\def\Pij{\nabla_{i}u\,\nabla_{j}u}
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\markboth{\hfil $\infty$-harmonic Functions \hfil EJDE--1995/07}%
{EJDE--1995/07\hfil Nobumitsu Nakauchi \hfil}
\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}\newline
Vol. {\bf 1995}(1995), No. 07, pp. 1-10. Published June 15, 1995.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 }
 \vspace{\bigskipamount} \\
A Remark on $\infty$-harmonic Functions on Riemannian Manifolds
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35J70, 35J20, 26B35. \newline\indent
{\em Key words and phrases:} $\infty$-harmonic function, 
$\infty$-Laplacian. 
\newline\indent
\copyright 1995 Southwest Texas State University  and University of
North Texas.\newline\indent
Submitted: February 20,1995.} }
\date{}
\author{Nobumitsu Nakauchi}
\maketitle

\begin{abstract}
In this note we prove an equality for $\,\infty$-harmonic functions 
on Riemannian manifolds. As a corollary, there is no non-constant 
$\,\infty$-harmonic function on positively (or negatively) curved 
manifolds.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}

\section{Introduction}
In \cite{Aro1}, \cite{Aro2}, Aronsson studied solutions of 
the boundary value problem for the degenerate elliptic equation 
\begin{eqnarray}
\sum_{i,j}\ 
   \nabla_{i}u\,\nabla_{j}u\,\nabla_{i}\nabla_{j}u = 0 
\label{eq:infty-harmonic}
\end{eqnarray} 
in a bounded subdomain $\,D\,$ of $\,{\Bbb R}^{n}\,$
with the boundary condition $\,u\,$ $=\,$ $\varphi\,$ on $\,\partial D\,$. 
His motivation is 
to consider the {\it absolutely minimizing Lipschitz extension problem}, 
which means the problem of finding an extension $\,u\,$ 
in $\,{\rm W}^{1,\infty}(D)\,$  
of any given Lipschitz function $\,\varphi\,$ on $\partial D\,$ 
satisfying the minimization property 
\begin{eqnarray*}
  \|\nabla u\|_{_{\scriptstyle {\rm L}^{\infty}(U) }} 
\ \leq\ 
  \|\nabla v\|_{_{\scriptstyle {\rm L}^{\infty}(U)}} 
\end{eqnarray*}
for any open set $\,U\,$ $\subset\,$ $D\,$ 
and for $\,v\,$ $\in\,$ ${\rm W}^{1,\infty}(U)\,$ 
such that $\,v\,-\,u\,$ $\in\,$ ${\rm W}^{1,\infty}_{0}(U)\,$. 
The equation (\ref{eq:infty-harmonic}) 
is the Euler-Lagrange equation 
of the functional $\,F_{\infty}(u)\,$ $=\,$ 
$\,\|\nabla u\|_{_{\scriptstyle {\rm L}^{\infty}}}\,$ 
in the following sense. 
A $\,p$-harmonic function $\,u\,$ is a solution of 
\begin{eqnarray}
   {\rm div}(\|\nabla u\|^{p\,-\,2}\,\nabla u)\ =\ 0\,, 
\label{eq:p-harmonic}
\end{eqnarray}
which is the Euler-Lagrange equation of 
the functional $\,F_{p}(u)\,$ $=\,$ 
$\,\|\nabla u\|_{_{\scriptstyle {\rm L}^{p}}}\,$. 
Rewrite (\ref{eq:p-harmonic}) to read 
\begin{eqnarray*}
    {1 \over {p\,-\,2} }\,
       \|\nabla u\|^{2}\,\bigtriangleup u 
 \ + \ 
    \sum_{i,j}\ 
       \nabla_{i}u\,\nabla_{j}u\,\nabla_{i}\nabla_{j}u 
 \ = \ 
    0\,.
\end{eqnarray*}
Formally passing to the limit as $\,p\,$ tends to infinity, 
the Euler-Lagrange equation~(\ref{eq:p-harmonic}) 
of the functional $\,F_{p}\,$ converges in some sense 
to the Euler-Lagrange equation (\ref{eq:infty-harmonic}) 
of the functional $\,F_{\infty}\,$. 
From the point of view by Aronsson, 
Jensen \cite{Jen} obtained 
existence and uniqueness results. 
(See also Bhattacharya, DiBenedetto and Manfredi \cite{B-D-M}.)
He proved 
\begin{enumerate}
\item any solution of 
 the absolutely minimizing Lipschitz extension problem
 is a viscosity solution of (\ref{eq:infty-harmonic}), and 
\item there exists a unique viscosity solution of 
 (\ref{eq:infty-harmonic}). 
 Any {\it bounded} such solution is locally Lipschitz continuous. 
\end{enumerate}
Aronsson's pioneering papers \cite{Aro1}, \cite{Aro2} 
investigated classical solutions. 
Recently Evans \cite{Eva} obtained 
a Harnack inequality for classical solutions. 

The absolutely minimizing Lipschitz extension problem 
is considered also on subdomains of {\it Riemannian manifolds} 
$\,M\,$. Then the associated equation corresponding to 
(\ref{eq:infty-harmonic}) is 
\begin{eqnarray}
   g^{ip}g^{jq}
   \nabla_{i}u\,\nabla_{j}u\,\nabla_{p}\nabla_{q}u = 0\,, 
\label{eq:infty-harmonic2}
\end{eqnarray}
where $\,g_{ij}\,$ (resp. $\,g^{ij}\,$) 
is the metric of $\,M\,$ (resp. the inverse matrix of $\,g_{ij}\,$), 
and $\,\nabla\,$ denotes the Levi-Civita connection of $\,g\,$. 
(Throughout this note, we use the Einstein summation convention; 
if the same index appears twice, 
once as a superscript and once as a subscript, 
then the index is summed over all possible values.)
In this note we are concerned with 
$\,{\rm W}^{2,2+\varepsilon}_{loc}\,$-solutions 
of (\ref{eq:infty-harmonic2})\ \,
($\,\varepsilon\,$ $>\,$ $0\,$). 
We say that $\,u\,$ is a 
{\it $\,{\rm W}^{2,2+\varepsilon}_{loc}\,$-solution} 
of (\ref{eq:infty-harmonic2}) in $\,D\,$ 
if the following two conditions hold: 
\begin{enumerate}
\item $u$ is locally Lipschitz continuous, and 
\item $u\in {\rm W}^{2,2+\varepsilon}_{loc}(D)$, and 
 $u$ satisfies (\ref{eq:infty-harmonic2}) a.e., 
\end{enumerate}
where $\,{\rm W}^{2,2+\varepsilon}_{loc}(D)\,$ 
denotes the Sobolev space of functions whose 
second derivatives belong to $\,{\rm L}^{2+\varepsilon}_{loc}(D)\,$. 
On this general setting, 
the curvature of $\,M\,$ provides an obstruction on existence of 
nontrivial $\,{\rm W}^{2,2+\varepsilon}_{loc}$-solutions of 
(\ref{eq:infty-harmonic2})\,. The purpose of this note is 
to prove the following equality. 

\begin{theorem} Let $\,M\,$ be a Riemannian manifold, 
and let $\,D\,$ be a domain in $\,M\,$. 
Let $\,u\,$ be a $\,{\rm W}^{2,2+\varepsilon}_{loc}$-solution 
of the equation {\rm (\ref{eq:infty-harmonic2})} 
in $\,D\,$. Then 
\begin{eqnarray}
g^{ip}g^{jq}g^{kr}g^{ls}
  R_{ijkl}\,
     \nabla_{p}u\,
     \nabla_{q}u\,
     \nabla_{r}u\,
     \nabla_{s}u 
   \ = \ 
     0 \hskip 4ex {\rm a.e.} \hskip 2ex {\rm in} \hskip 2ex D\,, 
\label{eq:main equality}
\end{eqnarray} 
where $\,R_{ijkl}\,$ is the Riemannian curvature tensor of $\,M\,$. 
\end{theorem}

Note that when $\,M\,$ $=\,$ ${\Bbb R}^{n}\,$, 
$\,R_{ijkl}\,$ $\equiv\,$ $0\,$; 
hence the equality (\ref{eq:main equality}) holds automatically 
in this case. 
>From  equality (\ref{eq:main equality}), 
we have $\,\nabla u\,$ $=\,$ $0\,$ 
at any point where the curvature is positive 
(or negative). So we have: 

\begin{corollary} Suppose that the sectional curvature of 
$\,M\,$ is positive {\rm (}or negative{\rm )} 
in $\,D\,$. 
Then any ${\rm W}^{2,2+\varepsilon}_{loc}$-solution  
of {\rm (\ref{eq:infty-harmonic2})} in $\,D\,$ 
is a constant function. 
\end{corollary}

We mention a related fact on harmonic functions. 
Let $\,u\,$ be a harmonic function 
on a Riemannian manifold $\,M\,$. 
Then $\,u\,$ is a constant function if 
one of the following two conditions holds: 
\begin{enumerate}
\item $\,M\,$ is compact (the maximum principle).
\item $\,M\,$ is complete and non-compact, 
 the Ricci curvature of $\,M\,$ is nonnegative, 
 and $\,u\,$ is bounded on $\,M\,$ (Yau \cite{Yau}). 
\end{enumerate}
These results need the assumption that 
$\,u\,$ is globally defined on compact or complete manifolds. 
On the other hand,  the above equality (\ref{eq:main equality}) holds 
when an $\,\infty$-harmonic function $\,u\,$ 
is defined on a {\it subdomain} of $\,M\,$; 
the structure of $\infty$-Laplacian 
gives a restriction on local existence of solutions. 

The author thinks that our theorem holds  without the assumption that 
solutions belong to the class $\,{\rm W}^{2,2+\varepsilon}_{loc}(D)\,$, 
though we use this assumption. 
Then Aronsson's minimization approach of the Lipschitz extention problem 
does not seem to work on any positively (or negatively) curved manifold. 

\section{A Bochner type formula} 
In this section we prove the following formula of Bochner type.

\begin{lemma} Let $\,u\,$ be a $\,{\rm C}^{3}_{loc}$-solution 
of {\rm (\ref{eq:infty-harmonic2})} 
on a subdomain $\,D\,$ of a Riemannian manifold $\,M\,$. 
Then the following equality holds. 
\begin{eqnarray}
 g^{ip}g^{jq} 
\nabla_{i}u\,\nabla_{j}u\,\nabla_{p}\nabla_{q}\,\|\nabla u\|^{2} 
\ + \ 
   { 1 \over 2 }\,\|\nabla \|\nabla u\|^{2}\,\|^{2} & &
\label{eq:Bochner equality} \\
+2\,g^{ip}g^{jq}g^{kr}g^{ls}
    R_{ikjl}\,\nabla_{p}u\,\nabla_{q}u\,\nabla_{r}u\,\nabla_{s}u 
 &=& 0 \hskip 4ex {\rm in\ \,}D\,,\nonumber 
\end{eqnarray}
where 
  $\,\|\nabla u\|^{2}\,=\, g^{ij}\nabla_{i}u \nabla_{j}u\,$ 
and 
  $\,\|\nabla \|\nabla u\|\,\|^{2}\, =\,g^{ij}\nabla_{i}\|\nabla u\|
\nabla_{j}\|\nabla u\|\,$. 
\end{lemma} 

\paragraph{Proof.} Note $\,\nabla g_{ij}\,=\,\nabla g^{ij}\,=\,0\,$, 
since $\,\nabla\,$ is the Levi-Civita connection. 
Applying $\,\nabla_{r}\,$ to both sides of 
(\ref{eq:infty-harmonic2}), we have 
\begin{eqnarray}
    g^{ip}g^{jq}
      \nabla_{i}u\,\nabla_{j}u\,\nabla_{r}\nabla_{p}\nabla_{q}u
 \ + \ 
    2\,g^{ip}g^{jq}
      \nabla_{i}u\,\nabla_{r}\nabla_{j}u\,\nabla_{p}\nabla_{q}u\,
 \ = \ 
   0\,. 
 \label{eq:equality 1} 
\end{eqnarray}
We see that
\begin{eqnarray}
   \nabla_{p}\nabla_{q}\nabla_{r}u
& = &  
   \nabla_{p}\nabla_{r}\nabla_{q}u
\label{eq:equality 2} \\ 
& = &  
   \nabla_{r}\nabla_{p}\nabla_{q}u
\ - \ 
   g^{ls} R_{prqs} \nabla_{l}u 
\hskip 2ex 
    (\mbox{by the Ricci formula})\,. 
\nonumber 
\end{eqnarray}
We get 
\begin{eqnarray}
\lefteqn{ 
    g^{ip}g^{jq}g^{kr} 
       \nabla_{i}u\,\nabla_{j}u\,\nabla_{p}\nabla_{k}u\,
                               \nabla_{q}\nabla_{r}u 
} \label{eq:equality 3} \\ 
&  & 
   \hskip 5ex 
 \ = \ 
    g^{kr} 
     { 1 \over 2 }\,
       \nabla_{k}(g^{ip}\nabla_{i}u \nabla_{p}u)\ 
     { 1 \over 2 }\,
       \nabla_{r}(g^{jq}\nabla_{j}u \nabla_{q}u)
\nonumber \\
&  & 
   \hskip 5ex 
 \ = \ 
   { 1 \over 4 }\,
    g^{kr} 
       \nabla_{k}\|\nabla u\|^{2}\,
       \nabla_{r}\|\nabla u\|^{2} 
\nonumber \\
&  & 
   \hskip 5ex 
 \ = \ 
   { 1 \over 4 }\,\|\nabla \|\nabla u\|^{2}\,\|^{2}\,. 
\nonumber 
\end{eqnarray}
Then we have 
\begin{eqnarray*}
\lefteqn{
    g^{ip}g^{jq} 
          \Pij\,\nabla_{p}\nabla_{q}\,\|\nabla u\|^{2}
} \nonumber \\
&=& 
    g^{ip}g^{jq} 
          \Pij\,\nabla_{p}\nabla_{q}(g^{kr}\nabla_{r}u\nabla_{k}u)
\nonumber \\ 
&=& 
    2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{p}\nabla_{q}\nabla_{r}u\,\nabla_{k}u \nonumber\\
&&+  
    2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{q}\nabla_{r}u\,\nabla_{p}\nabla_{k}u 
\nonumber \\ 
& =& 
    2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{r}\nabla_{p}\nabla_{q}u\,\nabla_{k}u \nonumber\\
&&- 2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,g^{ls}R_{prqs}\,\nabla_{l}u\,\nabla_{k}u 
\nonumber \\ 
&  &  
 +  2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{q}\nabla_{r}u\,\nabla_{p}\nabla_{k}u 
        \hskip 7ex 
                       (\mbox{by (\ref{eq:equality 2})}\,)
\nonumber \\ 
& =& 
  -\,4\,g^{ip}g^{jq}g^{kr} 
        \nabla_{i}u\,\nabla_{k}u\,\nabla_{r}\nabla_{j}u\,
                               \nabla_{p}\nabla_{q}u \nonumber\\
&& -2\,g^{ip}g^{jq}g^{kr}g^{ls} 
       R_{prqs}\,\nabla_{i}u\,\nabla_{j}u\,\nabla_{k}u\,\nabla_{l}u 
\nonumber \\ 
&  & 
 +  
    2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{q}\nabla_{r}u\,\nabla_{p}\nabla_{k}u 
        \hskip 7ex 
                       (\mbox{by (\ref{eq:equality 1})}\,)
\nonumber \\
& = & 
    -\,2\,g^{ip}g^{jq}g^{kr}g^{ls} 
    R_{pqrs}\,\nabla_{i}u\,\nabla_{j}u\,\nabla_{k}u\,\nabla_{l}u\nonumber\\
&&-2\,g^{ip}g^{jq}g^{kr} 
          \Pij\,\nabla_{q}\nabla_{r}u\,\nabla_{p}\nabla_{k}u 
        \hskip 7ex (\mbox{by exchange of indices}\,)
\nonumber \\
& = & 
    -\,2\,g^{ip}g^{jq}g^{kr}g^{ls} 
       R_{pqrs}\,\nabla_{i}u\,\nabla_{j}u\,\nabla_{k}u\,\nabla_{l}u 
 \ - \ 
    { 1 \over 2 }\,\|\nabla \|\nabla u\|^{2}\,\|^{2} 
        \hskip 4ex 
                       (\mbox{by (\ref{eq:equality 3})}\,)\,.
\end{eqnarray*}

\section{Proof of Theorem 1 for $\,{\rm C}^{3}_{loc}$-solutions}
Take any $\,\eta\,$ $\in\,$ ${\rm C}_{0}^{\infty}(D)\,$. 
Then from (\ref{eq:Bochner equality}),
we have 
\begin{eqnarray} 
\lefteqn{ 
\int_{D}\ 
   \eta\,
     g^{ip}g^{jq}
   \nabla_{i}u\,\nabla_{j}u\,\nabla_{p}\nabla_{q}\,\|\nabla u\|^{2} 
\ + \ 
   { 1 \over 2 }\,
\int_{D}\ 
       \|\nabla \|\nabla u\|^{2}\,\|^{2}\,\eta  
} 
\label{eq:equality 4} \\
&  & 
    \hskip 10ex 
 \ + \ 
2\,
 \int_{D}\ 
    \eta\,
      g^{ip}g^{jq}g^{kr}g^{ls} 
    R_{ikjl}\,\nabla_{p}u\,\nabla_{q}u\,\nabla_{r}u\,\nabla_{s}u 
 \ = \ 0\,. 
\nonumber 
\end{eqnarray} 
Note here 
\begin{eqnarray} 
g^{jq}
   \nabla_{j}u\,\nabla_{q}\|\nabla u\|^{2}
& = & 
g^{jq}
   \nabla_{j}u\,\nabla_{q}(g^{ip}\nabla_{i}u\nabla_{p}u)
 \label{eq:equality 5} \\
& = & 
 2\,g^{ip}g^{jq}
       \nabla_{j}u\,\nabla_{i}u\,\nabla_{q}\nabla_{p}u
\ =\ 
0\,. 
\nonumber 
\end{eqnarray} 
Using integration by parts, we get 
\begin{eqnarray} 
\lefteqn{ 
\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u\,\nabla_{j}u\,\nabla_{p}\nabla_{q}\,\|\nabla u\|^{2} 
} \label{eq:equality 6} \\
& = & 
-\int_{D}\ 
  g^{ip}g^{jq}
   \nabla_{p}\eta\, 
   \nabla_{i}u\,\nabla_{j}u\,\nabla_{q}\,\|\nabla u\|^{2}
\nonumber \\
&  & 
 -  
\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{p}\nabla_{i}u\,\nabla_{j}u\,\nabla_{q}\,\|\nabla u\|^{2} 
\nonumber \\
&  & 
 - 
\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u\,\nabla_{p}\nabla_{j}u\,\nabla_{q}\,\|\nabla u\|^{2} 
\nonumber \\
& = & 
-\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u\,\nabla_{p}\nabla_{j}u\,\nabla_{q}\,\|\nabla u\|^{2}
        \hskip 7ex 
                       (\mbox{by (\ref{eq:equality 5})}\,)
\nonumber \\
& = & 
-\int_{D}\ 
   \eta\,
   { 1 \over 2 }
  g^{jq}
   \nabla_{j}(g^{ip}\nabla_{i}u\,\nabla_{p}u)\,\nabla_{q}\,\|\nabla u\|^{2}
\nonumber \\
& = & 
 -\,  
   { 1 \over 2 }
\int_{D}\ 
  \eta\,
  g^{jq}
    \nabla_{j}\|\nabla u\|^{2}\,
       \nabla_{q}\,\|\nabla u\|^{2}
\nonumber \\
& = & 
 -\,  
   { 1 \over 2 }
\int_{D}\ 
    \|\nabla \|\nabla u\|^{2}\,\|^{2}\,\eta\,.
\nonumber 
\end{eqnarray} 
>From (\ref{eq:equality 4}) and (\ref{eq:equality 6}), we have 
\begin{eqnarray} 
\int_{D}\ 
    \eta\,
g^{ip}g^{jq}g^{kr}g^{ls}
      R_{ikjl}\,\nabla_{p}u\,\nabla_{q}u\,\nabla_{r}u\,\nabla_{s}u 
 \ = \ 0\,.
\label{eq:equality 7}
\end{eqnarray} 
Since $\,\eta\,$ is an arbitrary test function 
in $\,{\rm C}_{0}^{\infty}(D)\,$, 
we have  
\begin{eqnarray*} 
g^{ip}g^{jq}g^{kr}g^{ls}
    R_{ikjl}\,\nabla_{p}u\,\nabla_{q}u\,\nabla_{r}u\,\nabla_{s}u 
 \ = \ 0  \hskip 3ex {\rm a.e.\ \,in}\ \,D\,.  \hskip 3ex  \Box 
\end{eqnarray*} 
\section{Proof of Theorem 1} 
In this section we complete our proof of Theorem 1
using an approximation. 
For any ${\rm W}^{2,2+\varepsilon}_{loc}$-solution $\,u\,$ 
of (\ref{eq:infty-harmonic2}), 
we take an approximating sequence 
$\,\{\,u^{(\nu)}\,\}_{\nu\,=\,1}^{\infty}\,$ $\subset\,$ 
${\rm C}^{3}_{loc}(D)\,$ 
such that 
for any compact set $\,K\,$ in $\,D\,$, 
\begin{enumerate}
\item $\,\varphi^{(\nu)}\,:\,$ $=\,$ $u^{(\nu)}\,$ $-\,$ $u\,$ 
 approaches zero ${\rm in}\ $ ${\rm W}^{2,2+\varepsilon}_{loc}(D)\,$ 
 as $\,\nu\,$ tends to infinity, and 
\item  the Lipschitz constants of $\,u^{(\nu)}\,$ 
 $(\nu\,$ $=\,$ $1,\,$ $2,\,...)\,$ 
 are uniformly bounded on $\,K\,:\ $ 
 hence $\,\|\nabla u^{(\nu)}\|_{_{\scriptstyle {\rm L}^{\infty}(K)}}\,$ 
 and 
 $\,\|\nabla \varphi^{(\nu)}\|_{_{\scriptstyle {\rm L}^{\infty}(K)}}\,$ 
 $(\nu\,$ $=\,$ $1,\,$ $2,\,...)\,$ 
 are uniformly bounded on $\,K\,$. 
\end{enumerate}
Since $\,u\,$ $=\,$ $u^{(\nu)}\,$ $-\,$ $\varphi^{(\nu)}\,$ 
satisfies (\ref{eq:infty-harmonic2}), we have 
$$
   g^{ip}g^{jq}
    \nabla_{i}(u^{(\nu)}\,-\,\varphi^{(\nu)})\,
    \nabla_{j}(u^{(\nu)}\,-\,\varphi^{(\nu)})\,
    \nabla_{p}\nabla_{q}(u^{(\nu)}\,-\,\varphi^{(\nu)}) = 0
$$
i.e.,
\begin{equation}
  g^{ip}g^{jq}
       \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                               \nabla_{p}\nabla_{q}u^{(\nu)}
 + F( \varphi^{(\nu)},\,u^{(\nu)} ) = 0
\label{eq:approximating infty-harmonic}
\end{equation}
where 
\begin{eqnarray*}
\lefteqn{ 
   F( \varphi^{(\nu)},\,u^{(\nu)} )
} \nonumber \\ 
  & = & 
   - 
   g^{ip}g^{jq}
    \nabla_{i}\varphi^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                                 \nabla_{p}\nabla_{q}u^{(\nu)}
  -  
   g^{ip}g^{jq}
     \nabla_{i}u^{(\nu)}\,\nabla_{j}\varphi^{(\nu)}\,
                                 \nabla_{p}\nabla_{q}u^{(\nu)}
\nonumber \\ 
 &  & 
  - 
   g^{ip}g^{jq}
     \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                  \nabla_{p}\nabla_{q}\varphi^{(\nu)}
 + 
   g^{ip}g^{jq}
     \nabla_{i}\varphi^{(\nu)}\,\nabla_{j}\varphi^{(\nu)}\,
                                            \nabla_{p}\nabla_{q}u^{(\nu)}
\nonumber \\ 
 &  & 
 + 
   g^{ip}g^{jq}
     \nabla_{i}u^{(\nu)}\,\nabla_{j}\varphi^{(\nu)}\,
                  \nabla_{p}\nabla_{q}\varphi^{(\nu)}
 + 
   g^{ip}g^{jq}
     \nabla_{i}\varphi^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                                          \nabla_{p}\nabla_{q}\varphi^{(\nu)}
\nonumber \\ 
 &  & 
 - 
   g^{ip}g^{jq}
     \nabla_{i}\varphi^{(\nu)}\,\nabla_{j}\varphi^{(\nu)}\,
                  \nabla_{p}\nabla_{q}\varphi^{(\nu)}\,.
\nonumber 
\end{eqnarray*}
Let $\,\psi\,$ $\in\,$ ${\rm W}^{1,1}_{0}(D)\,$. 
Multiply by $\,-\,\nabla_{r}\psi\,$ both sides 
of (\ref{eq:approximating infty-harmonic})
and use integration by parts, then we have 
\begin{eqnarray*}
  \int_{D}\ 
   \psi\,
   g^{ip}g^{jq}
      \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
      \nabla_{r}\nabla_{p}\nabla_{q}u^{(\nu)}
 & & \\  
+ 2 \int_{D}\  \psi\,
   g^{ip}g^{jq}
      \nabla_{i}u^{(\nu)}\,\nabla_{r}\nabla_{j}u^{(\nu)}\,
              \nabla_{p}\nabla_{q}u^{(\nu)} 
& & \\ 
- \int_{M}\ 
  F( \varphi^{(\nu )},\,u^{(\nu)} )\,
    \nabla_{r}\psi 
& = & 0\,. \end{eqnarray*}
Let $\,\psi\,$ $=\,$ $\eta\,g^{kr}\nabla_{k}u^{(\nu)}\,$ 
and sum them up with respect to $\,r\,$. Then we get 
\begin{eqnarray}
 \int_{D}\ 
   \eta\,
   g^{ip}g^{jq}g^{kr}
      \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
      \nabla_{r}\nabla_{p}\nabla_{q}u^{(\nu)}\,\nabla_{k}u^{(\nu)}
& & \label{eq:equality 8} \\  
+ 2 \int_{D}\ 
   \eta\,
   g^{ip}g^{jq}g^{kr}
      \nabla_{i}u^{(\nu)}\,
        \nabla_{k}u^{(\nu)}\,
           \nabla_{r}\nabla_{j}u^{(\nu)}\, 
             \nabla_{p}\nabla_{q}u^{(\nu)} 
& &\nonumber \\ 
- \int_{M}\ 
  F(\varphi^{(\nu )},\,u^{(\nu)})\,
    g^{kr}\nabla_{r}(\eta\,\nabla_{k}u^{(\nu)}) 
& = & 0 
\nonumber \end{eqnarray}
We see 
\begin{eqnarray*}
\lefteqn{
  \int_{D}\ 
    \eta\, 
      g^{ip}g^{jq} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                       \nabla_{p}\nabla_{q}\,\|\nabla u^{(\nu)}\|^{2}
} \nonumber \\
& = & 
  \int_{D}\ 
    \eta\, 
    g^{ip}g^{jq} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                 \nabla_{p}\nabla_{q}(g^{kr}\nabla_{r}u\nabla_{k}u)
\nonumber \\ 
& =& 
    2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                \nabla_{p}\nabla_{q}\nabla_{r}u^{(\nu)}\,\nabla_{k}u^{(\nu)} 
\nonumber \\ 
&  & 
    + 2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                \nabla_{q}\nabla_{r}u^{(\nu)}\,\nabla_{p}\nabla_{k}u^{(\nu)} 
\nonumber \\ 
& =& 
    2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                \nabla_{r}\nabla_{p}\nabla_{q}u^{(\nu)}\,\nabla_{k}u^{(\nu)} 
\nonumber \\ 
&  & 
  -2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                g^{ls}R_{prqs}\,\nabla_{l}u^{(\nu)}\,\nabla_{k}u^{(\nu)} 
\nonumber \\ 
&  & 
  +2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                \nabla_{q}\nabla_{r}u^{(\nu)}\,\nabla_{p}\nabla_{k}u^{(\nu)} 
        \hskip 7ex 
                       (\mbox {by (\ref{eq:equality 2}})\,)
\nonumber \\ 
& =& 
 -4 \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{k}u^{(\nu)}\,
                  \nabla_{r}\nabla_{j}u^{(\nu)}\,
                               \nabla_{p}\nabla_{q}u^{(\nu)} 
\nonumber \\ 
&  & 
   + 2 
  \int_{D}\ 
        F(\varphi^{(\nu)},\,u^{(\nu)})\,
           g^{kr}
              \nabla_{k}(\eta\,\nabla_{r}u^{(\nu)}) 
\nonumber \\ 
&  & 
   - 2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr}g^{ls} 
         R_{prqs}\,\nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                         \nabla_{k}u^{(\nu)}\,\nabla_{l}u^{(\nu)} 
\nonumber \\ 
&  & 
    + 2 
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
              \nabla_{q}\nabla_{r}u^{(\nu)}\,\nabla_{p}\nabla_{k}u^{(\nu)} 
        \hskip 7ex 
                       (\mbox{by (\ref{eq:equality 8})}\,)
\nonumber \\
& =& 
 -2  \int_{D}\ 
    \eta\, 
      g^{ip}g^{jq}g^{kr}g^{ls} 
         R_{pqrs}\,\nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                         \nabla_{k}u^{(\nu)}\,\nabla_{l}u^{(\nu)} 
\nonumber \\ 
&  & 
 - 2
  \int_{D}\ 
    \eta\, 
       g^{ip}g^{jq}g^{kr} 
          \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
            \nabla_{q}\nabla_{r}u^{(\nu)}\,\nabla_{p}\nabla_{k}u^{(\nu)} 
        \hskip 7ex 
\nonumber \\
&  &   
   +2 
  \int_{D}\ 
        F(\varphi^{(\nu)},\,u^{(\nu)})\,
           g^{kr}
              \nabla_{k}(\eta\,\nabla_{r}u^{(\nu)}) 
 \hskip 7ex 
                       (\mbox {by exchange of indices})
\nonumber \\
& =& 
  -2 \int_{D}\ 
    \eta\, 
      g^{ip}g^{jq}g^{kr}g^{ls} 
         R_{pqrs}\,\nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                    \nabla_{k}u^{(\nu)}\,\nabla_{l}u^{(\nu)}\nonumber \\ 
& &-{ 1 \over 2 } \int_{D}\ \|\nabla \|\nabla u^{(\nu)}\|^{2}\,\|^{2} 
   + 2 
  \int_{D}\ 
        F(\varphi^{(\nu)},\,u^{(\nu)})\,
           g^{kr}
              \nabla_{k}(\eta\,\nabla_{r}u^{(\nu)}) 
 \hskip 4ex 
                       (\mbox{by (\ref{eq:equality 3})}\,)\,.
\nonumber 
\end{eqnarray*}
Therefore we obtain 
an integral form of the Bochner equality for $\,u^{(\nu )}\,$: 
\begin{eqnarray}
\int_{M}\ 
 \eta\,
  g^{ip}g^{jq} 
   \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
   \nabla_{p}\nabla_{q}\,\|\nabla u^{(\nu)}\|^{2} 
\ + \ 
   { 1 \over 2 } 
\int_{M}\ 
  \|\nabla \|\nabla u^{(\nu)}\|^{2}\,\|^{2}\, 
     \eta  & & \nonumber \\
-2 \int_{M}\ 
 F(\varphi^{(\nu )},\,u^{(\nu)})\,
  g^{kr}
   \nabla_{k}(\eta\,\nabla_{r}u^{(\nu)}) 
 & & \label{eq:Bochner^(nu)} \\
+2 \int_{M}\ 
     \eta\, 
  g^{ip}g^{jq}g^{kr}g^{ls} 
     R_{ikjl}\,\nabla_{p}u^{(\nu)}\,\nabla_{q}u^{(\nu)}\,
              \nabla_{r}u^{(\nu)}\,\nabla_{s}u^{(\nu)} 
&=&  0\,. \nonumber 
\end{eqnarray}
for any $\,\eta\,$ $\in\,$ ${\rm C}^{\infty}_{0}(D)\,$. 
Note here 
\begin{eqnarray} 
g^{jq}
   \nabla_{j}u^{(\nu)}\,\nabla_{q}\|\nabla u^{(\nu)}\|^{2}
& = & 
g^{jq}
   \nabla_{j}u^{(\nu)}\,
          \nabla_{q}(g^{ip}\nabla_{i}u^{(\nu)}\nabla_{p}u^{(\nu)})
 \label{eq:equality 9} \\
& = & 
 2\,g^{ip}g^{jq}
       \nabla_{j}u^{(\nu)}\,\nabla_{i}u^{(\nu)}\,
                        \nabla_{q}\nabla_{p}u^{(\nu)}
 \nonumber \\
& = & 
 -\,2\,F(\varphi^{(\nu)},\,u^{(\nu)})
 \hskip 7ex 
                  (\mbox{by (\ref{eq:approximating infty-harmonic})}\,)\,.
\nonumber 
\end{eqnarray} 
Then using integration by parts, we get 
\begin{eqnarray} 
\lefteqn{ 
\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
             \nabla_{p}\nabla_{q}\,\|\nabla u^{(\nu)}\|^{2} 
} \label{eq:equality 10} \\
& =& 
-\int_{D}\ 
  g^{ip}g^{jq}
   \nabla_{p}\eta\, 
   \nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                        \nabla_{q}\,\|\nabla u^{(\nu)}\|^{2}
\nonumber \\
&  & 
-\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{p}\nabla_{i}u^{(\nu)}\,\nabla_{j}u^{(\nu)}\,
                                   \nabla_{q}\,\|\nabla u^{(\nu)}\|^{2} 
\nonumber \\
&  & 
-\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u^{(\nu)}\,\nabla_{p}\nabla_{j}u^{(\nu)}\,
                                   \nabla_{q}\,\|\nabla u^{(\nu)}\|^{2} 
\nonumber \\
& =& 
2\int_{D}\ 
  g^{ip}\nabla_{p}\eta\,\nabla_{i}u^{(\nu)}\,
          F(\varphi^{(\nu)},\,u^{(\nu)}) 
=2\int_{D}\ 
   \eta\,
  g^{ip}
   \nabla_{p}\nabla_{i}u^{(\nu)}\,
                       F(\varphi^{(\nu)},\,u^{(\nu)}) 
\nonumber \\
&  & 
-\int_{D}\ 
   \eta\,
  g^{ip}g^{jq}
   \nabla_{i}u^{(\nu)}\,\nabla_{p}\nabla_{j}u^{(\nu)}\,
                                   \nabla_{q}\,\|\nabla u^{(\nu)}\|^{2}
 \hskip 7ex 
                       (\mbox{by (\ref{eq:equality 9})}\,)
\nonumber \\
&=& 
2\int_{D}\ 
  g^{ip}\nabla_{i}u^{(\nu)}\,\nabla_{p}\eta\,
          F(\varphi^{(\nu)},\,u^{(\nu)}) 
\ + \ 
  2 
\int_{D}\ 
   \eta\,
    \bigtriangleup u^{(\nu)}\,
                       F(\varphi^{(\nu)},\,u^{(\nu)}) 
\nonumber \\
&  & 
 -{ 1 \over 2 } 
\int_{D}\ 
   \|\nabla \|\nabla u^{(\nu)}\|\,\|^{2}\,\eta\,, 
\nonumber \end{eqnarray}
because
$$  g^{ip}\nabla_{i}u^{(\nu)}\,\nabla_{p}\nabla_{j}u^{(\nu)}
       \,=\, 
          { 1 \over 2 } \,
            \nabla_{j}(g^{ip}\nabla_{i}u^{(\nu)}\,\nabla_{p}u^{(\nu)})
       \,=\, 
         {\displaystyle { 1 \over 2 } }\,
                  \nabla_{j}\|\nabla u^{(\nu)}\|^{2}\,. 
$$
Then by (\ref{eq:Bochner^(nu)}) and (\ref{eq:equality 10}), 
we obtain 
\begin{eqnarray}
\lefteqn{
2 \int_{D}\ 
\eta\,
g^{ip}g^{jq}g^{kr}g^{ls} 
    R_{ijkl}\nabla_{p}u^{(\nu)}\,\nabla_{q}u^{(\nu)}\,
                                 \nabla_{r}u^{(\nu)}\,\nabla_{s}u^{(\nu)} 
}\nonumber \\ 
& =&
-2 \int_{D}\ 
  g^{ip}\nabla_{i}u^{(\nu)}\,\nabla_{p}\eta\,
                                 F(\varphi^{(\nu)},\,u^{(\nu)})
\label{eq:equality 11}  \\ 
&  &
-2 \int_{D}\ 
    \eta\,
          \bigtriangleup u^{(\nu)}\,
                  F(\varphi^{(\nu)},\,u^{(\nu)}) 
\nonumber \\ 
&  &
+2 \int_{D}\ 
     g^{kr}\nabla_{r}(\eta\,\nabla_{k}u^{(\nu)})\,
                            F(\varphi^{(\nu )},\,u^{(\nu)})\,.
\nonumber 
\end{eqnarray}
Since $\,\|\nabla u^{(\nu)}\|\,$ is bounded uniformly on $\,K\,$, 
we get 
\begin{eqnarray*}
\lefteqn{ 
|\,{\rm the\ right\ hand\ side\ of\ \,} (\ref{eq:equality 11})\,| 
} \nonumber \\ 
& \leq & 
       C \int_{K}\ 
            \|F(\varphi^{(\nu)},\,u^{(\nu)})\| 
  + \ 
       C \int_{K}\ 
            \|\nabla \nabla u^{(\nu)}\|\, 
            \|F(\varphi^{(\nu)},\,u^{(\nu)})\| 
\nonumber \\ 
& \leq & 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla u^{(\nu)}\|
 + \ 
       C \int_{K}\ 
            \|\nabla \nabla \varphi^{(\nu)}\| 
\nonumber \\ 
& & 
  + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|^{2}\,
            \|\nabla \nabla u^{(\nu)}\|
 + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla \varphi^{(\nu)}\|
\nonumber \\ 
& & 
  + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|^{2}\,
            \|\nabla \nabla \varphi^{(\nu)}\|
  + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla u^{(\nu)}\|^{2}
\nonumber \\ 
& & 
  + \ 
       C \int_{K}\ 
            \|\nabla \nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla u^{(\nu)}\|
  + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|^{2}\,
            \|\nabla \nabla u^{(\nu)}\|^{2}
\nonumber \\ 
& & 
 + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla u^{(\nu)}\|
\nonumber \\ 
& & + \ 
       C \int_{K}\ 
            \|\nabla \varphi^{(\nu)}\|^{2}\,
            \|\nabla \nabla \varphi^{(\nu)}\|\,
            \|\nabla \nabla u^{(\nu)}\|
\nonumber \\ 
& = \ : & 
  {\rm I}_{1} 
 \ + \ 
  {\rm I}_{2} 
 \ + \ 
  {\rm I}_{3} 
 \ + \ 
  {\rm I}_{4} 
 \ + \ 
  {\rm I}_{5} 
 \ + \ 
  {\rm I}_{6} 
 \ + \ 
  {\rm I}_{7} 
 \ + \ 
  {\rm I}_{8} 
 \ + \ 
  {\rm I}_{9} 
 \ + \ 
  {\rm I}_{10}\,. 
\nonumber 
\end{eqnarray*}
Since $\,\varphi^{(\nu)}\,$ converges to 
in $\,{\rm W}^{2,2+\varepsilon}(K)\,$ 
as $\,\nu\,$ tends to infinity, 
$\,\nabla \varphi^{(\nu)}\,$ 
and 
$\,\nabla \nabla \varphi^{(\nu)}\,$ 
approaches zero 
in $\,{\rm L}^{2}(K)\,$. 
Then 
\begin{eqnarray*}
 {\rm I}_{1}
\ \leq\ 
 C\,
   \left\{
     \int_{K}\ 
          \|\nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla u^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{2}
\ \leq\ 
 C\,
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          1^{2}
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{3}
\ \leq\ 
 C\,
   \sup_{K}\ \|\nabla \varphi^{(\nu)}\|\,
   \left\{
     \int_{K}\ 
          \|\nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla u^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{4}
\ \leq\ 
 C\,
   \left\{
     \int_{K}\ 
          \|\nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{5}
\ \leq\ 
 C\,
   \sup_{K}\ \|\nabla \varphi^{(\nu)}\|\,
   \left\{
     \int_{K}\ 
          \|\nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/ 2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{7}
\ \leq\ 
 C\,
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla u^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{9}
\ \leq\ 
 C\,
   \sup_{K}\ \|\nabla \varphi^{(\nu)}\|\,
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla u^{(\nu)}\|^{2} 
   \right\}^{1 \over 2}
\ \rightarrow\ 
  0\,, 
\end{eqnarray*}
\begin{eqnarray*}
 {\rm I}_{10}
\ \leq\ 
 C\,
   \sup_{K}\ \|\nabla \varphi^{(\nu)}\|^{2}\,
   \left\{
     \int_{K}\ 
          \|\nabla \nabla \varphi^{(\nu)}\|^{2} 
   \right\}^{1/2}
   \left\{
     \int_{K}\ 
          \|\nabla \nabla u^{(\nu)}\|^{2} 
   \right\}^{1/2}
\ \rightarrow\ 
  0\,. 
\end{eqnarray*}
Furthermore, 
since $\,\varphi^{(\nu)}\,$ converges to zero 
in $\,{\rm W}^{2,2+\varepsilon}(K)\,$, 
we have 
$$ {\rm I}_{6}
\leq C
   \sup_{K}\|\nabla \varphi^{(\nu)}\|^{1-\varepsilon}
   \left\{
     \int_{K} 
          \|\nabla \varphi^{(\nu)}\|^{2+\varepsilon} 
   \right\}^{\varepsilon/(2+\varepsilon) }
   \left\{
     \int_{K} 
          \|\nabla \nabla u^{(\nu)}\|^{2+\varepsilon} 
   \right\}^{ 2/(2+\varepsilon) }
 \rightarrow   0\,, 
$$ 
$$
 {\rm I}_{8}\leq 
  C \sup_{K}\|\nabla \varphi^{(\nu)}\|^{2-\varepsilon}
   \left\{
     \int_{K}
          \|\nabla \varphi^{(\nu)}\|^{2+\varepsilon} 
   \right\}^{\varepsilon /(2+\varepsilon) }
   \left\{
     \int_{K} 
          \|\nabla \nabla u^{(\nu)}\|^{2+\varepsilon} 
   \right\}^{ 2/(2+\varepsilon)}
 \rightarrow 0\,.
$$ 
Thus the right hand side of (\ref{eq:equality 11}) 
converges to zero as $\,\nu\,$ tends to infinity. 
Then, letting $\,\nu\,$ go to infinity in (\ref{eq:equality 11}), 
we have (\ref{eq:equality 7}). 
This completes the proof. 
             \hspace{5ex}    $\Box$ 

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\end{thebibliography}

\noindent
{\sc Nobumitsu Nakauchi \newline
Department of Mathematics \newline
Faculty of Science \newline 
Yamaguchi University \newline
Yamaguchi 753, Japan \newline} 
E-mail: nakauchi\@@ccy.yamaguchi-u.ac.jp

\newpage

\section*{December 11, 1996  Addendum}

In this article, Theorem 1 follows from general properties 
of the Riemannian curvature tensor, and Corollary 1 is incorrect. 
The Bochner formula does not seem to work in this situation. 
\medskip

Lemma 1 can be used in proving the following Liouville theorem
for ${\rm C}^{3}$-solutions.
\medskip

\noindent{\bf Theorem A.} {\it Let $M$ be a complete noncompact 
Riemannian manifold of nonnegative (sectional) curvature. 
Let $u$ be a bounded $\infty$-harmonic function of 
${\rm C}^{3}$-class on $M$. 
Then $u$ is a constant function. }
\medskip

The curvature assumption in Theorem A is necessary only for applying the 
Hessian comparison theorem in the proof (Here we use the operator 
${\rm Q}^{ij} = g^{ip}g^{jq}\nabla_{p}\nabla_{q}$). 
Theorem A also follows from arguments in [Cheng, S.Y.,
{\em Liouville theorem for harmonic maps}, 
 Proc. Symp. Pure Math. 36(1980), 147-151]. See also the article 
[Hong, N.C., {\it Liouville theorems for exponentially harmonic 
functions on Riemannian manifolds}, Manuscripta Math. 77(1992), 41-46]. 

\bigskip

Sincerely yours,

Nobumitsu Nakauchi

\end{document}