# Liouville theorem for $V$-harmonic maps under non-negative $(m,V)$-Ricci
curvature for non-positive $m$

Kazuhiro Kuwae, Songzi Li, Xiang-Dong Li and Yohei Sakurai Department of
Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
(kuwae@fukuoka-u.ac.jp). Supported in part by JSPS Grant-in-Aid for Scientific
Research (KAKENHI) 22H04942 and by funds (No.: 197004 and 215001) from the
Central Research Institute of Fukuoka University.School of Mathematics, Renmin
University of China, 59, Zhongguancun Da Jie, Beijing, 100872, China
(sli@ruc.edu.cn). Supported by NSFC No. 11901569 and by fund (No.:2018030249)
from Renmin University of China.Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, 55, Zhongguancun East Road, Beijing, 100190,
China (xdli@amt.ac.cn), and School of Mathematical Sciences, University of
Chinese Academy of Sciences, Beijing, 100049, China. Supported by National Key
R$\&$D Program of China (No. 2020YF0712700), NSFC No. 11771430, No. 11688101,
and Key Laboratory RCSDS, CAS, No. 2008DP173182.Department of Mathematics,
Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama-City, Saitama,
338-8570, Japan (ysakurai@rimath.saitama-u.ac.jp). Supported in part by JSPS
Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric
Analysis for Materials Design” 17H06460).

###### Abstract

Let $V$ be a $C^{1}$-vector field on an $n$-dimensional complete Riemannian
manifold $(M,g)$. We prove a Liouville theorem for $V$-harmonic maps
satisfying various growth conditions from complete Riemannian manifolds with
non-negative $(m,V)$-Ricci curvature for
$m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into Cartan-Hadamard
manifolds, which extends Cheng’s Liouville theorem proved in [11] for
sublinear growth harmonic maps from complete Riemannian manifolds with non-
negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a
Liouville theorem for $V$-harmonic maps from complete Riemannian manifolds
with non-negative $(m,V)$-Ricci curvature for
$m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into regular geodesic balls
of Riemannian manifolds with positive upper sectional curvature bound, which
extends the results of Hildebrandt-Jost-Wideman [17] and Choi [12]. Our
probabilistic proof of Liouville theorem for several growth $V$-harmonic maps
into Hadamard manifolds enhances an incomplete argument in [59]. Our results
extend the results due to Chen-Jost-Qiu [9] and Qiu [53] in the case of
$m=+\infty$ on the Liouville theorem for bounded $V$-harmonic maps from
complete Riemannian manifolds with non-negative $(\infty,V)$-Ricci curvature
into regular geodesic balls of Riemannian manifolds with positive sectional
curvature upper bound. Finally, we establish a connection between the
Liouville property of $V$-harmonic maps and the recurrence property of
$\Delta_{V}$-diffusion processes on manifolds. Our results are new even in the
case $V=\nabla f$ for $f\in C^{2}(M)$.

Keywords: $(m,V)$-Ricci curvature, $V$-Laplacian, $\Delta_{V}$-diffusion
process, radial process, sublinear growth, $V$-harmonic function, $V$-harmonic
maps.

Mathematics Subject Classification (2020): Primary 53C21, 53C43; Secondary
53C20, 58J65, 60J60.

##  1 Statement of Main Theorem

Let $(M,g,f)$ denote an $n$-dimensional weighted Riemannian manifold, namely,
$(M,g)$ is an $n$-dimensional complete Riemannian manifold, and $f\in
C^{\infty}(M)$. Let ${\sf d}:={\sf d}_{g}$ denote the Riemannian distance and
let $B_{r}(p):=\\{x\in M\mid{\sf d}(p,x)<r\\}$ denote the open ball with
center $p\in M$ and radius $r>0$. Throughout this paper, we always assume
$n\geq 2$. Let $\Delta_{f}:=\Delta-\langle\nabla f,\nabla\cdot\rangle$ be a
diffusion operator called _$f$ -Laplacian_ or _Witten Laplacian_ , which is
symmetric with respect to the measure
$\mu(\mathrm{d}x):=e^{-f(x)}\nu_{g}(\mathrm{d}x)$, where $\nu_{g}$ is the
volume measure of $(M,g)$. For $m\in\,[\,-\infty,\,+\infty\,]$, the
$m$-weighted Ricci curvature for $(M,g,f)$ or $\Delta_{f}$ is defined as
following ([3], [38], [34]):

$\displaystyle\operatorname{Ric}_{f}^{m}:=\operatorname{Ric}_{g}+{\rm
Hess}\,f-\frac{\mathrm{d}f\otimes\mathrm{d}f}{m-n},$

where when $m=\pm\infty$, the last term is interpreted as the limit $0$, and
when $m=n$, we only consider a constant function $f$, and set
$\operatorname{Ric}_{f}^{n}:=\operatorname{Ric}_{g}$.

Under the classical curvature condition

$\displaystyle\operatorname{Ric}_{f}^{m}\geq Kg$

for $K\in\mathbb{R}$ and $m\in[\,n,\,+\infty\,]$, comparison geometry has been
developed by many authors (see e.g. [3], [4], [13], [14], [34], [40], [51],
[52], [62], and references therein). In this case, the parameter $m$ is
interpreted as an upper bound for dimension $n$. In recent years, comparison
geometry for the complementary case of $m\in]-\infty,n\,[$ has begun to be
investigated (see e.g., [28], [29], [30], [39], [42], [43], [44], [45], [46],
[49], [55], [63], [64]). In these papers, $m$ is considered as a parameter
less than $1$ or non-positive, and it is called effective dimension. In this
case, $m$ might not seem an upper bound for dimension $n$ if we keep the usual
order for real line $\mathbb{R}$. In view of the coincidence
${\operatorname{Ric}}_{V}^{\infty}={\operatorname{Ric}}_{V}^{-\infty}$, if we
change the order like $n\leq m_{1}\leq+\infty\text{\;``$=$''}\\!-\infty\leq
m_{2}\leq 1$, then the parameter $m_{2}$ can be understood as an upper bound
for the dimension $n$. Wylie-Yeroshkin [64] introduced a curvature condition.

$\displaystyle\operatorname{Ric}_{f}^{1}\geq(n-1)\kappa e^{-\frac{4f}{n-1}}g$

for $\kappa\in\mathbb{R}$ in view of the study of weighted affine connection,
and obtained a Laplacian comparison theorem, Bonnet-Myers type diameter
comparison theorem, Bishop-Gromov type volume comparison theorem, and rigidity
results for the equality cases. The first and the third named authors [28]
have provided a generalized condition

$\displaystyle\operatorname{Ric}_{f}^{m}\geq(n-m)\kappa e^{-\frac{4f}{n-m}}g$

with $m\in\,]-\infty,\,1\,]$, and extended the comparison geometric results in
[64].

In [60], Yau developed the method of gradient estimate and proved the strong
or bounded Liouville theorem on complete Riemannian manifolds. More precisely,
let $(M,g)$ be a complete Riemannian manifold with Ricci curvature bounded
from below by $-K$, i.e., ${\operatorname{Ric}}_{g}\geq-Kg$, where $K\geq 0$
is a constant, and $u$ be a harmonic function (i.e. a solution to $\Delta
u=0$) which is bounded from below, then the following gradient estimate holds

$\displaystyle|\nabla u|\leq\sqrt{(n-1)K}\left(u-\inf_{M}u\right).$ (1.1)

Taking $K=0$, the gradient estimate (1.1) implies the strong or bounded
Liouville theorem, namely, every non-negative or bounded harmonic function on
a complete Riemannian manifold with non-negative Ricci curvature must be
constant. In [34], the third named author extended Yau’s gradient estimate to
$f$-harmonic functions, namely, let $(M,g,f)$ be a complete Riemannian
manifold with ${\operatorname{Ric}}_{f}^{m}\geq-Kg$, where $K\geq 0$ and
$m\in[\,n\,,+\infty\,[$ are two constants, and $u$ be a solution to
$\Delta_{f}u=0$ which is bounded from below, then

$\displaystyle|\nabla u|\leq\sqrt{(m-1)K}\left(u-\inf_{M}u\right).$ (1.2)

Taking $K=0$, the gradient estimate (1.2) implies an extension of Yau’s strong
or bounded Liouville theorem, namely, every non-negative or bounded
$\Delta_{f}$-harmonic function on a complete Riemannian manifold with non-
negative $m$-dimensional Bakry-Emery Ricci curvature (i.e.,
$\operatorname{Ric}_{f}^{m}\geq 0$) must be constant.

In the case $m=+\infty$, Brighton [7] proved the following Liouville Theorem
for bounded $\Delta_{f}$-harmonic functions as a corollary of gradient
estimate for $\Delta_{f}$-harmonic functions under non-negative lower bounds
of ${\operatorname{Ric}}_{f}^{\infty}$.

######  Theorem 1.1 ([7, Corollary 2]).

Suppose ${\operatorname{Ric}}_{f}^{\infty}\geq 0$. If $u$ is a bounded
$\Delta_{f}$-harmonic function, then $u$ is constant.

Theorem 1.1 can be derived from the Hamilton type Harnack inequality. In [35],
the third named author proved the Hamilton type dimension free Harnack
inequality for bounded positive solutions to the heat equation of
$\Delta_{f}$. More precisely, let $u$ be a positive solution to the heat
equation $\partial_{t}u=\Delta_{f}u$ with $u\leq A$ for some constant $A>0$.
Suppose that $\operatorname{Ric}_{f}^{\infty}\geq-Kg$ for some constant $K\geq
0$. Then the following Harnack inequality holds

$\displaystyle|\nabla\log u|^{2}\leq{2K\over 1-e^{-2Kt}}\log(A/u),$ (1.3)

which improves Hamilton type dimension free Harnack inequality for bounded
positive solutions to the heat equation of $\Delta_{f}$, i.e.,

$\displaystyle|\nabla\log u|^{2}\leq\left({1\over t}+K\right)\log(A/u).$ (1.4)

In particular, when $K=0$, and letting $t\rightarrow\infty$, we can derived
that all bounded $\Delta_{f}$-harmonic functions must be constant. That is to
say, we can recapture Brighton’s result from Harnack inequalities $(\ref{Li-
Harnack})$ and $(\ref{Hamilton-Harnack})$ for the heat equation of
$\Delta_{f}$. In [31], the second and third named authors proved the Li-Yau-
Hamilton type Harnack inequality for positive solutions to the heat equation
$\partial_{t}u=\Delta_{f}u$ on complete Riemannian manifolds with
${\operatorname{Ric}}_{f}^{m}\geq-(m-1)Kg$ with $m\in[\,n,\,+\infty\,[$. In
[32], the second and third named authors proved $(\ref{Li-Harnack})$ for
bounded positive solutions to the heat equation $\partial_{t}u=\Delta_{f}u$ on
complete $(K,\infty)$-super Ricci flow. See also [32, 33] for the Li-Yau type
and the Li-Yau-Hamilton type Harnack inequalities for positive solution to the
heat equation $\partial_{t}u=\Delta_{f}u$ on Ricci flow and variant of super
Ricci flows. Note that Hua [19] also provided another proof of Theorem 1.1.

On the other hand, S. Y. Cheng [11] generalized Yau’s gradient estimate to
harmonic maps. Recall that a smooth map $u:M\rightarrow N$ between two
Riemannian manifolds $(M,g)$ and $(N,g)$ is said to be a _harmonic map_ if
$\tau(u)=0$, where $\tau(u)$ is the tension field of $u$ and
$\mathrm{d}u:TM\to TN$ defined by $(\mathrm{d}u)_{x}:T_{x}M\to T_{u(x)}N$ is
the differential of $u$, see Section $3$ below or [56] and related references.

######  Theorem 1.2 ([11, Theorem]).

Let $(M,g)$ be a complete Riemannian manifold with
$\operatorname{Ric}_{g}\geq-Kg$, $K\geq 0$, and $N$ a Cartan-Hadamard
manifold, i.e., a complete connected and simply connected Riemannian manifold
having non-positive sectional curvature. Let $u:M\to N$ be a harmonic map.
Assume $u(B_{a}(p))\subset B_{b}(o)$ for some $p\in M$ and $o\in N$ and some
$a,b>0$. Then we have

$\displaystyle\sup_{B_{a/2}(p)}|\mathrm{d}u|^{2}\leq
C_{n}\cdot\frac{b^{4}}{a^{4}}\max\left\\{\frac{Ka^{4}}{\beta},\frac{a^{2}(1+Ka)}{\beta},\frac{a^{2}b^{2}}{\beta^{2}}\right\\},$
(1.5)

where $C_{n}$ is a constant depending only on $n$ and
$\beta=b^{2}-\sup_{B_{a}(p)}{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{2}(u,o)$. Here
${\sf d}_{N}$ denotes the Riemannian distance of $(N,h)$.

As a corollary, S. Y. Cheng [11] established a Liouville theorem for sublinear
growth harmonic maps:

######  Corollary 1.3 ([11, Liouville theorem for harmonic maps]).

Let $(M,g)$ be a complete Riemannian manifold with $\operatorname{Ric}_{g}\geq
0$, and $N$ a Cartan-Hadamard manifold, i.e., a complete connected and simply
connected Riemannian manifold having non-positive sectional curvature. Let
$u:M\to N$ be a harmonic map. If $u$ satisfies the sublinear growth

$\varlimsup_{a\to\infty}\frac{\sup_{x\in B_{a}(p)}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(x),o)}{a}=0$

for some $p\in M$ and $o\in N$. Then $u$ is a constant map.

Hildbrandt-Jost-Widemann [17] and Choi [12] further extended Cheng’s work [11]
as the following theorem:

######  Theorem 1.4 (cf. [12, (11)]).

Let $(M,g)$ be a complete smooth Riemannian manifold with
$\operatorname{Ric}_{g}\geq-Kg$, $K\geq 0$, and $N$ a complete smooth
Riemannian manifold having sectional curvature ${\rm Sect}_{N}\leq\kappa$,
$\kappa>0$. Let $u:M\to N$ be a harmonic map. Assume $u(M)\subset B_{b}(o)$
lies inside the cut locus of $o\in N$ and some $b<\pi/(2\sqrt{\kappa})$. Then

$\displaystyle\sup_{B_{a/2}(p)}|\mathrm{d}u|^{2}$ $\displaystyle\leq
4\max\left\\{\frac{2^{10}}{C_{2}^{2}a^{2}\beta^{2}},\frac{2K}{C_{2}}+\frac{8C_{1}(1+a)}{C_{2}a^{2}}+\frac{64}{C_{2}a^{2}}\right\\}.$
(1.6)

for some constants $C_{1},C_{2}>0$ independent of $n,\kappa$ and $b$. Here
$\beta:=b-\sup_{B_{a}(p)}\varphi\circ u$ and
$\varphi(y)=1-\cos\sqrt{\kappa}{\color[rgb]{0,0,0}{{\sf d}}}_{N}(y,o)$.

A geodesic ball $B_{R}(o)\subset N$ with ${\rm Sect}_{N}\leq\kappa$ is said to
be a _regular geodesic ball_ if $B_{R}(o)\cap{\rm Cut}(o)=\emptyset$ and
$R<\pi/2\sqrt{\kappa^{+}}$ with $\kappa^{+}:=\max\\{\kappa,0\\}$.

######  Corollary 1.5 ([12, Theorem]).

Let $(M,g)$ be a complete Riemannian manifold with $\operatorname{Ric}_{g}\geq
0$, and $N$ a complete Riemannian manifold with sectional curvature ${\rm
Sect}_{N}\leq\kappa$, $\kappa>0$. Let $u:M\to N$ be a harmonic map whose range
is contained in a regular geodesic ball $B_{b}(o)$. Then $u$ is a constant
map.

The purpose of this paper is to extend Cheng’s Liouville theorem for sublinear
growth $V$-harmonic maps $u:M\to N$ between Riemannian manifolds $(M,g)$ and
$(N,h)$ under the relaxed condition ${\operatorname{Ric}}_{V}^{m}\geq 0$ for
$m\in[-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ and $(N,h)$ is a Cartan-
Hadamard manifold, and to extend Hildbrandt-Jost-Widemann and Choi’s Liouville
theorem for $V$-harmonic maps $u:M\to N$ between Riemannian manifolds $(M,g)$
and $(N,h)$ under reasonable condition. Here, not only for the symmetric
diffusion operator $\Delta_{f}$, we work on the more general setting of a
Riemannian manifold $(M,g,V)$ with a $C^{1}$-vector field $V$ which is not
necessarily to be the gradient $V=\nabla f$ for some $f\in C^{2}(M)$. Such a
Riemannian manifold is equipped with canonical $V$-Laplacian and $V$-Ricci
curvature. More precisely, the _non-symmetric $V$-Laplacian_ is defined by

$\Delta_{V}:=\Delta-\langle V,\nabla\cdot\rangle$

and the $m$-dimensional _$V$ -Ricci curvature_, or the _$(m,V)$ -Ricci
curvature_, is defined as follows:

${\operatorname{Ric}}_{V}^{m}:={\operatorname{Ric}}_{g}+\frac{1}{2}\mathcal{L}_{V}g-\frac{V^{*}\otimes
V^{*}}{m-n},$

where $m\in\,[\,-\infty,+\infty]$ and set $V=0$ under $m=n$,
$\mathcal{L}_{V}g(X,Y):=\langle\nabla_{X}V,Y\rangle+\langle\nabla_{Y}V,X\rangle$
is the Lie derivative of $g$ with respect to $V$, and $V^{*}$ denotes its dual
$1$-form. We would like to point out that, in the case where
$m\in[\,n,\,+\infty\,[$, $\operatorname{Ric}_{V}^{m}$ has been introduced in
previous works of Qian [52] and Bakry-Qian [4]. In [34] (Section 8.7, p.
1347), for $V=-B$, the third named author used the notation
$\operatorname{Ric}_{m,n}^{S}(L)=\operatorname{Ric}_{g}-\nabla^{S}B-{B\otimes
B\over m-n}$ (instead of $\operatorname{Ric}_{V}^{m}$) to denote the
$m$-dimensional Bakry-Emery Ricci curvature for non-symmetric elliptic
diffusion operator $L=\Delta+B$, where $\nabla^{S}B(\xi,\eta)={1\over
2}\\{\langle\nabla_{\xi}B,\eta\rangle+\langle\nabla_{\eta}B,\xi\rangle\\}$.
Moreover, it has been pointed out in [34] that it would be very possible to
extend the various Liouville theorems and Harnack inequalities to non-
symmetric diffusion operators. Indeed, as the Laplacian comparison theorem
holds for $L=\Delta+B$ under the condition
$\operatorname{Ric}_{m,n}^{S}(L)\geq Kg$ for $K\in\mathbb{R}$ and
$m\in[\,n,\,+\infty\,[$, see Bakry-Qian [4], the proofs of the strong
Liouville theorem and the Li-Yau Harnack inequality for symmetric elliptic
diffusion operators in [34] can be extended to non-symmetric diffusion
operators. For the detail, see [36].

In the case of $V=\nabla f$, it coincides with ${\operatorname{Ric}}_{f}^{m}$.
For $m_{1}\in\,]-\infty,1\,]$ and $m_{2}\in\,[\,n,+\infty\,[$ we have the
following order

$\displaystyle{\operatorname{Ric}}_{V}^{1}\geq{\operatorname{Ric}}_{V}^{m_{1}}\geq{\operatorname{Ric}}_{V}^{-\infty}={\operatorname{Ric}}_{V}^{\infty}\geq{\operatorname{Ric}}_{V}^{m_{2}}.$
(1.7)

So the condition ${\operatorname{Ric}}_{V}^{m}\geq Kg$ is weaker than
${\operatorname{Ric}}_{V}^{\infty}\geq Kg$ for $m\in\,]\,-\infty,\,1\,]$ and
$K\in\mathbb{R}$.

We now describe more detail of our setting. Let $(M,g)$ and $(N,h)$ be two
complete smooth Riemaniann manifolds. A smooth map $u:M\rightarrow N$ is said
to be _$V$ -harmonic_ if $\tau_{V}(u):=\tau(u)-(\mathrm{d}u)(V)=0$. When
$V=\nabla f$ for $f\in C^{2}(M)$, any $V$-harmonic map is called the _$f$
-harmonic map_. The notion of $V$-harmonic map covers the various notions of
harmonicity for maps, e.g., Hermitian harmonic maps, Weyl harmonic maps and
affine harmonic maps (see [10] for these notions on harmonicity). In [9,
Theorem 2], Chen-Jost-Qiu proved the bounded Liouville property for
$V$-harmonic maps from complete Riemannian manifolds with
${\operatorname{Ric}}_{V}^{\infty}\geq 0$ into regular geodesic ball with
${\rm Sect}_{N}\leq\kappa$ for $\kappa\geq 0$ under the sublinear growth
condition for $V$. Later, Qiu [53, Theorem 2] also proved the bounded
Liouville property as in [9, Theorem 2] without assuming the sublinear growth
condition for $V$. We will prove not only the bounded Liouville property for
$V$-harmonic maps from complete Riemannian manifolds into regular geodesic
ball but also Liouville property for sublinear growth $V$-harmonic map into
Cartan-Hadamard manifold $(N,h)$ under ${\operatorname{Ric}}_{V}^{m}\geq 0$
with $m\in[\,-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$.

Hereafter, we always fix points $p\in M$ and $o\in N$, and also
$m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,[$.

######  Definition 1.6.

A map $u:M\to N$ is said to be of _sublinear growth_ if

$\displaystyle\varlimsup_{a\to\infty}m_{u}(a)/a=0,$

for some/any $o\in N$, where $m_{u}(a):=\sup_{x\in
B_{a}(p)}{\color[rgb]{0,0,0}{{\sf d}}}_{N}(u(x),o)$, i.e.,
$m_{u}(a)=o(a)\;\;(a\to\infty)$.

We prepare several growth conditions for a smooth map $u:M\to N$:

  1. (G1)

$u$ is of sublinear growth, i.e., $m_{u}(a)=o(a),\quad(a\to\infty)$.

  2. (G2)

$m_{u}(a)=o(\sqrt{a}),\quad(a\to\infty)$.

  3. (G3)

$m_{u}(a)=o(\sqrt{\log a}),\quad(a\to\infty)$.

It is easy to see that (G1) (resp. (G2)) is weaker than (G2) (resp. (G3)).
Note here that any bounded map $u:M\to N$ satisfies (G3), i.e., the
boundedness of $u$ is stronger than any (G$i$).

Fix a point $p\in M$. We set $v(r):=\sup_{x\in B_{r}(p)}|V|_{x}$ for $r>0$ and
$v(0):=|V|_{p}$. Then $v$ is a non-negative continuous function on
$[0,+\infty[$ and $|V|_{x}\leq v(r_{p}(x))$ holds for $x\in M$. Here
$r_{p}(x):={\sf d}(x,p)$ is the radial function from a referene point $p\in
M$. Now we consider the following conditions:

  1. (A1)

$v\in L^{1}([0,+\infty[)$, or $\langle
V,\dot{\gamma}_{t}\rangle_{\gamma_{t}}\geq 0$ a.e. $t$ holds for any unit
speed geodesic $\gamma$ starting from $p$;

  2. (A2)

There exists $D\geq 1$ such that
$\int_{0}^{r}v(s)\mathrm{d}s\leq\frac{n-m}{4}\log D(1+r)$;

  3. (A3)

There exists $D\geq 1$ such that
$\int_{0}^{r}v(s)\mathrm{d}s\leq\frac{n-m}{4}\log D(1+r^{2})$.

The condition (A2) (resp. (A3)) with $D=1$ is satisfied if
$v(r)\leq\frac{n-m}{4(1+r)}$ (resp. $v(r)\leq\frac{(n-m)r}{2(1+r^{2})}$) for
$r\geq 0$.

Our first main result extends Cheng’s Liouville theorem to sublinear growth
$V$-harmonic maps on manifolds with ${\operatorname{Ric}}_{V}^{m}\geq 0$ and
${\rm Sect}_{N}\leq 0$.

######  Theorem 1.7.

Let $(M,g,V)$ be a complete Riemaniann manifold with
${\operatorname{Ric}}_{V}^{m}\geq 0$ for some $C^{1}$-vector field $V$ and
some constant $m\in\,[-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$, and $(N,h)$ is
a Cartan-Hadamard manifold, i.e., $N$ is a complete connected and simply
connected Riemannian manifold with ${\rm Sect}_{N}\leq 0$. Let $u:M\to N$ be a
$V$-harmonic map. For each $i=1,2,3$, we suppose (A$i$) for $p\in M$ and
(G$i$) for $u$ provided $m\in[-\infty,\,0\,]\,\cup\,\\{+\infty\\}$, otherwise,
we only suppose (G1). Then $u$ is a constant map.

For each $i=1,2,3$, we consider the following condition
${\bf(B\text{\boldmath$i$})}$ on the shape of Laplacian comparison.

  1. (B$i$)

There exists $D\geq 1$ such that $r_{p}(x)\Delta_{V}r_{p}(x)\leq
D(1+r_{p}(x)^{i-1})$ for $x\notin{\rm Cut}(p)$.

Note here that the radial function $x\mapsto r_{p}(x)$ is secondly
differentiable at $x\notin{\rm Cut}(p)$. The condition (A$i$) implies (B$i$)
for each $i=1,2,3$ (see Theorem 2.3 below).

######  Remark 1.8.

The statement of Theorem 1.7 remains valid by replacing (A$i$) with (B$i$) for
each $i=1,2,3$ (see the proof of Theorem 2.3).

Our second main result extends Hildbrandt-Jost-Widemann and Choi’s Liouville
theorem as follows:

######  Theorem 1.9.

Let $(M,g,V)$ be a complete Riemaniann manifold with
${\operatorname{Ric}}_{V}^{m}\geq 0$ for some $C^{1}$-vector field $V$ and
some constant $m\in\,[-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$, and $N$ be a
complete Riemannian manifold with ${\rm Sect}_{N}\leq\kappa$ with $\kappa>0$.
Suppose (A3) for $p\in M$ when $m\in[-\infty,\,0\,]\,\cup\,\\{+\infty\\}$. Let
$u:M\to N$ be a $V$-harmonic map whose range is contained in a regular
geodesic ball $B_{R}(o)$. Then $u$ is a constant map.

######  Corollary 1.10 (Chen-Jost-Qiu [9, Theorem 2] and Qiu [53, Theorem 2]).

Let $(M,g,V)$ be a complete Riemaniann manifold with
${\operatorname{Ric}}_{V}^{\infty}\geq 0$ for some $C^{1}$-vector field $V$,
and $N$ be a complete Riemannian manifold with ${\rm Sect}_{N}\leq\kappa$ with
$\kappa>0$. Let $u:M\to N$ be a $V$-harmonic map whose range is contained in a
regular geodesic ball $B_{R}(o)$. Then $u$ is a constant map.

######  Corollary 1.11.

Let $(M,g,V)$ be a complete Riemaniann manifold with
${\operatorname{Ric}}_{V}^{m}\geq 0$ for some $C^{1}$-vector field $V$ and
some constant $m\in\,]-\infty,\,0\,]\,\cup\,[\,n,+\infty\,[$, and $N$ be a
complete Riemannian manifold with ${\rm Sect}_{N}\leq\kappa$ with $\kappa>0$.
Suppose that $V$ is bounded. Let $u:M\to N$ be a $V$-harmonic map whose range
is contained in a regular geodesic ball $B_{R}(o)$. Then $u$ is a constant
map.

We now discuss the connection between the Liouville property of harmonic maps
and the recurrence property of diffusion processes on manifolds. In the case
of harmonic functions, it is well-known that all bounded subharmonic functions
must be constant if and only if the Brownian motion on manifold is recurrent.
This remains true for general $V$-subharmonic functions and the
$\Delta_{V}$-diffusion process on manifolds. To help the reader who is not
familiar in this result, let us recall some notions and facts in potential
theory. Let ${\bf X}^{V}=(\Omega,X_{t},{\mathbb{P}}_{x})$ be the
$\Delta_{V}$-diffusion process associated to the $V$-Laplacian $\Delta_{V}$
and $\zeta:=\inf\\{t>0\mid X_{t}\notin M\\}$ its life time (see Section 3
below). A Borel function $g$ on $M$ is said to be _excessive_ if $g$ is non-
negative on $M$ and ${\mathbb{E}}_{x}[g(X_{t}):t<\zeta]\leq g(x)$ for any
$t>0$ and $\lim_{t\to 0}{\mathbb{E}}_{x}[g(X_{t}):t<\zeta]=g(x)$ and $x\in M$.
${\bf X}^{V}$ is said to be _recurrent_ if any excessive function is constant
(see [16, (2.4) Proposition]). Our third main result is the following:

######  Theorem 1.12.

Let $(M,g)$, $(N,h)$ be two complete Riemannian manifolds. Suppose that the
$\Delta_{V}$-diffusion process ${\bf X}^{V}$ is recurrent and ${\rm
Sect}_{N}\leq\kappa$ with $\kappa>0$. Let $u:M\to N$ be a $V$-harmonic map
whose range is contained in a regular geodesic ball $B_{R}(o)$. Then $u$ is a
constant map.

######  Remark 1.13.

We give the following remark.

  1. (i)

As far as we know, Theorem 1.7, Theorem 1.9 and Theorem 1.12 are new in the
literature, even in the symmetric case $V=\nabla f$ for some $f\in C^{2}(M)$.

  2. (ii)

Under ${\rm Ric}_{V}^{m}\geq 0$, the condition (A3) implies (B3) (see Theorem
2.3). Moreover, the statement of Theorem 1.9 remains valid by replacing (A3)
with (B3), because Theorem 1.9 is based on Theorem 1.7.

  3. (iii)

The assertion of Theorem 1.12 under $V=0$ is well-known. Indeed, if the
Brownian motion ${\bf X}^{V}$ is recurrent, then every bounded harmonic
function is constant. So the bounded Liouville property into regular geodesic
balls of Riemannian manifolds having positive upper sectional curvature bounds
holds by Kendall [22, 23, 24].

  4. (iv)

Theorem 1.12 is an extension of [9, Theorem 5], which states that if $(M,g)$
is a compact Riemannian manifold without boundary and $(N,h)$ is a complete
Riemannian manifold with sectional curvature bounded above by a positive
constant $\kappa$, then every $V$-harmonic map $u:M\rightarrow N$ with range
in a regular ball must be a constant map. We would like to point out that the
requirement of the compactness of $M$ is indeed for the validity of the
following strong maximum principle: If a smooth $\Delta_{V}$-harmonic function
$u$ attains a maximum $u(x_{0})$ at $x_{0}\in M$, then $u\equiv u(x_{0})$.
This is nothing but the bounded Liouville property for $\Delta_{V}$-harmonic
functions. The proof of Theorem 1.9 shows that the bounded Liouville property
holds for $\Delta_{V}$-harmonic functions on complete Riemannian manifolds
with recurrent $\Delta_{V}$-diffusion process. In view of this, Theorem 1.12
gives the connection between the bounded Liouville property for $V$-harmonic
maps with range in regular geodesic ball of complete Riemannian manifolds with
positive upper sectional curvature bounds and the recurrence property of
$\Delta_{V}$-diffusion process on the complete source Riemannian manifolds.

As a corollary of Theorem 1.12, the recurrence of $\Delta_{f}$-diffusion
process on complete Riemannian surface proved in [28, Theorem 2.6] under the
lower boundedness of $f$ and ${\operatorname{Ric}}_{f}^{1}\geq 0$ leads us to
the following:

######  Corollary 1.14.

Let $(M,g)$ be a complete Riemannian surface with a lower bounded function
$f\in C^{2}(M)$, and $(N,h)$ a complete smooth Riemannian manifold. Assume
${\operatorname{Ric}}_{f}^{1}\geq 0$ and ${\rm Sect}_{N}\leq\kappa$ with
$\kappa>0$. Let $u:M\to N$ be an $f$-harmonic map whose range is contained in
a regular geodesic ball $B_{R}(o)$. Then $u$ is a constant map.

######  Example 1.15.

Let $M=\mathbb{R}^{n}$ be an $n$-dimensional Euclidean space. Consider a
smooth lower bounded function $f(x):=(n-m)\log(n-m+|x|^{2})$ for
$m\in\,]-\infty,\,1\,]$. Then we see that for $x,v\in\mathbb{R}^{n}$,

$\displaystyle{\operatorname{Ric}}_{f}^{\infty}(v,v)(x)$
$\displaystyle=\frac{2(n-m)}{(n-m+|x|^{2})^{2}}\left((n-m+|x|^{2})|v|^{2}-\langle
x,v\rangle^{2}\right),$ $\displaystyle{\operatorname{Ric}}_{f}^{m}(v,v)(x)$
$\displaystyle=\frac{2(n-m)}{(n-m+|x|^{2})^{2}}\left((n-m+|x|^{2})|v|^{2}-\langle
x,v\rangle^{2}\right)\geq 0.$

When $f(x):=n\log(n+|x|^{2})$, then ${\operatorname{Ric}}_{f}^{0}(v,v)(x)\geq
0$, hence one can apply Theorem 1.7 for any $f$-harmonic maps having (G1)
condition into Hadamard manifolds. In this case, the condition
$\text{\bf(A1)}^{*}$ below holds, because $f$ is radially increasing and
rotationally symmetric around $0$. Since $\text{\bf(A1)}^{*}$ implies
$\text{\bf(A3)}^{*}$, hence ${\bf(B3)}$ holds (see the prof of Theorem 2.3
below), Theorem 1.9 can be applied.

######  Example 1.16.

Let $M=\mathbb{R}^{n}$ be an $n$-dimensional Euclidean space. Consider a
smooth lower bounded function $f(x):=(n-1)\log(n-1+|x|^{2})$. Then we see that
for $x,v\in\mathbb{R}^{n}$

$\displaystyle{\operatorname{Ric}}_{f}^{1}(v,v)(x)$
$\displaystyle=\frac{2(n-1)}{(n-1+|x|^{2})^{2}}\left((n-1+|x|^{2})|v|^{2}-\langle
x,v\rangle^{2}\right)\geq 0.$

Then by [28, Theorem 2.6], the $\Delta_{f}$-diffusion process is recurrent
under the lower boundedness of $f$, $n=2$ and
${\operatorname{Ric}}_{f}^{1}\geq 0$. In this case, any $f$-harmonic map into
regular geodesic ball in a complete Riemannian manifold $(N,h)$ with ${\rm
Sect}_{N}\leq\kappa$ ($\kappa>0$) is a constant map by Theorem 1.12.

When $N=\mathbb{R}$, Theorem 1.7 extends Theorem 1.1, because of the order
${\operatorname{Ric}}_{V}^{m}\geq{\operatorname{Ric}}_{V}^{\infty}$ for
$m\in]-\infty,\,0\,]$. Moreover, it extends the classical S. Y. Cheng’s
Liouville Theorem [11] for sublinear growth harmonic maps between Riemannian
manifolds. Indeed, based on the Laplacian comparison theorem due to Bakry and
Qian [4], S. Y. Cheng’s Liouville Theorem for sublinear growth $f$-harmonic
functions has been extended in a non-submitted paper in the 2007 Habilitation
Thesis of the third named author to symmetric diffusion operator $\Delta_{f}$
on complete Riemannian manifolds with $\operatorname{Ric}_{f}^{m}\geq 0$ for
$m\in[\,n,\,+\infty\,[$. The content of [11] is extended to the setting of
metric measure space (see Hua-Kell-Xia [20] and Kuwada-Kuwae [27]) for
sublinear growth harmonic functions. But our result is not covered by them.
Theorem 1.9 also extends the classical results on the bounded Liouville
property for harmonic maps from complete smooth Riemannian manifolds with non-
negative Ricci curvature into regular geodesic balls of complete smooth
Riemannian manifolds with positive upper sectional curvature bounds proved by
Hildebrandt-Jost-Wideman [17] and Choi [12]. Moreover, Theorems 1.7 and 1.9
extend the bounded Liouville properties for $V$-harmonic maps proved in [9,
53, Theorem 2] under ${\rm Ric}_{V}^{\infty}\geq 0$. Note that there was no
such Liouville theorem for sublinear growth $V$-harmonic maps into Cartan-
Hadamard manifolds in the past literature. In Theorem 1.7, we do not assume
the boundedness condition for the $V$-harmonic map as in [9, 53, Theorem 2]
when $(N,h)$ is a Cartan-Hadamard manifold. As in Remark 1.13(iii), Theorem
1.12 is an extension of [9, Theorem 5]. Note that Corollary 1.14 can not be
deduced from Theorem 1.9.

Our proof of Theorem 1.7 based on the method by [59] is stochastic, but the
proof in [59, §2 Lower bound] is questionable, because in [59] there is no
explicit proof for the sublinear growth condition for the norm $|\mathrm{d}u|$
of differential map $\mathrm{d}u:TM\to TN$ for harmonic map $u:M\to N$, and
the method of [59, §3 Upper bound] does not work well by using the Laplacian
comparison theorem under ${\operatorname{Ric}}_{V}^{m}\geq 0$ with
$m\in]-\infty,\,1\,]$ (see Section 2 for such Laplacian comparison theorems).
To overcome this difficulty, in addition to the non-negative bound for
$(m,V)$-Ricci curvature, we consider conditions (A$i$) for $V$-Laplacian
acting on the radial function (see Section 2). Moreover, we prove that for
each $i=1,2,3$ the growth condition (G$i$) for the $V$-harmonic map $u$
implies the same growth condition (G$i$) for $|\mathrm{d}u|$ under (B3) (see
Lemma 5.1). This fills the incompleteness of the arguments in [59, §2 Lower
bound]. The proof of Theorem 1.9 is based on Theorem 1.7 and the inheritance
of the bounded Liouville property for functions to maps firstly shown by W.
Kendall (see [22, 23, 24]). Finally, we emphasize that our results obtained by
stochastic method are new, that is, there were no such results obtained by
geometric analysis in the past literature.

##  2 Laplacian comparison theorem

Let $V$ be a $C^{1}$-vector field on a complete Riemannian manifold $(M,g)$.
In general, $V$ is not necessarily a gradient type like $V=\nabla f$. However,
we can make a function $f_{V}$ which plays a similar role like $f$ provided
$V=\nabla f$. Define

$\displaystyle V_{\gamma}(r):$ $\displaystyle=\int_{0}^{r}\langle
V_{\gamma_{s}},\dot{\gamma}_{s}\rangle\mathrm{d}s$

for a unit speed geodesic $\gamma:[\,0,\,+\infty\,[\to M$, and

$\displaystyle f_{V}(x):$
$\displaystyle=\inf\left\\{\left.\int_{0}^{r_{p}(x)}\langle
V_{\gamma_{s}},\dot{\gamma}_{s}\rangle\mathrm{d}s\;\right|\left.\begin{array}[]{ll}&\gamma:\text{unit
speed geodesic}\\\
&\gamma_{0}=p,\gamma_{r_{p}(x)}=x\end{array}\right.\right\\}.$ (2.3)

Note that $V_{\gamma}$ depends on the choice of unit speed geodesic $\gamma$,
and $f_{V}(x)$ depends on $p$ with $f_{V}(p)=0$ and it is well-defined for
$x\in M$. It is easy to see that
$f_{V}(x)=\int_{0}^{r_{p}(x)}Vr_{p}(\gamma_{s})\mathrm{d}s$ for $x\notin{\rm
Cut}(p)$, where $\gamma$ is the unique unit speed geodesic with $\gamma_{0}=p$
and $\gamma_{r_{p}(x)}=x$. Moreover, for $x\notin{\rm Cut}(p)$,
$f_{V}(x)=V_{\gamma}(r_{p}(x))$ for the unique unit speed geodesic $\gamma$
with $\gamma_{0}=p$ and $\gamma_{r_{p}(x)}=x$. Hence
$f_{V}(\gamma_{t})=V_{\gamma}(t)$ for any unit speed geodesic $\gamma$ with
$\gamma_{0}=p$ and $\gamma_{t}\notin{\rm Cut}(p)$. When $V=\nabla f$ is a
gradient vector field for some $f\in C^{2}(M)$, then one can see

$\displaystyle V_{\gamma}(t)$ $\displaystyle=\int_{0}^{t}\langle\nabla
f,\dot{\gamma}_{s}\rangle\mathrm{d}s=\int_{0}^{t}\frac{\mathrm{d}}{\mathrm{d}s}f(\gamma_{s})\mathrm{d}s=f(\gamma_{t})-f(\gamma_{0}).$

Throughout this section, we fix a point $p\in M$ and a constant $C=C_{p}>0$,
which may depend on $p$. For $x\in M$, we define

$\displaystyle
s_{p}(x):=\inf\left\\{\left.C_{p}\int_{0}^{r_{p}(x)}e^{-\frac{2V_{\gamma}(t)}{n-m}}\mathrm{d}t\;\right|\left.\begin{array}[]{ll}&\gamma:\text{unit
speed geodesic}\\\
&\gamma_{0}=p,\gamma_{r_{p}(x)}=x\end{array}\right.\right\\}.$

When $V=\nabla f$ with $f\in C^{2}(M)$ and
$C_{p}=\exp\left(-\frac{2f(p)}{n-m}\right)$, then

$\displaystyle
s_{p}(x)=\inf\left\\{\left.\int_{0}^{r_{p}(x)}e^{-\frac{2f(\gamma_{t})}{n-m}}\mathrm{d}t\;\right|\left.\begin{array}[]{ll}&\gamma:\text{unit
speed geodesic}\\\
&\gamma_{0}=p,\gamma_{r_{p}(x)}=x\end{array}\right.\right\\}.$

Since $(M,g)$ is complete, $s_{p}(x)$ is finite and well-defined from the
basic properties of Riemannian geodesics.

The following theorem is a special case of [30, Theorem 2.5] (see also [29,
Theorem 2.3]).

######  Theorem 2.1 (Laplacian Comparison Theorem).

Fix a point $p\in M$, $\kappa\in\mathbb{R}$ and $m\in]-\infty,\,1\,]$. Suppose

$\displaystyle{\operatorname{Ric}}_{V}^{m}\geq(n-m)\kappa
e^{-\frac{4f_{V}(x)}{n-m}}C_{p}^{2}g$ (2.4)

for $x\notin{\rm Cut}(p)\cup\\{p\\}$. Then

$\displaystyle(\Delta_{V}r_{p})(x)\leq(n-m)\cot_{\kappa}(s_{p}(x))e^{-\frac{2f_{V}(x)}{n-m}}C_{p}$
(2.5)

holds for such $x\notin({\rm Cut}(p)\cup\\{p\\})$. When $V=\nabla f$ for some
$f\in C^{2}(M)$ and $C_{p}=\exp\left(-\frac{2f(p)}{n-m}\right)$, then

$\displaystyle{\operatorname{Ric}}_{V}^{m}\geq(n-m)\kappa
e^{-\frac{4f(x)}{n-m}}g$ (2.6)

for $m\in]-\infty,\,1\,]$ yields that

$\displaystyle(\Delta_{V}r_{p})(x)\leq(n-m)\cot_{\kappa}(s_{p}(x))e^{-\frac{2f(x)}{n-m}}$
(2.7)

holds for $x\notin({\rm Cut}(p)\cup\\{p\\})$. Here
$\cot_{\kappa}(r)=\frac{\mathfrak{s}_{\kappa}^{\prime}(r)}{\mathfrak{s}_{\kappa}(r)}$
with

$\displaystyle\mathfrak{s}_{\kappa}(r):=\left\\{\begin{array}[]{cc}\frac{\sin(\sqrt{\kappa}r)}{\sqrt{\kappa}}&\kappa>0,\\\
r&\kappa=0,\\\
\frac{\sinh(\sqrt{-\kappa}r)}{\sqrt{-\kappa}}&\kappa<0.\end{array}\right.$

Recall that $f_{V}$ depends on $p$. We define
$\overline{f}_{V}(r):=\sup_{B_{r}(p)}f_{V}$ and
$\underline{f}_{V}(r):=\inf_{B_{r}(p)}f_{V}$. Then
$\overline{f}_{V}(r_{p}(x))$ and $\underline{f}_{V}(r_{p}(x))$ are
rotationally symmetric functions around $p$. We now consider the following
conditions, which are weaker than the conditions (A1), (A2) and (A3),
respectively:

  1. (A1)∗

$f_{V}$ is bounded, or $t\mapsto f_{V}(\gamma_{t})$ is increasing for any unit
speed geodesic $\gamma$ with $\gamma_{0}=p$,

  2. (A2)∗

There exists $D\geq 1$ such that
$\overline{f}_{V}(r)-\underline{f}_{V}(r)\leq\frac{n-m}{2}\log D(1+r)$.

  3. (A3)∗

There exists $D\geq 1$ such that
$\overline{f}_{V}(r)-\underline{f}_{V}(r)\leq\frac{n-m}{2}\log D(1+r^{2})$.

Indeed, in view of $|f_{V}(x)|\leq\int_{0}^{r_{p}(x)}v(s)\mathrm{d}s$,
$f_{V}(\gamma_{t})=\int_{0}^{t}\langle
V,\dot{\gamma}_{s}\rangle_{\gamma_{s}}\mathrm{d}s$ for unit speed geodesic
$\gamma$ joining $\gamma_{0}=p$ and $\gamma_{r_{p}(x)}=x\notin{\rm Cut}(p)$,
and $-\int_{0}^{r}v(s)\mathrm{d}s\leq\underline{f}_{V}(r)\leq
0\leq\overline{f}_{V}(r)\leq\int_{0}^{r}v(s)\mathrm{d}s$, we see that (A1),
(resp. (A2), (A3)) implies $\text{\bf(A1)}^{*}$, (resp. $\text{\bf(A2)}^{*}$,
$\text{\bf(A3)}^{*}$).

Recall the condition (B$i$) for each $i=1,2,3$.

######  Lemma 2.2.

The condition (B3) is equivalent to the following condition
$\text{\bf(B3)}^{*}$.

  1. $\text{\bf(B3)}^{*}$

There exist $a\in]\,0,\,{\color[rgb]{0,0,0}{{\sf d}}}(p,{\rm Cut}(p))\,[$ and
$D\geq 1$ such that $\Delta_{V}r_{p}(x)\leq D(1+r_{p}(x))$ for all $x\in
M\setminus({\rm Cut}(p)\cup B_{a}(p))$.

###### Proof.

It is easy to see that (B3) implies $\text{\bf(B3)}^{*}$ by taking any small
$a\in]\,0,\,{\color[rgb]{0,0,0}{{\sf d}}}(p,{\rm Cut}(p))\,[$. Since
$\lim_{x\to p}r_{p}(x)\Delta_{V}r_{p}(x)=n-1$, taking a sufficiently small
$a\in]\,0,\,{\color[rgb]{0,0,0}{{\sf d}}}(p,{\rm Cut}(p))\,[$, we can obtain
the converse implication $\text{\bf(B3)}^{*}$$\Longrightarrow$(B3). ∎

Theorem 2.1 tells us the following:

######  Theorem 2.3.

Suppose ${\rm Ric}_{V}^{m}\geq 0$ for $m\in]-\infty,\,1\,]$. If (A1) (resp.
(A2), (A3)) holds, then the condition (B1) (resp. (B2), (B3)) holds.

###### Proof.

Note first that (A1) (resp. (A2), (A3)) implies $\text{\bf(A1)}^{*}$ (resp.
$\text{\bf(A2)}^{*}$, $\text{\bf(A3)}^{*}$). Take $x\notin{\rm Cut}(p)$. Then
there exists a unique unit speed geodesic $\gamma_{t}$ with $\gamma_{0}=p$ and
$\gamma_{r_{p}(x)}=x$ such that

$\displaystyle
s_{p}(x)=C_{p}\int_{0}^{r_{p}(x)}\exp\left(-\frac{2f_{V}(\gamma_{t})}{n-m}\right)\mathrm{d}t.$

It is easy to see that

$\displaystyle
f_{V}(\gamma_{t})-f_{V}(x)\leq\left\\{\begin{array}[]{ll}2\|f_{V}\|_{\infty}\quad\text{
or }\quad 0&\text{ under $\text{\bf(A1)}^{*}$},\\\
\overline{f}_{V}(r_{p}(x))-\underline{f}_{V}(r_{p}(x))\leq\frac{n-m}{2}\log
D(1+r_{p}(x))&\text{ under $\text{\bf(A2)}^{*}$},\\\
\overline{f}_{V}(r_{p}(x))-\underline{f}_{V}(r_{p}(x))\leq\frac{n-m}{2}\log
D(1+r_{p}^{2}(x))&\text{ under $\text{\bf(A3)}^{*}$}.\end{array}\right.$

The comparison (2.5) becomes

$\displaystyle r_{p}(x)\Delta_{V}r_{p}(x)$
$\displaystyle\leq\frac{n-m}{\frac{1}{r_{p}(x)}\int_{0}^{r_{p}(x)}\exp\left(\frac{2(f_{V}(x)-f_{V}(\gamma_{t}))}{n-m}\right)\mathrm{d}t}$
$\displaystyle\leq\left\\{\begin{array}[]{lc}(n-m)e^{\frac{2\|f_{V}\|_{\infty}}{n-m}}\quad\text{
or }\quad n-m&\text{ under $\text{\bf(A1)}^{*}$},\\\ (n-m)D(1+r_{p}(x))&\text{
under $\text{\bf(A2)}^{*}$,}\\\ (n-m)D(1+r_{p}^{2}(x))&\text{ under
$\text{\bf(A3)}^{*}$}\end{array}\right.$

for $x\notin{\rm Cut}(p)$. ∎

######  Theorem 2.4.

Suppose ${\rm Ric}_{V}^{m}\geq-Kg$ with $K>0$ for $m\in]-\infty,\,1\,]$.
Assume that $f_{V}$ is bounded. Then we have (B2).

###### Proof.

Since $f_{V}$ is bounded, we have (2.4) with $K=-(n-m)\kappa
e^{-\frac{4\inf_{M}f_{V}}{n-m}}C_{p}^{2}$. Then, for $x\notin{\rm Cut}(p)$,
(2.5) implies

$\displaystyle r_{p}(x)\Delta_{V}r_{p}(x)$
$\displaystyle\leq(n-m)\sqrt{-\kappa}\,r_{p}(x)\coth(\sqrt{-\kappa}s_{p}(x))e^{-\frac{2f_{V}(x)}{n-m}}C_{p}$
$\displaystyle\leq(n-m)r_{p}(x)\left(\sqrt{-\kappa}+\frac{1}{s_{p}(x)}\right)e^{-\frac{2f_{V}(x)}{n-m}}C_{p}\quad(\because\;\coth
x\leq 1+x^{-1},\;x>0)$
$\displaystyle=(n-m)\left(\sqrt{-\kappa}\,r_{p}(x)+\frac{r_{p}(x)}{s_{p}(x)}\right)e^{-\frac{2f_{V}(x)}{n-m}}C_{p}$
$\displaystyle\leq(n-m)\left(\sqrt{-\kappa}\,e^{-\frac{2\inf_{M}f_{V}}{n-m}}C_{p}r_{p}(x)+\frac{r_{p}(x)}{s_{p}(x)}e^{-\frac{2f_{V}(x)}{n-m}}C_{p}\right)$
$\displaystyle\leq(n-m)\left(\sqrt{-\kappa}\,e^{-\frac{2\inf_{M}f_{V}}{n-m}}C_{p}r_{p}(x)+e^{\frac{2(\sup_{M}f_{V}-\inf_{M}f_{V})}{n-m}}\right).$

This yields the condition (B2). ∎

######  Corollary 2.5.

Assume that $V=\nabla f$ for some $f\in C^{2}(M)$ and $f$ is bounded. Suppose
${\rm Ric}_{f}^{m}\geq-Kg$ with $K>0$ for $m\in]-\infty,\,1\,]$. Then the
condition (B2) holds.

The following is shown in [53, (2.6)].

######  Theorem 2.6 ([53, (2.6)]).

Suppose ${\rm Ric}_{V}^{\infty}\geq-Kg$ for $K\geq 0$. Then for any
sufficiently small $a\in]\,0,\,{\color[rgb]{0,0,0}{{\sf d}}}(p,{\rm
Cut}(p))\,[$ there exists $C>0$ such that $\Delta_{V}r_{p}(x)\leq C+Kr_{p}(x)$
for all $x\in M\setminus({\rm Cut}(p)\cup B_{a}(p))$. In particular,
$\text{\bf(B3)}^{*}$, hence (B3) holds by Lemma 2.2.

######  Remark 2.7.

As a special case of Theorem 2.6, ${\rm Ric}_{V}^{\infty}\geq 0$ implies
${\bf(B3)}$. In particular, our Theorem 1.9 extends Chen-Jost-Qiu [9, Theorem
2] and Qiu [53, Theorem 2].

######  Corollary 2.8.

Suppose that ${\rm Ric}_{V}^{m}\geq-Kg$ for $K\geq 0$ with
$m\in]-\infty,\,1\,]$ and $V$ is a bounded $C^{1}$-vector field. Then the
condition (B3) holds.

###### Proof.

${\rm Ric}_{V}^{m}\geq-Kg$ implies ${\rm
Ric}_{V}^{\infty}(v,v)\geq-K|v|^{2}+\frac{\langle
V,v\rangle^{2}}{m-n}\geq-\left(K+\frac{|V|_{\infty}^{2}}{n-m}\right)|v|^{2}$,
i.e., ${\rm
Ric}_{V}^{\infty}\geq-\left(K+\frac{|V|_{\infty}^{2}}{n-m}\right)g$. Then the
conclusion follows from Theorem 2.6. ∎

######  Remark 2.9.

If ${\operatorname{Ric}}_{V}^{m}\geq(m-1)\kappa g$ with $\kappa<0$ for
$m\in[\,n,\,+\infty\,[$, then we know the following Laplacian comparison

$\displaystyle r_{p}(x)\Delta_{V}r_{p}(x)$
$\displaystyle\leq(m-1)\sqrt{-\kappa}\,r_{p}(x)\coth(\sqrt{-\kappa}\,r_{p}(x))$
$\displaystyle\leq(m-1)(1+\sqrt{-\kappa}\,r_{p}(x))\quad\text{ for }\quad
x\notin{\rm Cut}(p)$

by [4, Theorem 4.2], [34, Theorem 1.1], [29, Theorem 2.3] and [42, Theorem
3.9]. In particular, the condition (B2) holds under
${\operatorname{Ric}}_{V}^{m}\geq Kg$ for $m\in[\,n,\,+\infty\,[$ and
$K\in\mathbb{R}$.

##  3 Construction of $\Delta_{V}$-diffusion processes and Kendall’s
expression of radial processes

In this section, we summarize the construction of $\Delta_{V}$-diffusion
processes on $M$ and note that the Kendall’s expression for radial processes
remains valid for $\Delta_{V}$-diffusion process.

Let $V$ be a $C^{1}$-vector field and $\Delta_{V}:=\Delta-\langle
V,\nabla\cdot\rangle$ denote the $V$-Laplacian. The construction of
$\Delta_{V}$-diffusion process can be done by way of the _horizontal diffusion
process_ $(U_{t})_{t\geq 0}$ generated by $\Delta_{O(M)}-{\bf H}_{V}$ on
$O(M)$ by solving the Stratonovich stochastic differential equation (SDE)

$\displaystyle\mathrm{d}U_{t}=\sqrt{2}\sum_{i=1}^{n}H_{e_{i}}(U_{t})\circ\mathrm{d}B_{t}^{i}-{\bf
H}_{V}(U_{t})\mathrm{d}t,\quad U_{0}=U\in O(M),$

where $B_{t}:=(B_{t}^{1},B_{t}^{2},\cdots,B_{t}^{n})$ is the $n$-dimensional
Brownian motion on a complete filtered probability space
$(\Omega,(\mathscr{F}_{t})_{t\geq 0},\mathscr{F},\mathbb{P})$. Here
$(e_{i})_{i=1}^{n}$ is the canonical orthonormal basis in $\mathbb{R}^{n}$,
$(H_{e_{i}})_{i=1}^{n}$ the corresponding family of horizontal vector fields
in the frame bundle $\mathscr{F}(M)$, and ${\bf H}_{V}(U):=H_{U^{-1}V}(U)$,
$U\in O(M)$ denotes the horizontal lift of $V$, where $U^{-1}V$ is the unique
vector $e\in\mathbb{R}^{n}$ such that $V_{\pi(U)}=Ue$. Moreover,

$\displaystyle\Delta_{O(M)}:=\sum_{i=1}^{n}H_{e_{i}}^{2}$

is the _horizontal Laplace operator_. Since ${\bf H}_{V}$ is $C^{1}$, it is
well known that the equation has a unique solution up to the life time
$\zeta:=\lim_{k\to\infty}\zeta_{k}$, where

$\displaystyle\zeta_{k}:=\inf\\{t>0\mid{\color[rgb]{0,0,0}{{\sf
d}}}(\pi(U),\pi(U_{t}))\geq k\\},\quad k\in\mathbb{N}$

and $\pi:O(M)\to M$ is the projection from the orthonormal frame to its point
on $M$. Let $X_{t}:=\pi(U_{t})$. Then $X_{t}$ solves the equation

$\displaystyle\mathrm{d}X_{t}=\sqrt{2}U_{t}\circ\mathrm{d}B_{t}-V(X_{t})\mathrm{d}t,\quad
X_{0}=x:=\pi(u_{0})$

up to the life time $\zeta$. By Itô’s formula, for any $f\in C_{0}^{2}(M)$,

$\displaystyle
m_{t}^{f}:=f(X_{t})-f(X_{0})-\int_{0}^{t}\Delta_{V}f(X_{s})\mathrm{d}s=\sqrt{2}\int_{0}^{t}\langle
U_{s}^{-1}\nabla f(X_{s}),\mathrm{d}B_{s}\rangle$ (3.1)

is a martingale; i.e., $X_{t}$ is the _diffusion process_ generated by
$\Delta_{V}$, and we call it the _$\Delta_{V}$ -diffusion process_. The
expression (3.1) can be extended for $f\in C^{\infty}(M)$. In this case,
$m_{t}^{f}$ is no longer a martingale, but for any relatively compact open set
$G$, $t\mapsto m^{f}_{t\land\tau_{G}}$ is a martingale. Indeed, there exists
$\rho_{G}\in C_{0}^{\infty}(M)$ such that $\rho_{G}\equiv 1$ on $G$, and
$f_{G}\in C_{0}^{\infty}(M)$ defined by $f_{G}:=f\rho_{G}$ satisfies that
$m^{f}_{t\land\tau_{G}}=m^{f_{G}}_{t\land\tau_{G}}$ is a martingale.

From now on, the probability measure $\mathbb{P}$ is taken for the underlying
process starting from point $x$, i.e. $\mathbb{P}:=\mathbb{P}_{x}$. The
details of the construction for $\Delta_{V}$-diffusion process can be found in
[61, §2.1]. We denote the $\Delta_{V}$-diffusion process by ${\bf
X}^{V}:=(\Omega,X_{t},(\mathscr{F}_{t})_{t\geq
0},\mathscr{F},{\mathbb{P}}_{x})$.

The following is shown in [18, Proposition 1.1.5].

######  Theorem 3.1 ([18, Proposition 1.1.5]).

The diffusion process ${\bf X}^{V}$ on $M$ is tight in the sense that for any
increasing sequence $\\{G_{n}\\}$ of relatively compact open sets satisfying
$\overline{G_{n}}\subset G_{n+1}\subset M$ for any $n\in\mathbb{N}$ and
$M=\bigcup_{n=1}^{\infty}G_{n}$,

$\displaystyle{\mathbb{P}}_{x}\left(\lim_{n\to\infty}\tau_{G_{n}}=\zeta\right)=1,$
(3.2)

equivalently

$\displaystyle{\mathbb{P}}_{x}\left(\lim_{n\to\infty}\sigma_{M\setminus
G_{n}}\geq\zeta\right)=1$ (3.3)

for all $x\in M$. Note here that $\tau_{G_{n}}=\sigma_{M\setminus
G_{n}}\land\zeta$ and $\sigma_{M\setminus G_{n}}\geq\zeta$ is equivalent to
$\sigma_{M\setminus G_{n}}=+\infty$.

######  Lemma 3.2.

Let $\\{G_{n}\\}$ be an increasing sequence $\\{G_{n}\\}$ of relatively
compact open sets satisfying $\overline{G_{n}}\subset G_{n+1}\subset M$ for
any $n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$. The following
assertions are equivalent and hold.

  1. (i)

The diffusion process ${\bf X}^{V}$ on $M$ has no killing inside, i.e.,

$\displaystyle\mathbb{P}_{x}(X_{\zeta-}\in M,\zeta<\infty)=0$ (3.4)

for all $x\in M$.

  2. (ii)

We have

$\displaystyle\mathbb{P}_{x}(\sigma_{M\setminus G_{n}}<\zeta\quad\text{ for
all }\quad n\in\mathbb{N})=1$ (3.5)

for all $x\in M$.

  3. (iii)

We have

$\displaystyle{\mathbb{P}}_{x}\left(\lim_{n\to\infty}\sigma_{M\setminus
G_{n}}=\zeta\right)=1$ (3.6)

for all $x\in M$.

###### Proof.

We first prove (i). Let $\\{G_{n}\\}$ be the exhaustion as stated above. Let
us consider a pre-semi-Dirichlet form $(\mathscr{E}_{G_{n}},C_{0}^{1}(G_{n}))$
on $L^{2}(G_{n};\nu_{g})$ defined by

$\displaystyle\mathscr{E}_{G_{n}}(f,g)=\int_{G_{n}}\langle\nabla f,\nabla
g\rangle\mathrm{d}\nu_{g}+\int_{G_{n}}\langle V,\nabla
f\rangle\,g\,\mathrm{d}\nu_{g}\quad\text{ for }\quad f,g\in C_{0}^{1}(G_{n}).$
(3.7)

Thanks to the boundedness $V$ on $G_{n}$, there exists
$\alpha_{0}=\alpha_{0}(G_{n})$ such that

$\displaystyle(\mathscr{E}_{G_{n}})_{\alpha_{0}}(u,u)\geq 0\quad\text{ for
}\quad u\in C_{0}^{1}(G_{n}),$ (3.8)

where
$(\mathscr{E}_{G_{n}})_{\alpha_{0}}(u,u):=\mathscr{E}_{G_{n}}(u,u)+\alpha_{0}\int_{G_{n}}|u|^{2}\mathrm{d}\nu_{g}$,
and there exists $K=K(G_{n})\geq 1$ such that

$\displaystyle|\mathscr{E}_{G_{n}}(u,v)|\leq
K(\mathscr{E}_{G_{n}})_{\alpha_{0}}(u,u)^{\frac{1}{2}}(\mathscr{E}_{G_{n}})_{\alpha_{0}}(v,v)^{\frac{1}{2}}\quad\text{
for all }\quad u,v\in C_{0}^{1}(G_{n}).$ (3.9)

Moreover, for any $\alpha>\alpha_{0}$, there exists
$0<\lambda_{1}\leq\lambda_{2}$ depending on $\alpha$ such that

$\displaystyle\lambda_{1}{\bf
D}_{1}(u,u)\leq(\mathscr{E}_{G_{n}})_{\alpha}(u,u)\leq\lambda_{2}{\bf
D}_{1}(u,u)\quad\text{ for all }\quad u\in C_{0}^{1}(G_{n}),$ (3.10)

where ${\bf D}_{1}(u,u):=\int_{G_{n}}|\nabla
u|^{2}\mathrm{d}\nu_{g}+\int_{G_{n}}|u|^{2}\mathrm{d}\nu_{g}$. From (3.10),
$(\mathscr{E}_{G_{n}},C_{0}^{1}(G_{n}))$ is closable on $L^{2}(G_{n};\nu_{g})$
and denote its closure by $(\mathscr{E}_{G_{n}},D(\mathscr{E}_{G_{n}}))$ and
it is a regular semi-Dirichleet form on $L^{2}(G_{n};\nu_{g})$ (cf. [50,
Theorems 1.5.2 and 1.5.3] in the Euclidean setting). The part process ${\bf
X}_{G_{n}}^{V}$ defined by $X_{t}^{G_{n}}:=X_{t}$ if $t<\tau_{G_{n}}$,
$X_{t}^{G_{n}}:=\partial$ if $t\geq\tau_{G_{n}}$, is a diffusion process
associated with $(\mathscr{E}_{G_{n}},D(\mathscr{E}_{G_{n}}))$ on
$L^{2}(G_{n};\nu_{g})$. Thanks to [50, Theorem 3.5.16], ${\bf X}_{G_{n}}^{V}$
has no killing inside in the sense that

$\displaystyle\mathbb{P}_{x}(X_{\tau_{G_{n}}-}\in
G_{n},\tau_{G_{n}}<+\infty)=0\quad\nu_{g}\text{-a.e.~{}}x\in G_{n}.$ (3.11)

From (3.11), we can deduce

$\displaystyle\mathbb{P}_{x}(X_{\zeta-}\in
M,\zeta<+\infty)=0\quad\nu_{g}\text{-a.e.~{}}x\in M.$ (3.12)

Indeed, under $\zeta<+\infty$, we have
$\lim_{n\to\infty}X_{\tau_{G_{n}}-}=X_{\zeta-}$ immediately from (3.2). Take
$n>\ell>k$. Then

$\displaystyle\mathbb{P}_{x}(X_{\tau_{G_{n}}-}\in G_{\ell},\zeta<+\infty)$
$\displaystyle\leq\mathbb{P}_{x}(X_{\tau_{G_{n}}-}\in
G_{\ell},\tau_{G_{n}}<+\infty)$
$\displaystyle\leq\mathbb{P}_{x}(X_{\tau_{G_{n}}-}\in
G_{n},\tau_{G_{n}}<+\infty)=0\quad\text{ for }\quad\nu_{g}\text{-a.e.~{}}x\in
G_{k}.$

Thus

$\displaystyle\mathbb{E}_{x}[{\bf
1}_{G_{\ell}}(X_{\tau_{G_{n}}-}):\zeta<+\infty]=0\quad\nu_{g}\text{-a.e.~{}}x\in
G_{k}.$

From this,

$\displaystyle\mathbb{E}_{x}[{\bf 1}_{G_{\ell}}(X_{\zeta-}):\zeta<+\infty]$
$\displaystyle\leq\mathbb{E}_{x}[\varliminf_{n\to\infty}{\bf
1}_{G_{\ell}}(X_{\tau_{G_{n}}-}):\zeta<+\infty]$
$\displaystyle\leq\varliminf_{n\to\infty}\mathbb{E}_{x}[{\bf
1}_{G_{\ell}}(X_{\tau_{G_{n}}-}):\zeta<+\infty]=0\quad\nu_{g}\text{-a.e.~{}}x\in
G_{k}.$

Letting $\ell\to\infty$ and $k\to\infty$, we obtain (3.12). Next we prove that
$\mathbb{P}_{x}(X_{\zeta-}\in M,\zeta<+\infty)$ is an excessive function with
respect to ${\bf X}^{V}$. Indeed, by $\zeta\circ\theta_{t}=(\zeta-t)_{+}$,

$\displaystyle p_{t}(\mathbb{P}_{\text{\tiny$\bullet$}}(X_{\zeta-}$
$\displaystyle\in M,\zeta<+\infty))(x)$
$\displaystyle=\mathbb{E}_{x}[\mathbb{P}_{X_{t}}(X_{\zeta-}\in
M,\zeta<+\infty)]$
$\displaystyle=\mathbb{P}_{x}\left(X_{(t+\zeta\circ\theta_{t})-}\in
M,t+\zeta\circ\theta_{t}<+\infty\right)$
$\displaystyle=\mathbb{P}_{x}(X_{\zeta-}\in
M,t+\zeta\circ\theta_{t}<+\infty,t<\zeta)+\mathbb{P}_{x}(X_{t-}\in
M,t+\zeta\circ\theta_{t}<+\infty,t\geq\zeta)$
$\displaystyle=\mathbb{P}_{x}(X_{\zeta-}\in M,t\leq\zeta<+\infty)$
$\displaystyle\nearrow\mathbb{P}_{x}(X_{\zeta-}\in
M,0<\zeta<+\infty)=\mathbb{P}_{x}(X_{\zeta-}\in M,\zeta<+\infty)\quad\text{ as
}\quad t\searrow 0.$

On the other hand, ${\bf X}^{V}$ satisfies the Meyer’s hypotheses (L), i.e.,
$R_{\alpha}(x,\mathrm{d}y)\ll\nu_{g}(\mathrm{d}y)$ for each $\alpha>0$ and
$x\in M$, where
$R_{\alpha}f(x)=\mathbb{E}_{x}\left[\int_{0}^{\infty}e^{-\alpha
t}f(X_{t})\mathrm{d}t\right]$ for $f\in\mathscr{B}(M)_{+}$. Indeed, since
${\bf X}_{G_{n}}^{V}$ is associated to
$(\mathscr{E}_{G_{n}},D(\mathscr{E}_{G_{n}}))$, for any
$f\in\mathscr{B}(M)_{+}$ with $f=0$ $\nu_{g}$-a.e.

$\displaystyle\lambda_{1}\|R_{\alpha}^{G_{n}}f\|_{L^{2}(G_{n};\nu_{g})}^{2}$
$\displaystyle\leq\lambda_{1}{\bf
D}_{1}(R_{\alpha}^{G_{n}}f,R_{\alpha}^{G_{n}}f)$
$\displaystyle\leq(\mathscr{E}_{G_{n}})_{\alpha}(R_{\alpha}^{G_{n}}f,R_{\alpha}^{G_{n}}f)$
$\displaystyle=(f,R_{\alpha}^{G_{n}}f)_{L^{2}(G_{n};\nu_{g})}=0$

implies $R_{\alpha}^{G_{n}}f=0$ $\nu_{g}$-a.e. Here
$R_{\alpha}^{G_{n}}f(x):=\mathbb{E}_{x}\left[\int_{0}^{\tau_{G_{n}}}e^{-\alpha
t}f(X_{t})\mathrm{d}t\right]$. Letting $n\to\infty$, we get $R_{\alpha}f=0$
$\nu_{g}$-a.e. on $M$. Hence $R_{\alpha}f=0$ on $M$ by the strong Feller
property of ${\bf X}^{V}$ (see [47],[2, Proposition 1.11]). Thus we obtain
Meyer’s hypotheses (L) for ${\bf X}^{V}$. Then, for any excessive function $h$
on $M$, $h=0$ $\nu_{g}$-a.e. implies $h=0$ on $M$ by [57, (10.25)]. Therefore,
applying this to the excessive function
$h:=\mathbb{P}_{\text{\tiny$\bullet$}}(X_{\zeta-}\in M;\zeta<\infty)$, we
obtain (i) from (3.12).

Next we prove (i)$\Longrightarrow$(ii). By (3.3),
$\lim_{n\to\infty}\sigma_{M\setminus G_{n}}>\zeta$ or
$\lim_{n\to\infty}\sigma_{M\setminus G_{n}}=\zeta$ $\mathbb{P}_{x}$-a.s. on
$\\{\zeta<+\infty\\}$. This implies that $\sigma_{M\setminus G_{n}}=+\infty$
for some $n\in\mathbb{N}$ or $\sigma_{M\setminus G_{n}}<\zeta$ for all
$n\in\mathbb{N}$ $\mathbb{P}_{x}$-a.s. on $\\{\zeta<+\infty\\}$, because
$\sigma_{M\setminus G_{n}}\geq\zeta$ is equivalent to $\sigma_{M\setminus
G_{n}}=+\infty$. Thanks to (3.4) for all $x\in M$, under $\zeta<+\infty$, the
former case, i.e., $\sigma_{M\setminus G_{n}}=+\infty$ for some
$n\in\mathbb{N}$ does not occur. Under $\zeta=+\infty$, by (3.3),
$\lim_{n\to\infty}\sigma_{M\setminus G_{n}}=+\infty$. Applying (3.4) for all
$x\in M$ again, $\sigma_{M\setminus G_{n}}=+\infty$ for some $n\in\mathbb{N}$
does not occur in this case. Therefore, we obtain (ii). The implication
(ii)$\Longrightarrow$(iii) is clear from (3.3).

Finally, we prove (iii)$\Longrightarrow$(i). Suppose
$\lim_{n\to\infty}\sigma_{M\setminus G_{n}}=\zeta$. If $\sigma_{M\setminus
G_{n}}=\zeta$ for some $n\in\mathbb{N}$, then $\sigma_{M\setminus
G_{n}}=\zeta=+\infty$. So under $\zeta<+\infty$, this does not occur. Hence
$\sigma_{M\setminus G_{n}}<\zeta$ for all $n\in\mathbb{N}$ under
$\zeta<+\infty$. Then, we see $\lim_{n\to\infty}X_{\sigma_{M\setminus
G_{n}}}=X_{\zeta-}$ and $X_{\sigma_{M\setminus G_{n}}}\in M\setminus G_{n}$
$\mathbb{P}_{x}$-a.s. on $\\{\zeta<+\infty\\}$, hence

$\displaystyle r_{p}(X_{\zeta-})=\lim_{n\to\infty}r_{p}(X_{\sigma_{M\setminus
G_{n}}})\geq\lim_{n\to\infty}{\color[rgb]{0,0,0}{{\sf
d}}}(p,G_{n}^{c})=+\infty$

holds $\mathbb{P}_{x}$-a.s. on $\\{\zeta<+\infty\\}$. This implies (3.4) for
all $x\in M$. ∎

A function $f$ on $M$ is said to be _$V$ -subharmonic_ if it belongs to
$C^{2}(M)$ and $\Delta_{V}f\geq 0$ on $M$. A function $f$ on $M$ is said to be
_$V$ -superharmonic_ if $-f$ is $V$-subharmonic. A function $f$ on $M$ is said
to be _$V$ -harmonic_ if it is $V$-subharmonic and $V$-superharmonic. A
function $f$ on $M$ is said to be _finely $V$-subharmonic_ if it is finely
continuous (see [5] for the definition of fine topology) and $t\mapsto
f(X_{t})$ is a local (${\mathbb{P}}_{x}$-)submartingale on $[\,0,\,\zeta\,[$
for all $x\in M$, i.e., there exists an increasing sequence of stopping times
$\\{T_{k}\\}$ with $T_{k}<\zeta$ for all $k\in\mathbb{N}$ and
$\lim_{k\to\infty}T_{k}=\zeta$ such that $t\mapsto f(X_{t\land T_{k}})$ is a
submartingale. A function $f$ on $M$ is said to be _finely $V$-superharmonic_
if $-f$ is finely $V$-subharmonic. A function $f$ on $M$ is said to be _finely
$V$-harmonic_ if it is finely $V$-subharmonic and finely $V$-superhamonic. If
$f\in C^{2}(M)$, then the both sides of (3.1) are local
(${\mathbb{P}}_{x}$-)martingales on $[\,0,\,\zeta\,[$. More precisely, let
$\\{G_{k}\\}$ be the increasing sequence of relatively compact open sets. Then
there exists $\rho_{k}\in C_{0}^{\infty}(M)$ such that $\rho_{k}\equiv 1$ on
${G}_{k}$. We set $f_{k}:=f\rho_{k}\in C_{0}^{2}(M)$. Then $M_{t}^{f_{k}}$ is
a martingale and $M_{t\land\tau_{G_{k}}}^{f}=M_{t\land\tau_{G_{k}}}^{f_{k}}$,
hence $M_{t}^{f}=M_{t}^{f_{k}}$ for $t<\tau_{G_{k}}$. This means that
$M_{t}^{f}$ is a local (${\mathbb{P}}_{x}$-)martingale on $[\,0,\,\zeta\,[$ by
Theorem 3.1, because $\tau_{G_{k}}<\zeta$. In particular, every
$V$-subharmonic (resp. $V$-superharmonic, $V$-harmonic) functions are finely
$V$-subharmonic (resp. $V$-superharmonic, $V$-harmonic).

The following theorem can be proved in a similar way with the case of Brownian
motion on $M$.

######  Theorem 3.3 (cf. [18, Theorem 3.5.1], [6, proof of Corollary 3.6],
[61, (3.7.10)]).

Suppose that $(X_{t})_{t\geq 0}$ is a $\Delta_{V}$-diffusion process on $M$.
Then there exists a one-dimensional Euclidean Brownian motion $\beta$ and a
non-degenerated continuous process $L_{t}$ which increases only when
$X_{t}\in{\rm Cut}(p)$ such that

$\displaystyle
r_{p}(X_{t})-r_{p}(X_{0})=\sqrt{2}\beta_{t}+\int_{0}^{t}\Delta_{V}r_{p}(X_{s}){\bf
1}_{\\{X_{s}\notin{\rm Cut}(p)\\}}\mathrm{d}s-L_{t},\qquad t<\zeta$ (3.13)

${\mathbb{P}}_{x}$-a.s. for all $x\in M$. Here $\zeta:=\inf\\{t>0\mid
X_{t}\notin M\\}$ is the life time of $(X_{t})_{t\geq 0}$.

######  Lemma 3.4.

Suppose (B3). Let $\\{G_{n}\\}$ be an increasing sequence $\\{G_{n}\\}$ of
relatively compact open sets satisfying $\overline{G_{n}}\subset
G_{n+1}\subset M$ for any $n\in\mathbb{N}$ and
$M=\bigcup_{n=1}^{\infty}G_{n}$.

$\displaystyle{\mathbb{E}}_{x}\left[r_{p}^{2}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]$ $\displaystyle\leq\mathfrak{D}(t),$ (3.14)
$\displaystyle{\mathbb{E}}_{x}\left[r_{p}^{4}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]$ $\displaystyle\leq
r_{p}(x)^{4}+4(3+D)\int_{0}^{t}{\mathfrak{D}}(s)\mathrm{d}s$
$\displaystyle\hskip
28.45274pt+4De^{4Dt}\int_{0}^{t}\left(r_{p}^{4}(x)+4(D+3)\int_{0}^{s}{\mathfrak{D}}(u)\mathrm{d}u\right)e^{-4Ds}\mathrm{d}s$
(3.15)

hold for all $x\in M$. Here

${\mathfrak{D}}(t):=r_{p}(x)^{2}+2(1+D)t+2De^{2Dt}\int_{0}^{t}(r_{p}^{2}(x)+2(1+D)s)e^{-2Ds}\mathrm{d}s.$

###### Proof.

By Theorem 3.3, the radial process $r_{p}(X_{t})$ is a semimartingale on
$[0,\zeta[$ in the sense that $t\mapsto r_{p}(X_{t\land\sigma_{M\setminus
G_{n}}})$ is a semimartingale for each $n\in\mathbb{N}$. By way of Itô’s
formula, we have

$\displaystyle r_{p}^{2}(X_{t})-r_{p}^{2}(X_{0})$
$\displaystyle=2\sqrt{2}\int_{0}^{t}r_{p}(X_{s})\mathrm{d}\beta_{s}+2t$
$\displaystyle\hskip
42.67912pt+2\int_{0}^{t}r_{p}(X_{s})\Delta_{V}r_{p}(X_{s}){\bf
1}_{M\setminus{\rm
Cut}(p)}(X_{s})\mathrm{d}s-2\int_{0}^{t}r_{p}(X_{s})\mathrm{d}L_{s},$
$\displaystyle r_{p}^{4}(X_{t})-r_{p}^{4}(X_{0})$
$\displaystyle=4\sqrt{2}\int_{0}^{t}r_{p}^{3}(X_{s})\mathrm{d}\beta_{s}+12\int_{0}^{t}r_{p}^{2}(X_{s})\mathrm{d}s$
$\displaystyle\hskip
42.67912pt+4\int_{0}^{t}r_{p}^{3}(X_{s})\Delta_{V}r_{p}(X_{s}){\bf
1}_{M\setminus{\rm
Cut}(p)}(X_{s})\mathrm{d}s-4\int_{0}^{t}r_{p}^{3}(X_{s})\mathrm{d}L_{s},\quad$

$t\in[\,0,\,\zeta\,[$, ${\mathbb{P}}_{x}$-a.s. for $x\in M$. From these
equalities, we can obtain (3.14) and (3.15). Indeed, by (3.5), we see from
(B3) that

$\displaystyle{\mathbb{E}}_{x}\left[r_{p}^{2}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]$ $\displaystyle\leq
r_{p}^{2}(x)+2t+2D{\mathbb{E}}_{x}\left[\int_{0}^{t\land\sigma_{M\setminus
G_{n}}}(1+r_{p}^{2}(X_{s}))\mathrm{d}s\right]$
$\displaystyle=r_{p}^{2}(x)+2(1+D)t+2D{\mathbb{E}}_{x}\left[\int_{0}^{t}r_{p}^{2}(X_{s}){\bf
1}_{\\{s<\sigma_{M\setminus G_{n}}\\}}\mathrm{d}s\right]$ $\displaystyle\leq
r_{p}^{2}(x)+2(1+D)t+2D\int_{0}^{t}{\mathbb{E}}_{x}\left[r_{p}^{2}(X_{s\land\sigma_{M\setminus
G_{n}}})\right]\mathrm{d}s.$

Applying the Gronwall’s inequality to the continuous function

$\displaystyle t\mapsto
f(t):={\mathbb{E}}_{x}\left[r_{p}^{2}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]<+\infty$

under (3.5), we obtain (3.14). For (3.15), we have similarly

$\displaystyle{\mathbb{E}}_{x}\left[r_{p}^{4}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]$ $\displaystyle\leq
r_{p}^{4}(x)+4(D+3)\int_{0}^{t}\mathfrak{D}(s)\mathrm{d}s+4D\int_{0}^{t}{\mathbb{E}}_{x}\left[r_{p}^{4}(X_{s\land\sigma_{M\setminus
G_{n}}})\right]\mathrm{d}s,$

and applying the Gronwall’s inequality to the continuous function

$\displaystyle t\mapsto
g(t):={\mathbb{E}}_{x}\left[r_{p}^{4}(X_{t\land\sigma_{M\setminus
G_{n}}})\right]<+\infty$

under (3.5), we obtain (3.15). ∎

######  Corollary 3.5.

Suppose (B3). Then ${\bf X}^{V}$ is conservative.

###### Proof.

By (3.6) and (3.14),

$\displaystyle\mathbb{E}_{x}[r_{p}^{2}(X_{\zeta-}):t\geq\zeta,\zeta<+\infty]\,$
$\displaystyle\stackrel{{\scriptstyle\text{\tiny\eqref{eq:nest2}}}}{{=}}\mathbb{E}_{x}[\lim_{n\to\infty}r_{p}^{2}(X_{\sigma_{M\setminus
G_{n}}}):t\geq\zeta,\zeta<+\infty]$
$\displaystyle=\mathbb{E}_{x}[\lim_{n\to\infty}r_{p}^{2}(X_{t\land\sigma_{M\setminus
G_{n}}}):t\geq\zeta,\zeta<+\infty]$
$\displaystyle\leq\varliminf_{n\to\infty}\mathbb{E}_{x}[r_{p}^{2}(X_{t\land\sigma_{M\setminus
G_{n}}})]\,$
$\displaystyle\stackrel{{\scriptstyle\text{\text{\tiny\eqref{eq:quadratic}}}}}{{\leq}}\mathfrak{D}(t)<+\infty.$

This and (3.4) together imply $\mathbb{P}_{x}(t\geq\zeta,\zeta<+\infty)=0$ for
all $x\in M$. Therefore, we obtain $\mathbb{P}_{x}(\zeta<+\infty)=0$ for all
$x\in M$. ∎

Thanks to the conservativeness of ${\bf X}^{V}$ under (B3), hereafter, we may
write $\tau_{G}=\sigma_{M\setminus G}$ for any relatively compact open subset
$G$ of $M$.

##  4 Bochner Identity

In this section, based on the Bochner identity for smooth map $u:M\to N$ in
terms of usual Laplace-Bertrami operator $\Delta$, we establish the Bochner
identity for $u:M\to N$ in terms of $\Delta_{V}$. We denote the Levi-Civita
connection on $M$ by $\nabla^{M}$. Let $\\{e_{i}\\}_{i=1}^{n}$ be the
canonical ONB of $TM$, i.e., $e_{i}=\left(\partial/\partial x^{i}\right)_{x}$,
and $u^{-1}TN$ the induced vector bundle from $TN$ by $u$, i.e., the fibre
bundle over $M$ with fibre $T_{u(x)}N$ for $x\in M$. Moreover
$\nabla^{u^{-1}TN}$ denotes the induced connection from the Levi-Civita
connection $\nabla^{N}$ on $N$. Let $\nabla$ be the connection over
$TM^{*}\otimes u^{-1}TN$ induced from the dual connection $\nabla^{*}$ over
$TM^{*}$ and the connection $\nabla^{u^{-1}TN}$ over $u^{-1}TN$ (see [48,
(3.23)] for the definition of $\nabla$). The _Hessian_ of $u:M\to N$ is the
symmetric bilinear map ${\rm Hess}\,u:TM\times TM\to TN$ defined by ${\rm
Hess}\,u(X,Y):=(\nabla_{X}\mathrm{d}u)(Y)=\nabla_{X}^{u^{-1}TN}(\mathrm{d}u(Y))-\mathrm{d}u(\nabla_{X}^{M}Y)$
for $X,Y\in\Gamma(TM)$ (see [48, pp. 93–94] for definitions
$\nabla^{\prime}:=\nabla^{N}$ and ${}^{\prime}\nabla:=\nabla^{u^{-1}TN}$). The
tension field of $u$ is given by $\tau(u):=\sum_{i=1}^{n}{\rm
Hess}\,u(e_{i},e_{i})=\sum_{i=1}^{n}(\nabla_{e_{i}}\mathrm{d}u)(e_{i})$ and
the energy density of $u$ is given by
$e(u):=\frac{1}{2}|\mathrm{d}u|^{2}=\frac{1}{2}\sum_{i=1}^{n}\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i})\rangle$.

######  Lemma 4.1 (Bochner Identity).

For any smooth map $u:M\to N$, we have

$\displaystyle\frac{1}{2}\Delta_{V}|\mathrm{d}u|^{2}$ $\displaystyle=|{\rm
Hess}\,u|^{2}+\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\tau_{V}(u))\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i,j=1}^{n}{\operatorname{Ric}}_{V}^{\infty}(e_{i},e_{j})\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
(4.1) $\displaystyle\hskip 56.9055pt-\sum_{i,j=1}^{n}\left\langle{}^{N}{\rm
Riem}(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j}))\mathrm{d}u(e_{j}),\mathrm{d}u(e_{i})\right\rangle.$

Here ${}^{N}{\rm Riem}$ is the Riemannian curvature tensor for $(N,h)$.

###### Proof.

The assertion seems to be known for experts. For completeness we give a proof.
The proof of (4.1) is based on the following Bochner identity for smooth map
$u$

$\displaystyle\frac{1}{2}\Delta|\mathrm{d}u|^{2}$ $\displaystyle=|{\rm
Hess}\,u|^{2}+\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\tau(u))\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i,j=1}^{n}{\operatorname{Ric}}_{g}(e_{i},e_{j})\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
(4.2) $\displaystyle\hskip 56.9055pt-\sum_{i,j=1}^{n}\left\langle{}^{N}{\rm
Riem}(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j}))\mathrm{d}u(e_{j}),\mathrm{d}u(e_{i})\right\rangle.$

The Bochner identity (4.2) for smooth map $u:M\to N$ is well-known for experts
(see [1, Proposition 1.5], [41, Corollary 2.10]). From (4.2), we can deduce
(4.1) as follows:

$\displaystyle\frac{1}{2}\Delta_{V}|\mathrm{d}u|^{2}$
$\displaystyle=\frac{1}{2}\Delta|\mathrm{d}u|^{2}-\frac{1}{2}\langle
V,\nabla|\mathrm{d}u|^{2}\rangle$
$\displaystyle\stackrel{{\scriptstyle\text{\tiny\eqref{eq:BochnerWeitzen}}}}{{=}}|{\rm
Hess}\,u|^{2}+\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\tau_{V}(u))\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i,j=1}^{n}{\operatorname{Ric}}_{V}^{\infty}(e_{i},e_{j})\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
$\displaystyle\hskip 56.9055pt-\sum_{i,j=1}^{n}\left\langle{}^{N}{\rm
Riem}(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j}))\mathrm{d}u(e_{j}),\mathrm{d}u(e_{i})\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\mathrm{d}u(V))\right\rangle$
$\displaystyle\hskip
56.9055pt+\frac{1}{2}\sum_{i,j=1}^{n}\left(\left\langle\nabla_{e_{i}}^{M}V,e_{j}\right\rangle+\left\langle\nabla_{e_{j}}^{M}V,e_{i}\right\rangle\right)\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
$\displaystyle\hskip 85.35826pt-\frac{1}{2}\langle
V,\nabla|\mathrm{d}u|^{2}\rangle.$

So it suffices to prove that the last three terms vanish. Observe

$\displaystyle|\mathrm{d}u|^{2}=\sum_{i=1}^{n}|\mathrm{d}u(e_{i})|^{2}.$

By definition of the connection $\nabla$ in $TM^{*}\otimes u^{-1}TN$,

$\displaystyle\frac{1}{2}V|\mathrm{d}u|^{2}$
$\displaystyle=\sum_{i=1}^{n}\langle\nabla_{V}\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i})\rangle$
$\displaystyle=\sum_{i=1}^{n}\left\langle\nabla\mathrm{d}u(V,e_{i}),\mathrm{d}u(e_{i})\right\rangle$
$\displaystyle=\sum_{i=1}^{n}\left\langle\nabla_{V}^{u^{-1}TN}\mathrm{d}u(e_{i})-\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle\qquad\text{(by
\cite[cite]{[\@@bibref{}{Nishikawa}{}{}, (3.25)]})}$
$\displaystyle=\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{V}^{u^{-1}TN}\mathrm{d}u(e_{i})\right\rangle-\sum_{i=1}^{n}\left\langle\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle.$

Then the sum of last three terms becomes

$\displaystyle\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\mathrm{d}u(V))\right\rangle-\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{V}^{u^{-1}TN}\mathrm{d}u(e_{i})\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i=1}^{n}\left\langle\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle-\sum_{i,j=1}^{n}\left\langle\nabla_{e_{i}}^{M}V,e_{j}\right\rangle\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\rangle$
$\displaystyle=\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\nabla_{e_{i}}^{u^{-1}TN}(\mathrm{d}u(V))-\nabla_{V}^{u^{-1}TN}\mathrm{d}u(e_{i})\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i=1}^{n}\left\langle\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle-\sum_{i,j=1}^{n}\left\langle\nabla_{e_{i}}^{M}V,e_{j}\right\rangle\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
$\displaystyle=\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u([e_{i},V])\right\rangle\qquad(\text{by
\cite[cite]{[\@@bibref{}{Nishikawa}{}{}, Lemma~{}3.7]}})$ $\displaystyle\hskip
28.45274pt+\sum_{i=1}^{n}\left\langle\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle-\sum_{i,j=1}^{n}\left\langle\nabla_{e_{i}}^{M}V,e_{j}\right\rangle\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\right\rangle$
$\displaystyle=\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(\nabla_{e_{i}}^{M}V-\nabla_{V}^{M}e_{i})\right\rangle$
$\displaystyle\hskip
28.45274pt+\sum_{i=1}^{n}\left\langle\mathrm{d}u(\nabla_{V}^{M}e_{i}),\mathrm{d}u(e_{i})\right\rangle-\sum_{i=1}^{n}\left\langle\mathrm{d}u(e_{i}),\mathrm{d}u(\nabla_{e_{i}}^{M}V)\right\rangle$
$\displaystyle=0.$

∎

We need the following lemma to prove Corollary 4.3.

######  Lemma 4.2.

Let $H$ be a real separable Hilbert space and suppose that $h_{ij}\in H$
satisfies $h_{ij}=h_{ji}$ for $i,j=1,2,\cdots,n$. Then we have

$\displaystyle\sum_{i,j=1}^{n}\|h_{ij}\|_{H}^{2}\geq\frac{1}{n}\left\|\sum_{i=1}^{n}h_{ii}\right\|_{H}^{2}.$

If further the symmetric matrix $(h_{ij})\in H^{n}\otimes H^{n}$ has a null
eigenvalue $0_{H}\in H$ with multiplicity $k\in\\{1,2,\cdots,n-1\\}$, then we
have

$\displaystyle\sum_{i,j=1}^{n}\|h_{ij}\|_{H}^{2}\geq\frac{1}{n-k}\left\|\sum_{i=1}^{n}h_{ii}\right\|_{H}^{2}.$

###### Proof.

The first assertion can be easily confirmed for the case $H=\mathbb{R}$. In
fact, in this case any real symmetric matrix $A=(a_{ij})$ satisfies
$\|A\|^{2}={\rm Tr}(A^{2})\geq\frac{{\rm Tr}(A)^{2}}{n}$. For general case,
letting $\\{e_{\ell}\\}_{\ell=1}^{\infty}$ be the complete orthonormal base in
$H$, the assertion follows from the Parseval’s identity
$\|h\|_{H}^{2}=\sum_{\ell=1}^{\infty}\langle h,e_{\ell}\rangle_{H}^{2}$ and
the assertion for $H=\mathbb{R}$. Next we prove the second assertion. In view
of the Fourier expansion $h=\sum_{\ell=1}^{\infty}\langle
h,e_{\ell}\rangle_{H}e_{\ell}$ in $H$, we can prove

$\displaystyle{\rm Ker}(h_{ij})=\bigcap_{\ell=1}^{\infty}{\rm Ker}(\langle
h_{ij},e_{\ell}\rangle_{H}),$

where

$\displaystyle{\rm
Ker}(h_{ij}):=\\{\phi\in\mathbb{R}^{n}\mid(h_{ij})\phi=0_{H^{n}}\\},\quad{\rm
Ker}(\langle h_{ij},e_{\ell}\rangle_{H}):=\\{\phi\in\mathbb{R}^{n}\mid(\langle
h_{ij},e_{\ell}\rangle_{H})\phi=0_{\mathbb{R}^{n}}\\}.$

Therefore, if $(h_{ij})$ has a null eigenvalue $0_{H}$ with multiplicity $k$,
then the real symmetric matrix $(\langle h_{ij},e_{\ell}\rangle_{H})$ has the
null eigenvalue $0$ with multiplicity $k$ for all $\ell\in\mathbb{N}$. The
converse also holds. The rest of the proof is similar to the proof of the
first assertion. ∎

######  Corollary 4.3.

Suppose that ${\operatorname{Ric}}_{V}^{m}\geq 0$ with
$m\in\,[-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$ and ${\rm Sect}_{N}\leq 0$.
Then any $V$-harmonic map $u:M\to N$ satisfies

$\displaystyle\Delta_{V}|\mathrm{d}u|^{2}\geq\frac{2m}{n(m-n)}|\mathrm{d}u(V)|^{2}\geq
0.$ (4.3)

If further the symmetric matrix $({\rm Hess}\,u(e_{i},e_{j}))_{ij}$ for
$V$-harmonic map $u:M\to N$ admits a null eigenvalue with multiplicity
$k\in\\{1,2,\cdots,n-1\\}$ independent of the choice of fibre, then
$\Delta_{V}|\mathrm{d}u|^{2}\geq\frac{2(m-k)}{(n-k)(m-n)}|\mathrm{d}u(V)|^{2}\geq
0$ under ${\operatorname{Ric}}_{V}^{m}\geq 0$ with $m\in]-\infty,\,k\,]$ and
${\rm Sect}_{N}\leq 0$.

###### Proof.

We may assume $m\in]-\infty,\,0\,]$ by (1.7). By Lemma 4.2,

$\displaystyle|{\rm Hess}\,u|^{2}=\sum_{i,j=1}^{n}|{\rm
Hess}\,u(e_{i},e_{j})|^{2}\geq\frac{1}{n}\left|\sum_{i=1}^{n}{\rm
Hess}\,u(e_{i},e_{i})\right|^{2}=\frac{|\tau(u)|^{2}}{n}.$

Since $u:M\to N$ is $V$-harmonic, $\operatorname{Ric}_{V}^{m}\geq 0$ and ${\rm
Sect}_{N}\leq 0$, we have

$\displaystyle\frac{1}{2}\Delta_{V}|\mathrm{d}u|^{2}$ $\displaystyle\geq|{\rm
Hess}\,u|^{2}+\sum_{i,j=1}^{n}{\operatorname{Ric}}_{V}^{\infty}(e_{i},e_{j})\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\rangle$
$\displaystyle=|{\rm
Hess}\,u|^{2}+\sum_{i,j=1}^{n}{\operatorname{Ric}}_{V}^{m}(e_{i},e_{j})\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\rangle$
$\displaystyle\hskip 28.45274pt+\frac{1}{m-n}\sum_{i,j=1}^{n}\langle
V,e_{i}\rangle\langle
V,e_{j}\rangle\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\rangle$
$\displaystyle\geq|{\rm Hess}\,u|^{2}+\frac{1}{m-n}\sum_{i,j=1}^{n}\langle
V,e_{i}\rangle\langle
V,e_{j}\rangle\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j})\rangle$
$\displaystyle\geq\frac{|\tau(u)|^{2}}{n}+\frac{1}{m-n}|\mathrm{d}u(V)|^{2}$
$\displaystyle=\frac{|\mathrm{d}u(V)|^{2}}{n}+\frac{1}{m-n}|\mathrm{d}u(V)|^{2}\qquad(\text{by
}\tau_{V}(u)=0)$ $\displaystyle=\frac{m}{n(m-n)}|\mathrm{d}u(V)|^{2}\geq 0.$

In the second inequality above, we use the fact that any $n\times n$ symmetric
non-negative definite real matrix $A=(a_{ij})$ satisfies
$\sum_{i,j=1}^{n}a_{ij}\langle h_{i},h_{j}\rangle_{H}\geq 0$ for any real
Hilbert space $(H,\langle\cdot,\cdot\rangle)$ and $h_{i}\in H$ ($1\leq i\leq
n$). If further $({\rm Hess}\,u(e_{i},e_{j}))_{ij}$ for $V$-harmonic map
$u:M\to N$ admits a null eigenvalue with multiplicity
$k\in\\{1,2,\cdots,n-1\\}$ independent of the choice of fibre, then $|{\rm
Hess}\,u|^{2}\geq\frac{|\tau(u)|^{2}}{n-k}$, consequently
$\Delta_{V}|\mathrm{d}u|^{2}\geq\frac{2(m-k)}{(n-k)(m-n)}|\mathrm{d}u(V)|^{2}\geq
0$ under $m\in]-\infty,\,k\,]$. ∎

######  Remark 4.4.

  1. (i)

For any smooth function $u:M\to\mathbb{R}$, the assertion of Lemma 4.1 is
well-known:

$\displaystyle\frac{1}{2}\Delta_{V}|\nabla u|^{2}=|{\rm
Hess}\,u|^{2}+\langle\nabla\Delta_{V}u,\nabla
u\rangle+{\operatorname{Ric}}_{V}^{\infty}(\nabla u,\nabla u).$ (4.4)

From (4.4), for $m\in\,[-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$, we have

$\displaystyle\frac{1}{2}\Delta_{V}|\nabla
u|^{2}\geq\frac{(\Delta_{V}u)^{2}}{n}+\frac{2\Delta_{V}u\langle V,\nabla
u\rangle}{n}+\langle\nabla\Delta_{V}u,\nabla
u\rangle+{\operatorname{Ric}}_{V}^{m}(\nabla u,\nabla u),$ (4.5)

which has been known for $m\in\,[\,n,+\infty\,]$. The inequality (4.5) also
yields that any $V$-harmonic function $u:M\to\mathbb{R}$ satisfies
$\Delta_{V}|\nabla u|^{2}\geq 0$ under ${\operatorname{Ric}}_{V}^{m}\geq 0$
with $m\in\,[-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$.

  2. (ii)

When $m\in\,[\,n,+\infty\,]$, the third named author proved the following
Bochner inequality for $V=\nabla f$ (see p. 1303 in [34])

$\displaystyle\frac{1}{2}\Delta_{V}|\nabla
u|^{2}\geq\operatorname{Ric}^{m}_{V}(\nabla u,\nabla
u)+\frac{(\Delta_{V}u)^{2}}{m}+\langle\nabla\Delta_{V}u,\nabla u\rangle.$
(4.6)

In [49], Ohta extended $(\ref{eq:BochnernegativeN})$ to the case
$m\in\,[\,-\infty,\,0\,[$, which is a special case of (4.7) below for
${\varepsilon}=1$. On the other hand, Kolesnikov-Milman [26] also obtained
(4.6) including the case $m=0$ with the convention $1/0-:=-\infty$. Our
inequality (4.5) is different from (4.6) at least for the case $m=0$. The
inequality (4.6) can be proved for general $V$. Indeed, by the same argument
as used for the proof of $(\ref{eq:BochnernegativeN})$ in [34, p. 1303] for
$V=\nabla f$ and $m\in[\,n,\,+\infty\,]$, the Bochner identity (4.2) for
smooth function $u:M\to\mathbb{R}$ and the elementary inequality
$(a+b)^{2}\geq\frac{a^{2}}{1+\alpha}-\frac{b^{2}}{\alpha}$ for
$a,b\in\mathbb{R}$ with $\alpha\in\,]-\infty,\,-1\,[\,\cup\,]\,0,\,+\infty\,[$
together show

$\displaystyle\frac{1}{2}\Delta_{V}|\nabla u|^{2}$
$\displaystyle\geq{\operatorname{Ric}}_{V}^{\infty}(\nabla u,\nabla
u)+\frac{(\Delta_{V}u+\langle V,\nabla
u\rangle)^{2}}{n}+\langle\nabla\Delta_{V}u,\nabla u\rangle$
$\displaystyle\geq{\operatorname{Ric}}_{V}^{m}(\nabla u,\nabla
u)+\frac{\langle V,\nabla
u\rangle^{2}}{m-n}+\frac{(\Delta_{V}u)^{2}}{n(1+\alpha)}-\frac{\langle
V,\nabla u\rangle^{2}}{n\alpha}+\langle\nabla\Delta_{V}u,\nabla u\rangle$
$\displaystyle={\operatorname{Ric}}_{V}^{m}(\nabla u,\nabla
u)+\frac{(\Delta_{V}u)^{2}}{m}+\langle\nabla\Delta_{V}u,\nabla u\rangle,$

where we take $\alpha=\frac{m}{n}-1<-1$ for $m\in]-\infty,\,0\,[$ in the last
equality.

  3. (iii)

When $V=\nabla f$ and $u:M\to\mathbb{R}$ is a smooth function, we can deduce
the following inequality for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$
and ${\varepsilon}\in\mathbb{R}$ with

$\displaystyle\varepsilon=0\,\text{ for
}\,m=0,\,\,\,\varepsilon\in\left]-\sqrt{\frac{m}{m-n}},\sqrt{\frac{m}{m-n}}\right[\,\text{
for }\,m\neq 0,n,\,\,\,\varepsilon\in\mathbb{R}\,\text{ for }\,m=n:$
$\displaystyle\frac{1}{2}\Delta_{f}|\nabla u|^{2}$
$\displaystyle=\operatorname{Ric}^{m}_{f}(\nabla u,\nabla
u)+\frac{1}{n}\left(1-{\varepsilon}^{2}\frac{m-n}{m}\right)\,(\Delta_{f}u)^{2}$
$\displaystyle\hskip
85.35826pt+e^{-\frac{2(1-{\varepsilon})f}{n}}\langle\nabla(e^{\frac{2(1-{\varepsilon})f}{n}}\,\Delta_{f}u),\nabla
u\rangle$ $\displaystyle\hskip
28.45274pt+\frac{1}{n}\left(\sqrt{\frac{m}{m-n}}\langle\nabla f,\nabla
u\rangle+{\varepsilon}\sqrt{\frac{m-n}{m}}\Delta_{f}u\right)^{2}+\left\|\nabla^{2}u-\frac{\Delta
u}{n}g\right\|^{2}$ $\displaystyle\geq\operatorname{Ric}^{m}_{f}(\nabla
u,\nabla
u)+\frac{1}{n}\left(1-{\varepsilon}^{2}\frac{m-n}{m}\right)(\Delta_{f}u)^{2}$
$\displaystyle\hskip
85.35826pt+e^{-\frac{2(1-{\varepsilon})f}{n}}\langle\nabla(e^{\frac{2(1-{\varepsilon})f}{n}}\,\Delta_{f}u),\nabla
u\rangle.$ (4.7)

The inequality (4.7) is meaningful even if $m=0$ and ${\varepsilon}=0$ with
the convention ${\varepsilon}^{2}\frac{m-n}{m}=0$. Our inequality (4.5) for
$V=\nabla f$ is a consequence of (4.7) for ${\varepsilon}=0$.

  4. (iv)

When $u:M\to N$ is $V$-harmonic, (4.1) is also proved in [9, Lemma 1].

######  Lemma 4.5.

Suppose that $(N,h)$ is a Cartan-Hadamard manifold. Let $u:M\to N$ be a
$V$-harmonic map. Then $\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)\geq 2|\mathrm{d}u|^{2}$ for any $o\in N$.

###### Proof.

By using [1, 1.177], the Hessian of the composite map of
${\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(\cdot,o):N\to\mathbb{R}$ and $u:M\to
N$ is given by

$\displaystyle{\rm Hess}\,{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)(e_{i},e_{j})={\rm Hess}^{N}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{j}))+\mathrm{d}({\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o))({\rm Hess}\,u(e_{i},e_{j})).$ (4.8)

By taking a trace to (4.8), we have

$\displaystyle\Delta{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)=\sum_{i=1}^{n}{\rm Hess}^{N}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i}))+\mathrm{d}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\tau(u)).$

Thus we obtain

$\displaystyle\Delta_{V}{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(u,o)$
$\displaystyle=\sum_{i=1}^{n}{\rm Hess}^{N}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i}))+\mathrm{d}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\tau_{V}(u))$ $\displaystyle=\sum_{i=1}^{n}{\rm
Hess}^{N}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(\cdot,o)(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i}))$
$\displaystyle\geq
2\sum_{i=1}^{n}\langle\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i})\rangle=2|\mathrm{d}u|^{2}.$

In the last inequality above, we use the Greene-Wu Hessian comparison theorem
under ${\rm Sect}_{N}\leq 0$ (see [15, Theorem A], [58, p. 227], [54, Chapter
IV, Lemma 2.9]), where we use the completeness and simple connectedness of
$(N,h)$. Note that ${\rm Hess}^{N}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(\cdot,o)^{2}(e_{\alpha}^{\prime},e_{\beta}^{\prime})=2\delta_{\alpha\beta}$
for $o\in N=\mathbb{R}^{\ell}$ and $\alpha,\beta=1,2,\cdots,\ell$ with
$e_{\alpha}^{\prime}=\left(\partial/\partial y^{\alpha}\right)_{y}$,
$y\in\mathbb{R}^{\ell}$. ∎

######  Remark 4.6.

In the assertion of Lemma 4.5, we can relax the simple connectedness of
$(N,h)$. Precisely to say, if ${\rm Sect}_{N}\leq 0$ and ${\rm Im}(u)\cap{\rm
Cut}(o)=\emptyset$, then $\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{2}(u,o)\geq 2|\mathrm{d}u|^{2}$. This is a consequence of Greene-
Wu’s Hessian comparison theorem (e.g. [54, Chapter IV, Lemma 2.9]).

##  5 Polynomial growth of differential map for $V$-harmonic maps

In this section, we show that for each $i=1,2,3$, if a smooth $V$-harmonic map
$u:M\to N$ has a growth condition (G$i$) under (B3) and ${\rm Ric}_{V}^{m}\geq
0$ with $m\in[-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$, then the differential
map $\mathrm{d}u:TM\to TN$ also has the same growth (G$i$). For this, we need
the following lemma.

######  Lemma 5.1.

Assume ${\rm Ric}_{V}^{m}\geq 0$ with
$m\in[-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$. Suppose that $u:M\to N$ is a
smooth $V$-harmonic map and (B3). Then for each $p\in M$, there exist
$C=C(p,D)>0$ and $A=A(p,D)>0$ such that

$\displaystyle\sup_{x\in B_{a}(p)}|\mathrm{d}u|^{2}(x)\leq
C(m_{u}(2a)+1)^{2}\qquad\text{ for }\qquad a>A.$

Here $m_{u}(2a):=\sup_{B_{2a}(p)}{\color[rgb]{0,0,0}{{\sf d}}}_{N}(u,o)$. In
particular, for each $i=1,2,3$ if (G$i$) holds for $u$, then it holds for
$|\mathrm{d}u|$.

###### Proof.

Let $\tilde{\varphi}(r)$ be a $C^{2}$-function defined on $[\,0,\,+\infty\,[$
such that $\tilde{\varphi}(r)>0$ for $r\in[0,2[$ and

$\displaystyle\tilde{\varphi}(r):=\left\\{\begin{array}[]{cl}1&\quad\text{ if
}\quad r\in[\,0,\,1\,],\\\ 0&\quad\text{ if }\quad
r\in[\,2,+\infty\,[\end{array}\right.$

with

$\displaystyle
0\geq\frac{\tilde{\varphi}^{\prime}(r)}{\sqrt{\tilde{\varphi}(r)}}\geq-C_{1}$

and

$\displaystyle\tilde{\varphi}^{\prime\prime}(r)\geq-C_{2}$

for some constants $C_{1},C_{2}>0$ (see [37, proof of Theorem 1.2] or [34,
proof of Theorem 1.5]). We set

$\displaystyle\varphi(x):=\tilde{\varphi}\left(\frac{r_{p}(x)}{a}\right).$

We then see that on $B_{2a}(p)\setminus{\rm Cut}(p)$

$\displaystyle|\nabla\varphi|\leq\frac{C_{1}}{a}\quad\text{ and
}\quad\frac{|\nabla\varphi|^{2}}{\varphi}\leq\frac{C_{1}^{2}}{a^{2}},$ (5.1)

and

$\displaystyle\Delta_{V}\varphi=\frac{\tilde{\varphi}^{\prime}(r_{p}/a)\Delta_{V}r_{p}}{a}+\frac{\tilde{\varphi}^{\prime\prime}(r_{p}/a)|\nabla
r_{p}|^{2}}{a^{2}}\geq-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}$ (5.2)

on $B_{2a}(p)$. Here we use (B3). Set $b:=\sqrt{5}(m_{u}(2a)+1)$,
$F(x)=\frac{|\mathrm{d}u|^{2}(x)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(x),o)}$ and $G:=\varphi F$. Note that $F\geq 0$ on
$B_{2a}(p)$ because $b^{2}-{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(u(x),o)\geq
5>0$ on $B_{2a}(p)$. Moreover,

$\displaystyle\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\leq\frac{m_{u}(2a)^{2}}{b^{2}-m_{u}(2a)^{2}}\leq\frac{1}{4}\quad\text{
on }\quad B_{2a}(p).$ (5.3)

Then, $G$ has a maximum at $\bar{x}\in B_{2a}(p)$. At this point $r_{p}$ may
not be twice differentiable. We will remedy this by the Calabi’s argument (see
[8]): Let $\gamma:[0,r_{p}(\bar{x})]\to M$ be the minimal geodesic joining
$\gamma_{0}=p$ and $\gamma_{r_{p}(\bar{x})}=\bar{x}$. Let ${\varepsilon}>0$ be
small. Along $\gamma$, $\gamma_{{\varepsilon}}$ is not conjugate to $\bar{x}$.
So there exists a geodesic cone $\mathscr{C}$ with vertex
$\gamma_{{\varepsilon}}$ and contains a neighborhood of $\bar{x}$ such that
$x\mapsto r_{\gamma_{{\varepsilon}}}(x)$ is smooth inside $\mathscr{C}$. Let
$\tilde{r}_{{\varepsilon}}(x):=r_{\gamma_{{\varepsilon}}}(x)+{\varepsilon}$.
Then triangle inequality states that

$\displaystyle r_{p}(x)\leq\tilde{r}_{{\varepsilon}}(x)\quad\text{ for }\quad
x\in\mathscr{C},\qquad r_{p}(\bar{x})=\tilde{r}_{{\varepsilon}}(\bar{x}).$
(5.4)

Then
$\tilde{G}_{{\varepsilon}}:=\tilde{\varphi}\left(\frac{\tilde{r}_{{\varepsilon}}}{a}\right)F$
is smooth near $\bar{x}$ and also attains the same maximum value at $\bar{x}$.
Letting ${\varepsilon}\to 0$, we may assume without loss of generality that
$r_{p}$ is already smooth near $\bar{x}$.

If $F(\bar{x})=0$, we have $|\mathrm{d}u|\equiv 0$ on $B_{2a}(p)$. Then we
have the conclusion. So we may assume $F(\bar{x})>0$, hence
$|\mathrm{d}u|(\bar{x})>0$. At $\bar{x}$, we have

$\displaystyle 0\geq\Delta_{V}(\varphi F)\quad\text{ and }\quad\nabla(\varphi
F)=0.$

The latter one is equivalent to $0=\frac{\nabla\varphi}{\varphi}+\frac{\nabla
F}{F}$ at $\bar{x}$, i.e.,

$\displaystyle\frac{\nabla{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}=-\frac{\nabla\varphi}{\varphi}-\frac{\nabla|\mathrm{d}u|^{2}}{|\mathrm{d}u|^{2}}$
(5.5)

at $\bar{x}$. These imply that at $\bar{x}$ with (5.2)

$\displaystyle 0$
$\displaystyle\geq(\Delta_{V}\varphi)F+2\langle\nabla\varphi,\nabla
F\rangle+\varphi(\Delta_{V}F)$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}F+2\langle\nabla\varphi,\nabla(\varphi
F)\rangle-2\frac{|\nabla\varphi|^{2}}{\varphi}F$ $\displaystyle\hskip
28.45274pt+\varphi\left(\frac{\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}+2\left\langle\nabla|\mathrm{d}u|^{2},\nabla\frac{1}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\right\rangle+|\mathrm{d}u|^{2}\Delta_{V}\frac{1}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\right)$
$\displaystyle=-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}F-2\frac{|\nabla\varphi|^{2}}{\varphi}F$
$\displaystyle\hskip
56.9055pt+\varphi\left\\{\frac{\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}+2\left\langle\nabla|\mathrm{d}u|^{2},\frac{\nabla{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{2}}\right\rangle\right.$ $\displaystyle\hskip
113.81102pt\left.+|\mathrm{d}u|^{2}\left(\frac{2|\nabla{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)|^{2}}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{3}}+\frac{\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{2}}\right)\right\\}.$

Substituting (5.5) into the the second term in the brace above, the last
equals

$\displaystyle-$
$\displaystyle\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}F-2\frac{|\nabla\varphi|^{2}}{\varphi}F$
$\displaystyle\hskip
28.45274pt+\varphi\frac{\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}-\frac{2\varphi}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\left(\frac{|\nabla|\mathrm{d}u|^{2}|^{2}}{|\mathrm{d}u|^{2}}-\frac{\langle\nabla|\mathrm{d}u|^{2},\nabla\varphi\rangle}{\varphi}\right)$
$\displaystyle\hskip
56.9055pt+\varphi\frac{|\mathrm{d}u|^{2}}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\left(\frac{2|\nabla{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)|^{2}}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{2}}+\frac{\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\right).$

By Lemma 4.5, we know $\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{2}(u,o)\geq 2|\mathrm{d}u|^{2}\geq 0$, the last equation is greater
than

$\displaystyle-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}F-2\frac{|\nabla\varphi|^{2}}{\varphi}F$
$\displaystyle-2\varphi
F\frac{|\nabla|\mathrm{d}u|^{2}|^{2}}{|\mathrm{d}u|^{4}}-2\varphi
F\left\langle\frac{\nabla|\mathrm{d}u|^{2}}{|\mathrm{d}u|^{2}},\frac{\nabla\varphi}{\varphi}\right\rangle$
$\displaystyle+\varphi F\left(\frac{8{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)|\mathrm{d}u|^{2}}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{2}}+\frac{2|\mathrm{d}u|^{2}}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\right).$

Multiplying $\varphi(\bar{x})$ in both sides, we have

$\displaystyle 0$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}\varphi
F-2\frac{|\nabla\varphi|^{2}}{\varphi}\varphi F$ $\displaystyle\hskip
28.45274pt-2\varphi\cdot\varphi
F\cdot\frac{|\nabla|\mathrm{d}u|^{2}|^{2}}{|\mathrm{d}u|^{4}}-2\varphi
F\left\langle\frac{\nabla|\mathrm{d}u|^{2}}{|\mathrm{d}u|^{2}},\nabla\varphi\right\rangle$
$\displaystyle\hskip 56.9055pt+\varphi\cdot\varphi
F\left(\frac{8{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}F+2F\right)$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}\varphi
F-2\frac{|\nabla\varphi|^{2}}{\varphi}\varphi F$ $\displaystyle\hskip
28.45274pt-2\varphi\cdot\varphi
F\cdot\frac{|\nabla|\mathrm{d}u|^{2}|^{2}}{|\mathrm{d}u|^{4}}-2\varphi
F\frac{|\nabla|\mathrm{d}u|^{2}|}{|\mathrm{d}u|^{2}}|\nabla\varphi|$
$\displaystyle\hskip 56.9055pt+\varphi\cdot\varphi
F\left(\frac{8{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}F+2F\right).$

Applying (5.1) and (5.5), we have

$\displaystyle 0$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}}{a^{2}}\varphi
F-2\frac{C_{1}^{2}}{a^{2}}\varphi F$ $\displaystyle\hskip
28.45274pt-2\varphi^{2}F\left(\frac{4{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)|\mathrm{d}u|^{2}}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))^{2}}+\frac{4{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u,o)|\mathrm{d}u||\nabla\varphi|}{(b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o))\varphi}+\frac{|\nabla\varphi|^{2}}{\varphi^{2}}\right)$
$\displaystyle\hskip 56.9055pt-2\varphi
F\left(\frac{|\nabla\varphi|^{2}}{\varphi}+\frac{2{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u,o)|\mathrm{d}u|}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\right)$ $\displaystyle\hskip 85.35826pt+\varphi
F\left(\frac{8{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\varphi F+2\varphi F\right)$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}+2C_{1}^{2}}{a^{2}}\varphi F$
$\displaystyle\hskip
28.45274pt-2\varphi^{2}F\left(4\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}F+4\sqrt{\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}}\sqrt{F}\frac{|\nabla\varphi|}{\varphi}+\frac{|\nabla\varphi|^{2}}{\varphi^{2}}\right)$
$\displaystyle\hskip 56.9055pt-2\varphi
F\cdot\frac{|\nabla\varphi|^{2}}{\varphi}-4\varphi
F|\nabla\varphi|\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u,o)|\mathrm{d}u|}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}$ $\displaystyle\hskip 85.35826pt+\varphi
F\left(\frac{8{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{2}{\,}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\varphi F+2\varphi F\right).$

Then

$\displaystyle 0$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}+2C_{1}^{2}}{a^{2}}\varphi F$
$\displaystyle\hskip 28.45274pt-8(\varphi F)^{2}\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}-8(\varphi F)\sqrt{\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}}\sqrt{\varphi
F}\frac{|\nabla\varphi|}{\sqrt{\varphi}}-4(\varphi
F)\frac{|\nabla\varphi|^{2}}{\varphi}$ $\displaystyle\hskip
56.9055pt-4(\varphi
F)\cdot\frac{|\nabla\varphi|}{\sqrt{\varphi}}\sqrt{\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}}\sqrt{\varphi F}+\varphi
F\left(8\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\varphi F+2\varphi F\right)$
$\displaystyle=-\frac{C_{1}D(1+2a^{2})+C_{2}+2C_{1}^{2}}{a^{2}}\varphi F$
$\displaystyle\hskip 28.45274pt-4(\varphi
F)\frac{|\nabla\varphi|^{2}}{\varphi}-12(\varphi
F)\cdot\frac{|\nabla\varphi|}{\sqrt{\varphi}}\sqrt{\frac{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}}\sqrt{\varphi F}+2(\varphi F)^{2}.$

Applying (5.1) and (5.3), and dividing by
$G(\bar{x})=\varphi(\bar{x})F(\bar{x})>0$, we have at $\bar{x}$

$\displaystyle 0$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}+6C_{1}^{2}}{a^{2}}-\frac{6C_{1}}{a}\sqrt{\varphi
F}+2\varphi F$
$\displaystyle\geq-\frac{C_{1}D(1+2a^{2})+C_{2}+6C_{1}^{2}}{a^{2}}-\frac{6C_{1}}{a}\sqrt{\varphi
F}+\varphi F.$

This implies that

$\displaystyle\sqrt{G(\bar{x})}\leq\sqrt{\frac{12C_{1}^{2}}{a^{2}}+\frac{C_{1}D(1+2a^{2})+C_{2}+6C_{1}^{2}}{a^{2}}}=O(1).$

Therefore,

$\displaystyle\frac{1}{b^{2}}\sup_{B_{a}(p)}|\mathrm{d}u|^{2}\leq\sup_{B_{a}(p)}\frac{\varphi|\mathrm{d}u|^{2}}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}\leq\sup_{B_{2a}(p)}\frac{\varphi|\mathrm{d}u|^{2}}{b^{2}-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}=G(\bar{x})=O(1),$

hence

$\displaystyle\sup_{B_{a}(p)}|\mathrm{d}u|^{2}\leq
b^{2}O(1)=5(m_{u}(2a)+1)^{2}O(1).$

∎

Finally, we show the following lemma for the upper bound.

######  Lemma 5.2.

We have the following:

  1. (i)

Suppose (B1). Then, there exists $D>0$ such that for any relatively compact
open set $G$

$\displaystyle{\mathbb{E}}_{x}\left[r_{p}^{2}(X_{t\land\tau_{G}})\right]\leq
r_{p}^{2}(x)+2(1+D)t$ (5.6)

holds for all $x\in M$.

  2. (ii)

Suppose (B2). Then, there exists $D>0$ such that for any relatively compact
open set $G$

$\displaystyle{\mathbb{E}}_{x}\left[r_{p}(X_{t\land\tau_{G}})-\log(1+r_{p}(X_{t\land\tau_{G}}))\right]\leq
r_{p}(x)-\log(1+r_{p}(x))+(1+D)t$ (5.7)

holds for all $x\in M$.

  3. (iii)

Suppose (B3). Then, there exists $D>0$ such that for any relatively compact
open set $G$

$\displaystyle{\mathbb{E}}_{x}\left[\log(1+r_{p}^{2}(X_{t\land\tau_{G}}))\right]\leq\log(1+r_{p}^{2}(x))+2(1+D)t$
(5.8)

holds for all $x\in M$.

###### Proof.

By Itô’s formula, we have

$\displaystyle r_{p}^{2}(X_{t})$
$\displaystyle=r_{p}^{2}(X_{0})+2\sqrt{2}\int_{0}^{t}r_{p}(X_{s})\mathrm{d}\beta_{s}+2t$
$\displaystyle\hskip
28.45274pt+2\int_{0}^{t}r_{p}(X_{s})\Delta_{V}r_{p}(X_{s}){\bf
1}_{M\setminus{\rm
Cut}(p)}(X_{s})\mathrm{d}s-2\int_{0}^{t}r_{p}(X_{s})\mathrm{d}L_{s},$
$\displaystyle r_{p}(X_{t})-\log(1+r_{p}(X_{t}))$
$\displaystyle=r_{p}(X_{0})-\log(1+r_{p}(X_{0}))$ $\displaystyle\hskip
14.22636pt+\sqrt{2}\int_{0}^{t}\frac{r_{p}(X_{s})\mathrm{d}\beta_{s}}{1+r_{p}(X_{s})}+\int_{0}^{t}\frac{\mathrm{d}s}{(1+r_{p}(X_{s}))^{2}}$
$\displaystyle\hskip
28.45274pt+\int_{0}^{t}\frac{r_{p}(X_{s})\Delta_{V}r_{p}(X_{s})}{1+r_{p}(X_{s})}{\bf
1}_{M\setminus{\rm
Cut}(p)}(X_{s})\mathrm{d}s-\int_{0}^{t}\frac{r_{p}(X_{s})\mathrm{d}L_{s}}{1+r_{p}(X_{s})},$
$\displaystyle\log(1+r_{p}^{2}(X_{t}))=\log$
$\displaystyle(1+r_{p}^{2}(X_{0}))+2\sqrt{2}\int_{0}^{t}\frac{r_{p}(X_{s})}{1+r_{p}^{2}(X_{s})}\mathrm{d}\beta_{s}$
$\displaystyle+2\int_{0}^{t}\frac{1-r_{p}^{2}(X_{s})}{(1+r_{p}^{2}(X_{s}))^{2}}\mathrm{d}s+2\int_{0}^{t}\frac{r_{p}(X_{s})\Delta_{V}r_{p}(X_{s})}{1+r_{p}^{2}(X_{s})}{\bf
1}_{M\setminus{\rm Cut}(p)}(X_{s})\mathrm{d}s$ $\displaystyle\hskip
56.9055pt-\int_{0}^{t}\frac{2r_{p}(X_{s})}{1+r_{p}^{2}(X_{s})}\mathrm{d}L_{s},$

$t\in[\,0,\,+\infty\,[$, ${\mathbb{P}}_{x}$-a.s. for $x\in M$. From these
formulae, we can obtain the conclusion. ∎

##  6 Lower bound

Let $(X_{t})_{t\geq 0}$ be the $\Delta_{V}$-diffusion process on $M$. In this
section, we show a lower bound of ${\mathbb{E}}_{x}[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o)]$ for a reference point $o\in N$
and $V$-harmonic map $u:M\to N$ with the increasing sequence $\\{G_{n}\\}$ of
relatively compact open sets satisfying $\overline{G_{n}}\subset
G_{n+1}\subset M$ for $n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$.
The proof of the lower estimate ${\mathbb{E}}_{x}[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(X_{t}),o)^{2}]\geq{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(x),o)^{2}+|{\rm d}u|^{2}(x)t$ for harmonic map $u:M\to N$ in terms
of Brownian motion $(X_{t})_{t\geq 0}$ on $M$ under
${\operatorname{Ric}}_{g}\geq 0$ in [59] is incomplete, because of the lack of
the gradient estimate for harmonic map $u$ like Lemma 5.1. In [59], such a
gradient estimate for harmonic function $u:M\to\mathbb{R}$ due to Li-Yau [37]
was only used.

######  Lemma 6.1.

Assume that ${\operatorname{Ric}}_{V}^{m}\geq 0$ with
$m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$ and ${\rm Sect}_{N}\leq 0$.
Let $u:M\to N$ be a $V$-harmonic map. Suppose (G1) for $u$ and (B3). Then,

$\displaystyle|\mathrm{d}u|^{2}(x)\leq{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{2}(X_{t})\right]$
(6.1)

holds for all $x\in M$.

###### Proof.

Applying Corollary 4.3 to the $V$-harmonic map $u$, we have

$\displaystyle\Delta_{V}|\mathrm{d}u|^{2}\geq 0\quad\text{ on }\quad M.$

Then Itô’s formula (3.1) tells us that there exists a local martingale
$m_{t}^{|\mathrm{d}u|^{2}}$ such that

$\displaystyle|\mathrm{d}u|^{2}(X_{t})-|\mathrm{d}u|^{2}(X_{0})=m_{t}^{|\mathrm{d}u|^{2}}+\int_{0}^{t}\Delta_{V}|\mathrm{d}u|^{2}(X_{s})\mathrm{d}s,\qquad
t\in\,[\,0,\,+\infty\,[$

holds $\mathbb{P}_{x}$-a.s. for all $x\in M$. In fact the local
${\mathbb{P}}_{x}$-martingale $m_{t}^{|\mathrm{d}u|^{2}}$ is given by

$\displaystyle m_{t}^{|\mathrm{d}u|^{2}}:=\sqrt{2}\int_{0}^{t}\left\langle
U_{s}^{-1}\nabla|\mathrm{d}u|^{2}(X_{s}),\mathrm{d}B_{s}\right\rangle.$

Let $G$ be a relatively compact open set. We see that

$\displaystyle|\mathrm{d}u|^{2}(X_{t\land\tau_{G}})-|\mathrm{d}u|^{2}(X_{0})=m_{t\land\tau_{G}}^{|\mathrm{d}u|^{2}}+\int_{0}^{t\land\tau_{G}}\Delta_{V}|\mathrm{d}u|^{2}(X_{s})\mathrm{d}s$

holds $\mathbb{P}_{x}$-a.s. for all $x\in M$. Since
${\mathbb{E}}_{x}[m_{t\land\tau_{G}}^{|\mathrm{d}u|^{2}}]=0$, we have

$\displaystyle|\mathrm{d}u|^{2}(x)\leq{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{2}(X_{t\land\tau_{G}})\right]$

for $x\in M$. Now we consider an increasing sequence $\\{G_{n}\\}$ of
relatively compact open sets satisfying $\overline{G_{n}}\subset
G_{n+1}\subset M$ for $n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$. We
prove that

$\displaystyle\sup_{n\in\mathbb{N}}{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{4}(X_{t\land\tau_{G_{n}}})\right]<\infty$
(6.2)

holds for all $x\in M$. By Lemma 5.1 and (G1) for $u$, we have (G1) for
$|\mathrm{d}u|$, hence there exists $A=A(p)>0$ such that

$\displaystyle|\mathrm{d}u|^{4}(x)\leq r_{p}^{4}(x)\quad\text{ for }\quad
r_{p}(x)>A.$

Then

$\displaystyle{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{4}(X_{t\land\tau_{G_{n}}})\right]$
$\displaystyle={\mathbb{E}}_{x}\left[|\mathrm{d}u|^{4}(X_{t\land\tau_{G_{n}}}):r_{p}(X_{t\land\tau_{G_{n}}})\leq
A\right]$ $\displaystyle\hskip
56.9055pt+{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{4}(X_{t\land\tau_{G_{n}}}):r_{p}(X_{t\land\tau_{G_{n}}})>A\right]$
$\displaystyle\leq\sup_{z\in
B_{A}(p)}|\mathrm{d}u|^{4}(z)+{\mathbb{E}}_{x}[r_{p}^{4}(X_{t\land\tau_{G_{n}}})].$

By (3.15)

$\displaystyle{\mathbb{E}}_{x}\left[|\mathrm{d}u|^{4}(X_{t\land\tau_{G_{n}}})\right]$
$\displaystyle\leq\sup_{z\in
B_{A}(p)}|\mathrm{d}u|^{4}(z)+r_{p}(x)^{4}+4(3+D)\int_{0}^{t}{\mathfrak{D}}(s)\mathrm{d}s$
$\displaystyle\hskip
28.45274pt+4De^{4Dt}\int_{0}^{t}\left(r_{p}^{4}(x)+4(D+3)\int_{0}^{s}{\mathfrak{D}}(u)\mathrm{d}u\right)e^{-4Ds}\mathrm{d}s$

holds for all $x\in M$. This implies (6.2). Therefore
$\\{|\mathrm{d}u|^{2}(X_{t\land\tau_{G_{n}}})\\}_{n=1}^{\infty}$ is uniformly
${\mathbb{P}}_{x}$-integrable for all $x\in M$. Since

$\displaystyle{\mathbb{P}}_{x}\left(|\mathrm{d}u|(X_{t})=\lim_{n\to\infty}|\mathrm{d}u|(X_{t\land\tau_{G_{n}}})\right)=1\quad\text{
for \ all }\quad x\in M$

by Corollary 3.5 and Theorem 3.1,
$\\{|\mathrm{d}u|(X_{t\land\tau_{G_{n}}})\\}_{n=1}^{\infty}$
$L^{2}({\mathbb{P}}_{x})$-converges to $|\mathrm{d}u|(X_{t})$ for all $x\in
M$, hence

$\displaystyle|\mathrm{d}u|^{2}(x)\leq\lim_{n\to\infty}{\mathbb{E}}_{x}[|\mathrm{d}u|^{2}(X_{t\land\tau_{G_{n}}})]={\mathbb{E}}_{x}[|\mathrm{d}u|^{2}(X_{t})]$

for all $x\in M$. ∎

######  Lemma 6.2.

Assume that ${\operatorname{Ric}}_{V}^{m}\geq 0$ for
$m\in\,[-\infty,\,0\,]\,\cup\,[\,n,+\infty\,]$ and $(N,h)$ is a Cartan-
Hadamard manifold. Let $u:M\to N$ be a $V$-harmonic map. Suppose (G1) for $u$
and (B3). Then, for any increasing sequence $\\{G_{n}\\}$ of relatively
compact open sets satisfying $\overline{G_{n}}\subset G_{n+1}\subset M$ for
$n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$,

$\displaystyle
2t|\mathrm{d}u|^{2}(x)\leq\sup_{n\in\mathbb{N}}{\mathbb{E}}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o)\right]$ (6.3)

for all $x\in M$.

###### Proof.

Itô’s formula (3.1) tells us that there exists a local martingale
$m_{t}=m_{t}^{{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(u,o)}$ such that

$\displaystyle{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t}),o)-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{0}),o)=m_{t}+\int_{0}^{t}\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{s}),o)\mathrm{d}s,\qquad t\in\,[\,0,\,\infty\,[$

holds $\mathbb{P}_{x}$-a.s. for all $x\in M$. In fact $m_{t}$ is given by

$\displaystyle m_{t}:=\sqrt{2}\int_{0}^{t}\left\langle
U_{s}^{-1}\nabla{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{s}),o),\mathrm{d}B_{s}\right\rangle.$

Let $G$ be a relatively compact open set. By
$\mathbb{E}_{x}[m_{t\land\tau_{G}}]=0$, we have

$\displaystyle\mathbb{E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G}}),o)\right]-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(x),o)=\mathbb{E}_{x}\left[\int_{0}^{t\land\tau_{G}}\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)(X_{s})\mathrm{d}s\right].$ (6.4)

By (6.1), Corollary 3.5 and Theorem 3.1, we can deduce that

$\displaystyle 2t|\mathrm{d}u|^{2}(x)$ $\displaystyle\leq
2\int_{0}^{t}\mathbb{E}_{x}[|\mathrm{d}u|^{2}(X_{s})]\mathrm{d}s$
$\displaystyle=2{\mathbb{E}}_{x}\left[\int_{0}^{t}|\mathrm{d}u|^{2}(X_{s})\mathrm{d}s\right]$
$\displaystyle=2\sup_{n\in\mathbb{N}}\mathbb{E}_{x}\left[\int_{0}^{t\land\tau_{G_{n}}}|\mathrm{d}u|^{2}(X_{s})\mathrm{d}s\right]$
$\displaystyle\leq\sup_{n\in\mathbb{N}}\mathbb{E}_{x}\left[\int_{0}^{t\land\tau_{G_{n}}}\Delta_{V}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)(X_{s})\mathrm{d}s\right]\quad\text{(by
Lemma~{}\ref{lem:LapalcianDistaHarm})}$
$\displaystyle\leq\sup_{n\in\mathbb{N}}\mathbb{E}_{x}[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o)]-{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(x),o)\quad(\text{by \eqref{eq:EqBoth}})$

for all $x\in M$. ∎

##  7 Proof of Theorem 1.7

###### Proof of Theorem 1.7.

Under ${\rm Ric}_{V}^{m}\geq 0$, for each $i=1,2,3$, the condition (A$i$)
implies (B$i$) by Theorem 2.3. Suppose that $u:M\to N$ is a $V$-harmonic map
having growth conditions. Since $o(\sqrt{a})$ (resp. $o(\sqrt{\log a})$)
implies $o(\sqrt{a-\log(1+a)})$ (resp. $o(\sqrt{\log(1+a^{2})}$), for any
${\varepsilon}>0$, there exists $A>0$ such that for any $a>A$,

$\displaystyle m_{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}(a)\leq{\varepsilon}\left\\{\begin{array}[]{cl}a&\text{
under {\bf(G1)}},\\\ \sqrt{a-\log(1+a)}&\text{ under {\bf(G2)}},\\\
\sqrt{\log(1+a^{2})}&\text{ under {\bf(G3)}}.\end{array}\right.$ (7.4)

Moreover, for $a\leq A$, $m_{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}(a)\leq m_{{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u,o)}(A)$. We find that

$\displaystyle{\bf E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o)\right]$ $\displaystyle={\bf
E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o):r_{p}(X_{t\land\tau_{G_{n}}})\leq
A\right]$ $\displaystyle\hskip 8.5359pt+{\bf
E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o):r_{p}(X_{t\land\tau_{G_{n}}})>A\right]$
$\displaystyle\leq\sup_{z\in B_{A}(p)}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(z),o)$ $\displaystyle\hskip 28.45274pt+{\bf
E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o):r_{p}(X_{t\land\tau_{G_{n}}})>A\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:sublinearet}}}{{\leq}}\sup_{z\in
B_{A}(p)}{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(u(z),o)$ $\displaystyle\hskip
11.38092pt+{\varepsilon}^{2}\left\\{\begin{array}[]{lc}{\bf
E}_{x}\left[r_{p}^{2}(X_{t\land\tau_{G_{n}}})\right]&\text{ under
}\text{\bf(G1)},\\\ {\bf
E}_{x}\left[r_{p}(X_{t\land\tau_{G_{n}}})-\log(1+r_{p}(X_{t\land\tau_{G_{n}}}))\right]&\text{
under }\text{\bf(G2)},\\\ {\bf
E}_{x}\left[\log(1+r_{p}^{2}(X_{t\land\tau_{G_{n}}}))\right]&\text{ under
}\text{\bf(G3)}\end{array}\right.$ $\displaystyle\leq\sup_{z\in
B_{A}(p)}{\color[rgb]{0,0,0}{{\sf d}}}_{N}^{\,2}(u(z),o)$ $\displaystyle\hskip
11.38092pt+{\varepsilon}^{2}\left\\{\begin{array}[]{lc}r_{p}^{2}(x)+2(1+D)t&\text{
under }\text{\bf(B1)},\\\ r_{p}(x)-\log(1+r_{p}(x))+(1+D)t&\text{ under
}\text{\bf(B2)},\\\ \log(1+r_{p}^{2}(x))+2(1+D)t&\text{ under
}\text{\bf(B3)}\end{array}\right.$

for all $x\in M$. In the last inequality, we use Lemma 5.2. By Lemma 6.2,

$\displaystyle 2t|\mathrm{d}u|^{2}(x)$
$\displaystyle\leq\sup_{n\in\mathbb{N}}{\bf
E}_{x}\left[{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(X_{t\land\tau_{G_{n}}}),o)\right]$
$\displaystyle\leq\sup_{z\in B_{A}(p)}{\color[rgb]{0,0,0}{{\sf
d}}}_{N}^{\,2}(u(z),o)+{\varepsilon}^{2}(r_{p}^{2}(x)+2(1+D)t)$ (7.5)

for all $x\in M$ under (B$i$) and the growth condition (G$i$) for each
$i=1,2,3$. Dividing (7.5) by $t$ and letting $t\to\infty$, we see

$\displaystyle 2|\mathrm{d}u|^{2}(x)\leq 2{\varepsilon}^{2}(1+D)$

for all $x\in M$. Since ${\varepsilon}>0$ is arbitrary, we have
$|\mathrm{d}u|^{2}(x)=0$ for all $x\in M$. ∎

######  Remark 7.1.

By Remark 4.6, the conclusion of Theorem 1.7 under (A$i$) and (G$i$) for each
$i=1,2,3$ also holds if ${\rm Ric}_{V}^{m}\geq 0$, ${\rm Sect}_{N}\leq 0$ and
$u:M\to N$ is a $V$-harmonic map satisfying ${\rm Im}(u)\cap{\rm
Cut}(o)=\emptyset$ for some point $o\in N$.

######  Remark 7.2.

In Theorem 1.7, under (A$i$) and (G$i$) for each $i=1,2,3$, if further the
symmetric matrix $({\rm Hess}\,u(e_{i},e_{j}))$ for the $V$-harmonic map $u$
admits a null eigenvalue with multiplicity $k\in\\{1,2,\cdots,$ $n-1\\}$
independent of the choice of fibre, then the same conclusion holds under
${\operatorname{Ric}}_{V}^{m}\geq 0$ with $m\in]-\infty,\,k\,]$ and ${\rm
Sect}_{N}\leq 0$ with $\pi_{1}(N)=\\{e\\}$.

##  8 Proofs of Theorem 1.9, Corollaries 1.10, 1.11 and Theorem 1.12

Our proof relies on the convex geometry on the regular geodesic ball on
complete Rimannian manifolds with positive upper curvature, first introduced
by Kendall [23, 24]. In our situation, more precisely, there exists a convex
function $\Phi:B_{R}(o)\times B_{R}(o)\rightarrow\mathbb{R}$, such that for
any geodesic $\gamma$ in $B_{R}(o)\times B_{R}(o)$, there exists a positive
constant $C$,

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\Phi(\gamma(t))\geq
C\left(\frac{\mathrm{d}}{\mathrm{d}t}\Phi(\gamma(t))\right)^{2}.$ (8.1)

Moreover, $\Phi$ is a nonnegative and bounded function. It is also symmetric
$\Phi(x,y)=\Phi(y,x)$, and vanishes only on its diagonal, i.e. $\Phi(x,y)=0$
if and only if $x=y$. For the concrete construction of $\Phi(x,y)$, see [24].

######  Lemma 8.1.

Fix $q\in B_{R}(o)$, then $\Phi(u(x),q)$ can be considered as a function on
$M$. Moreover, we have

$\displaystyle\Delta_{V}\Phi(u(x),q)\geq 0.$ (8.2)

###### Proof.

Recall that for harmonic map $u$, we have

$\displaystyle\Delta_{V}\Phi(u(x),q)=\sum^{n}_{i=1}{\rm
Hess}^{N}\Phi(\cdot,q)(\mathrm{d}u(e_{i}),\mathrm{d}u(e_{i})).$ (8.3)

Here ${\rm Hess}^{N}\Phi(\cdot,q)$ is the Hessian of $\Phi(\cdot,q)$ with
respect to $(N,h)$. Notice that for a geodesic $\gamma(t)\in B_{R}(o)$, we
have

$\left.\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\right|_{t=t_{0}}\Phi(\gamma(t),q)={\rm
Hess}^{N}\Phi(\cdot,q)(\dot{\gamma}(t_{0}),\dot{\gamma}(t_{0})).$

Now we consider the geodesic $\gamma(t)$, such that for $\gamma(t_{0})=u(x)$
and $\dot{\gamma}(t_{0})=X$, where $X\in T_{u(x)}N$. Then,

$\displaystyle\left.\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\right|_{t=t_{0}}\Phi(\gamma(t),q)$
$\displaystyle={\rm Hess}^{N}\Phi(\cdot,q)(X,X),$
$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=t_{0}}\Phi(\gamma(t),q)$
$\displaystyle=\langle\nabla^{N}\Phi(\cdot,q),X\rangle.$

Therefore by (8.1) and (8.3), we derive that

$\displaystyle\Delta_{V}\Phi(u(x),q)\geq
C\sum^{n}_{i=1}\langle\nabla^{N}\Phi(u(x),q),\mathrm{d}u(e_{i})\rangle^{2}\geq
0.$

∎

Now we are in the position to prove Theorem 1.9.

###### Proof of Theorem 1.9.

Recall that ${\bf X}^{V}$ is conservative by Corollary 3.5. According to
(8.2), we know that $\Phi(u,q)$ can be seen as a bounded $V$-subharmonic
function on $M$. Then it suffice to prove a special case of Theorem 1.7, with
$N=\mathbb{R}$ and $v(x):=\Phi(u(x),q)$ is a non-negative, bounded
$V$-subharmonic function on $M$.

As before, let $X_{t}$ be the diffusion on $M$ whose generator is $\Delta_{V}$
and with initial position $X_{0}=x$. Now apply Itô’s formula for $v(X_{t})$,
we derive that

$\displaystyle
m_{t}:=v(X_{t})-v(X_{0})-\int^{t}_{0}\Delta_{V}v(X_{s})\mathrm{d}s$

is a local (${\mathbb{P}}_{x}$-)martingale. Thus, by Lemma 8.1 $v(X_{t})$ is a
non-negative, bounded, local submartingale. Let $\\{G_{n}\\}$ the increasing
sequence of relatively compact open sets satisfying $\overline{G_{n}}\subset
G_{n+1}\subset M$ for $n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$.
Then $n\mapsto v(X_{\tau_{G_{n}}})$ is a discrete time submartingale. By the
martingale convergence theorem, there exists a unique limit of
$v(X_{\tau_{G_{n}}})$, $\mathbb{P}_{x}$-a.s.

Now let ${\rm Im}(u)\subset B_{R}(o)$ be the image of $u$ and we may assume
that it is a compact subset of $B_{R}(o)$. Then for each $\omega\in\Omega$,
there exists a subsequence $n_{k}=n_{k}(\omega)$ such that the limit

$\lim_{k\to\infty}u(X_{\tau_{G_{n_{k}}}}(\omega))=L(\omega)\in B_{R}(o)\subset
N$

exists.

Therefore, we may conclude that for each $\omega\in\Omega$,

$\lim_{n\to\infty}v(X_{\tau_{G_{n}}}(\omega))=\lim_{k\to\infty}\Phi(u(X_{\tau_{G_{n_{k}}}}(\omega)),q)=\Phi(L(\omega),q).$

Moreover, we also have

$\lim_{n\to\infty}v(X_{\tau_{G_{n}}}\circ\theta_{t}\,)=\Phi(L\circ\theta_{t},q).$

On the other hand, for large $n\in\mathbb{N}$, $t<\tau_{G_{n}}$, hence
$t+\tau_{G_{n}}\circ\theta_{t}=\tau_{G_{n}}$ $\mathbb{P}_{x}$-a.s. so that

$\lim_{n\to\infty}v(X_{\tau_{G_{n}}})\circ\theta_{t}=\lim_{n\to\infty}v(X_{t+\tau_{G_{n}}\circ\,\theta_{t}})=\lim_{n\to\infty}v(X_{\tau_{G_{n}}})=\Phi(L,q)$

holds $\mathbb{P}_{x}$-a.s. for all $x\in M$. Thus

$\Phi(L\circ\theta_{t},q)=\Phi(L,q)$

holds for any $q\in B_{R}(o)$. Taking $q=L(\omega)$, we have

$\Phi(L\circ\theta_{t}(\omega),L(\omega))=0\qquad\text{ for }\qquad
t\in[\,0,+\infty\,[$

implying that $L\circ\theta_{t}(\omega)=L(\omega)$. Therefore
$L\in\bigcap_{0<t<\infty}\mathscr{I}_{t}$, where $\mathscr{I}_{t}$ is the
invariant $\sigma$-field of $\theta_{t}$. We claim that $\mathscr{I}$ is
trivial, leading to that $L$ is constant $L\equiv L_{\infty}$,
$\mathbb{P}_{x}$-a.s..

To prove the triviality of $\mathscr{I}$, first notice that for any
$A\in\mathscr{I}$, we have
$\mathbb{P}_{x}(A\,|\,\mathscr{F}_{t})=\mathbb{P}_{X_{t}}(A)$, meaning that
the function $x\mapsto\mathbb{P}_{x}(A)$ is finely $V$-harmonic on $M$, i.e.
$t\mapsto\mathbb{P}_{X_{t}}(A)$ is a local (${\mathbb{P}}_{x}$-)martingale.
Indeed, for any increasing sequence $\\{G_{n}\\}$ of relatively compact open
sets satisfying $\overline{G_{n}}\subset G_{n+1}\subset M$ for
$n\in\mathbb{N}$ and $M=\bigcup_{n=1}^{\infty}G_{n}$,
$Y_{t}:=\mathbb{P}_{X_{t\land\tau_{G_{n}}}}(A)$ is a bounded martingale: for
$s<t$

$\displaystyle\mathbb{E}_{x}[Y_{t}\,|\,\mathscr{F}_{s}]$
$\displaystyle=\mathbb{E}_{x}\left[\mathbb{P}_{X_{t\land\tau_{G_{n}}}}(A)\,\left|\,\mathscr{F}_{s}\right]\right.$
$\displaystyle=\mathbb{E}_{x}\left[\mathbb{P}_{x}(A\,|\,\mathscr{F}_{t\land\tau_{G_{n}}})\,\left|\,\mathscr{F}_{s}\right]\right.$
$\displaystyle=\mathbb{P}_{x}(A\,|\,\mathscr{F}_{s\land\tau_{G_{n}}})=Y_{s}.$

Meanwhile by taking $N=\mathbb{R}$ in Theorems 3.1 and 3.2 in Kendall’s paper
[25], we see that $x\mapsto\mathbb{P}_{x}(A)$ is in fact smooth on $M$.
Indeed, $\mathbb{P}_{x}(A)$ is finely $V$-harmonic and the vector field $V$ is
of course locally bounded. It is worth to point out that in the proof of [25,
Theorem 3.1], we do not need to assume the continuity of the map.

Therefore, the local (${\mathbb{P}}_{x}$-)martingale property and the
smoothness give rise to that $\mathbb{P}_{x}(A)$ is a $V$-harmonic function.
By Theorem 1.7, we derive that $\mathbb{E}_{x}[{\bf 1}_{A}]\equiv c$ for some
constant $0\leq c\leq 1$.

On the other hand, we have $c^{2}=c$. To see this, the invariance of $A$ and
Markov property give rise to

$\displaystyle c$ $\displaystyle=\mathbb{E}_{x}[{\bf
1}_{A}]=\mathbb{E}_{x}[\mathbb{E}_{x}[{\bf
1}^{2}_{A}\,|\,\mathscr{I}\,]\,]=\mathbb{E}_{x}[{\bf 1}_{A}\mathbb{E}_{x}[{\bf
1}_{A}\,|\,\mathscr{I}\,]\,]=\mathbb{E}_{x}[{\bf
1}_{A}\lim_{t\to\infty}\mathbb{E}_{x}[{\bf 1}_{A}\,|\,\mathscr{I}_{t}\,]\,]$
$\displaystyle=\mathbb{E}_{x}[{\bf 1}_{A}\lim_{t\to\infty}\mathbb{E}_{x}[{\bf
1}_{A}\circ\theta_{t}\,|\,\mathscr{I}_{t}\,]\,]=\mathbb{E}_{x}[{\bf
1}_{A}\lim_{t\to\infty}\mathbb{E}_{X_{t}}[{\bf 1}_{A}]\,]=c^{2},$

implying that either $c=0$ or $c=1$.

Thus,

$\displaystyle\lim_{n\to\infty}\Phi(u(X_{\tau_{G_{n}}}),L_{\infty})=\Phi(L(\omega),L_{\infty})=0.$

Consequently, submartingale property of
$n\mapsto\Phi(u(X_{\tau_{G_{n}}}),L_{\infty})$ gives rise to

$\displaystyle\Phi(u(x),L_{\infty})\leq\lim_{n\rightarrow\infty}\mathbb{E}_{x}\left[\Phi(u(X_{\tau_{G_{n}}}),L_{\infty})\right]=0,$

leading to $\Phi(u(x),L_{\infty})=0$, $\mathbb{P}_{x}$-a.s. for all $x\in M$,
since $\Phi(x,y)$ is nonnegative. Therefore by the vanishing property, we have

$u\equiv L_{\infty}.$

∎

###### Proof of Corollary 1.10.

By Remark 2.7, ${\rm Ric}_{V}^{\infty}\geq 0$ implies (B3) and Theorem 1.9
remains valid by replacing (A3) with (B3). Then we obtain the conclusion. ∎

###### Proof of Corollary 1.11.

By Corollary 2.8, ${\rm Ric}_{V}^{m}\geq 0$ with the boundedness of $V$
implies (B3) and Theorem 1.9 remains valid by replacing (A3) with (B3). Then
we obtain the conclusion. ∎

To prove Theorem 1.12, we need the following:

######  Lemma 8.2.

Suppose that the $\Delta_{V}$-diffusion process ${\bf
X}^{V}=(\Omega,X_{t},{\mathbb{P}}_{x})$ is recurrent. Then every bounded
finely $V$-subharmonic (or $V$-superharmonic) function is constant. In
particular, every bounded smooth $V$-subharmonic (or $V$-superharmonic)
function is constant. Conversely, if every bounded finely $V$-subharmonic
function is constant, then ${\bf X}^{V}$ is recurrent.

###### Proof.

Suppose that ${\bf X}^{V}$ is recurrent. The recurrence of ${\bf X}^{V}$
implies ${\mathbb{P}}_{x}(\zeta=+\infty)=1$ for $x\in M$ (see [16, (3.4)
Lemma]). First note that any smooth bounded $V$-subharmonic function is finely
$V$-subharmonic. This is easily confirmed by way of Itô’s formula. Let $f$ be
a bounded finely $V$-subharmonic (resp. $V$-superharmonic) function, then
$g:=\|f\|_{\infty}-f$ (resp. $g=\|f\|_{\infty}+f$) is a non-negative finely
$V$-superharmonic function. Let $\\{G_{n}\\}$ be the increasing sequence of
relatively compact open sets satisfying $\overline{G_{n}}\subset
G_{n+1}\subset M$ for $n\in N$ and $M=\bigcup_{n=1}^{\infty}G_{n}$. Then
$t\mapsto g(X_{t\land\tau_{G_{n}}})$ is a bounded supermartingle, hence
${\mathbb{E}}_{x}[g(X_{t}):t<\tau_{G_{n}}]\leq g(x)$ for any $t>0$ and $x\in
M$. Letting $k\to\infty$, we have ${\mathbb{E}}_{x}[g(X_{t})]\leq g(x)$ for
any $t>0$ and $x\in M$. Since $g$ is finely continuous, $g$ is an excessive
function with respect to ${\bf X}^{V}$. Therefore we obtain the first
conclusion. Conversely suppose that any bounded finely $V$-subharmonic
function is constant. Let $g$ be a bounded excessive function with respect to
${\bf X}^{V}$. Then $\mathbb{E}_{x}[g(X_{t\land\tau_{G_{n}}})]\leq g(x)$ for
all $t>0$ and $x\in M$ by [5, Chapter II, (2.8) Proposition]. From this, we
can deduce that $t\mapsto g(X_{t\land\tau_{G_{n}}})$ is a bounded
supermartingale, that is, $g$ is finely $V$-superharmonic. By assumption, $g$
is constant. For general excessive function $g$, for each $\ell>0$,
$g\land\ell$ is a bounded excessive function and it is a constant. If $g$
takes different values $g(x_{1})$ and $g(x_{2})$ with $g(x_{1})<g(x_{2})$,
then $g(x)\land\left(\frac{g(x_{1})+g(x_{2})}{2}\right)$ has the constant
value $c$. Then one can obtain $c=g(x_{1})=\frac{g(x_{1})+g(x_{2})}{2}$ so
that $g(x_{1})=g(x_{2})$. This is a contradiction. Therefore $g$ is constant,
hence ${\bf X}^{V}$ is recurrent. ∎

###### Proof of Theorem 1.12.

By Lemma 8.2, $\Phi(u,o)$ is constant. Thanks to the concrete expression of
$\Phi$ shown in [24, Theorem 3], ${\color[rgb]{0,0,0}{{\sf d}}}_{N}(u,o)$ is
constant. Then one can conclude that $u(M)\subset\partial B_{r}(o)\subset
B_{R}(o)$ for some $r>0$. Fix an $x_{0}\in M$. Since $B_{R}(o)\cap{\rm
Cut}(o)=\emptyset$, $u(x_{0})$ and $o$ can be joined by a unique minimal
geodesic in $B_{R}(o)$. On this geodesic arc, we can choose a point
$\hat{o}\in B_{R}(o)$ which is different from $o$ such that we can find
another geodesic ball $B_{\hat{r}}(\hat{o})$ satisfying $B_{r}(o)\subset
B_{\hat{r}}(\hat{o})\subset B_{R}(o)$. Applying Lemma 8.1 to
$\Phi(u,\hat{o})$, it is constant so that ${\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u,\hat{o})\equiv\tilde{r}$, that is, $u(M)\subset\partial
B_{\tilde{r}}(\hat{o})$ for some $\tilde{r}>0$. Then we see
$\tilde{r}={\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(x_{0}),\hat{o})=r-{\color[rgb]{0,0,0}{{\sf d}}}_{N}(o,\hat{o})$.
Obviously, $u(x_{0})\in\partial B_{r}(o)\cap\partial B_{\tilde{r}}(\hat{o})$.
Suppose there exists a different $u(x_{1})\neq u(x_{0})$ such that
$u(x_{1})\in\partial B_{r}(o)\cap\partial B_{\tilde{r}}(\hat{o})$. Since
$u(x_{1})\neq u(x_{0})$, $u(x_{1})$ does not lie on the minimal geodesic
joining $o$ and $u(x_{0})$. Then $r={\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(x_{1}),o)<{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(u(x_{1}),\hat{o})+{\color[rgb]{0,0,0}{{\sf
d}}}_{N}(\hat{o},o)=\tilde{r}+{\color[rgb]{0,0,0}{{\sf d}}}_{N}(\hat{o},o)$.
This is a contradiction. Hence $\partial B_{r}(o)\cap\partial
B_{\tilde{r}}(\hat{o})=\\{u(x_{0})\\}$, consequently, $u(M)\subset\partial
B_{r}(o)\cap\partial B_{\tilde{r}}(\hat{o})=\\{u(x_{0})\\}$ implies
$u(M)=\\{u(x_{0})\\}$. ∎

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