitarytrivialization}
\pi_j^{-1}(\bar\D_{z^1}) \simeq \bar\D_{z^1} \times \bar\D_{z^2} \ \ \ (\mbox{resp. }\pi_i^{-1}(\bar\D_{z^2}) \simeq \bar\D_{z^1} \times \bar\D_{z^2})
\end{equation}

is a local unitary trivialization of $\pi_j:  \bar{\mathcal D}_j \rightarrow \Sigma_j$ (resp. $\pi_i: \bar {\mathcal D}_i \rightarrow \Sigma_i$).

We will refer to the coordinates $(z^1,z^2)$ discussed in (A) above as the standard product coordinates on a set  $\Omega \times \bar\D_{z^2} $ of type (A).

We will refer to the coordinates $(z^1,z^2)$ discussed in (B) above as the standard product coordinates and the holomorphic coordinates on a set $\bar{\D}_{z^1}^* \times \bar{\D}_{z^2}^*$ of type (B).

Holomorphic coordinates in a set of type (A) is defined later (cf. Definition <ref>).

§ THE POINCARÉ-TYPE METRIC AND ITS ESTIMATES

§.§ Poincaré-type metric

Recall the Poincaré metric

\begin{equation} \label{poincaremetric}