数理解析研究所講究録 1293 巻 2002 年 175-182

175-INTEGRALS OF
ANOTE ON DIVERGENCE OF
SUBHARMONIC FUNCTIONS AND ITS APPLICATIONS$L^{p}$KENSHO TAKEGOSHI
Throughout this note we always denotes $(M, g)$ anon-compact complete Riemannian manifold of dimension $m$ and$\Delta_{g}$the divergence property of

the Laplacian defined by$L^{p}$$\Delta_{g}:=\mathrm{R}\mathrm{a}\mathrm{c}\mathrm{e}_{g}\nabla\nabla$-integral of anon-trivial solution$u$. Our interests are

satisfying the inequality

either $uAgu\geq k$ or $\Delta_{g}\log u\geq k$ for alocally integralble function$k$

on $M$ , and its sev-

eral applications in differential geometry ;for instance conformal deformation of metrics,
parabolicity of manifolds and Liouville theorem for harmonic maps. The non-negativity of$k$is the most important case. However the condition can be relaxed by the non-negativityof the interal $\int_{M}kdv_{g}$ under assuming the integrability of the negative part of . Such$k$an observation brings us afew interesting applications and is originated in Yau, [Y]. The
proof of results stated below will appear elsewhere.

1. To formulate our result we begin with the following theorem which contains avanishingof gradient length of certain functions whose
Theorem 1.1.

Let$\mathrm{I}(r)$, and $E(r)$ (resp.

$L^{p}$-integrals have amoderate growth.

$K_{\pm}(r)$) be non-negative absolutely continuousand non-decreasing functions (resp. non-negative continuous functions) on

Suppose $K_{-}(+ \infty):=\lim_{rarrow+\infty}K_{-}(r)<+\infty$ ,(1)$\frac{d}{dr}\mathrm{I}(r)>0$$K_{+}(r)+C_{1}E(r)\leq \mathrm{K}_{-}(\mathrm{r})$for almost all $r\geq r_{0}>>1$ and two constants$\mathrm{R}_{+}:=[1, +\infty)$.

and$+c_{2}\sqrt{\frac{d}{dr}\mathrm{I}(r)\frac{d}{dr}E(r)}$$C_{1}\geq 0$, $C_{2}>0$ . Then the following assertions

are valid:The case $C_{1}>0$ : if $K_{-}(+ \infty)\leq K_{+}(+\infty):=\lim_{rarrow+\infty}K_{+}(r)\leq+\infty$ , then either
Mathematics Subject Classification ;$35\mathrm{B}50,35\mathrm{J}60$,$53\mathrm{A}30,53\mathrm{C}21$,$58\mathrm{G}03$

Typeset by$\mathrm{A}\mathrm{A}\phi*\mathrm{I}\mathrm{E}\mathrm{K}$176$\mathrm{E}(\mathrm{r})\equiv 0$or$1/ \frac{d}{dr}\mathrm{I}(r)\in L^{1}(\mathrm{R}_{+})$$\lim_{rarrow+}\inf_{\infty}E(r)/\int_{1}^{r}dt/\frac{d}{dt}\mathrm{I}(t)>0$

Applying Theorem 1.1 to$K_{\pm}(r)\equiv 0$. The case $C_{1}=0$ : if $K_{-}(+\infty)<K_{+}(+\infty)\leq+\infty$ , then.
(see below), $E(r)= \int_{B_{x}(r)}|u|^{p-2}|\nabla u|^{2}dv_{g}$ and$\mathrm{I}(r)=\mathrm{I}_{x}^{p}(u, r)$respectively we can obtain the following result which implies not only Theorem

2.1 but also Theorem 2.2 in [K] simultaneously, and is ageneralization of Theorem 1, b)
in [S] for the Laplacian
Theorem 1.2.
$uA9u\geq 0$$\Delta_{g}$Supposein view of Lemma given in 5. Appendix.
$u$is anon-constant smooth function $satisffi\dot{n}g$ the inequality

on M. Then for any $p>1$ , $r>0$ and $x\in M$ , the function$\mathrm{I}_{x}^{p}(u, r):=\int_{B_{x}(r)}|u|^{p}dv_{g}$satisfies$1/ \frac{d}{dr}\mathrm{I}_{x}^{p}(u, r)\in L^{1}(\mathrm{R}_{+})$$\mathrm{I}_{x}^{p}(u, r)$defined by, where $B_{x}(r)$ is the geodesicball centered at $x\in M$ and of radius .
$r$Remark 1.

Since the distance function

on $M$ and satisfies$|\nabla r_{x}|\equiv 1$$\int_{S_{x}(r)}|u|^{p}\sigma_{r}$from apoint $x\in M$ is Lipschitz continuouswithin the cut locus of , letting$x$

Hausdorff measure of the geodesic sphere
integral$r_{x}$$S_{x}(r)$induced by ,for almost all $r>0$ by the$g$$\sigma_{r}$be the $m-1$ -dimensional$\frac{d}{dr}\mathrm{I}_{x}^{p}(u, r)$coincides with the-area formula (cf. [F], 3.2.12, Theorem$\mathrm{c}\mathrm{o}$and 3.2.46).

Let
$u$

$f$

: $(M, g)arrow(M, g)$ be aconformal transformation of $(M,g)$ withthe conformal factor of$f$$\dim_{\mathrm{R}}M\geq 3$

defined by $f^{*}g=u^{4/(m-2)}g$ . It is known that

$u$

and

satisfies thefollowing non-linear equality$c_{m}\Delta_{g}u-s_{g}u+K_{fg}.u^{(m+2)/(m-2)}\equiv 0$

for $c_{m}:=4(m-1)/(m-2)$ andpreserves the scalar curvature$s_{g}$$K_{f^{*}g}$on $M$

is the scalar curvature of$f^{*}g$. If$s_{g}\leq 0$

and

$f$

, i.e., $s_{g}=Kf*g$ ’then we can see$(u-1) \Delta_{g}(u-1)=\frac{-s_{g}u(u-1)(u^{4/(m-2)}-1)}{c_{m}}\geq 0$

on $M$ .

Therefore Theorem 1.2 has the following interesting application which implies auniquenessof solution for the scalar curvature equation and is known for the case p $=2$ (cf. [BRS],
Theorem 1.5).

177
Theorem 1.3.
sion $m\geq 3$ and$s_{g}$of$g$

Let (M, g) be anon-compact complete Riemannian manifold of dimenaconformal transformation of $(M, g)$ . Suppose (i) the scalar curvature$f$

is non-positive on $M$ , (ii)$f$preserves$1/ \max\{\frac{d}{dr}\mathrm{I}_{x}^{p}(u-1, r), 1\}\not\in L^{1}(\mathrm{R}_{+})$$s_{g}$, and (iii) tie conformal factor$u$satisfies

for aconstant $p>1$ and apoint $x\in M$ . Then$u\equiv 1$,

i.e., $f$ is isometric.

Remark 2.

In Theorem 1.3 the constant function 1is atrivial solution of the above scalar

curvature equation under the condition $s_{g}=Kf*g$ . However for two non-trivial solutions

and$u_{2}$of the equation, we do not know whether$L^{1}(\mathrm{R}_{+})$$u_{1}\equiv u_{2}$if$u_{1}$$1/ \max\{\frac{d}{dr}\mathrm{I}_{x}^{p}(u_{1}-u_{2}, r), 1\}\not\in$for aconstant $p>1$ . But under this condition it is not so hard to see that Theorem

1.2 implies $|u_{1}-u_{2}|\leq 1$ on $M$ , and moreover $\inf_{M}|u_{1}-u_{2}|=0$ if $(M, g)$ is not parabolic(see Corollary 2.2 below).

As acorollary, we get the following gap theorem of solution for the scalar curvature equation
immediately.
Corollary 1.4.

Let $(M, g)$ and$f$be as above. Suppose the conditions (i) and (ii) of

Theorem 1.3 are satisfied and moreover (iii)’ $(M, g)$ has polynomial volume growth, i.e.,$\lim_{rarrow+\infty}\mathrm{V}\mathrm{o}\mathrm{l}(B_{y}(r))/r"<+\infty$

the distance function$r_{y}$

for some $\alpha>0$ and $|u(x)-1|<C/(1+r_{y}(x))^{\beta}$ , $x\in M$ , forfrom afixed point $y\in M$ , $C>0$ and $\beta>0$ , Then$f$

is isometric.

2. (A#, ) is said to be parabolic if $(M, g)$ admits no positive Green’s function. Varopoulos$g$showed that $(M, g)$ is parabolic if$\mathrm{V}\mathrm{o}\mathrm{l}(B_{x}(r))$$r/V_{x}(r)\not\in L^{1}(\mathrm{R}_{+})$

is the volume of $B_{x}(r)$ relative to$g$for some point $x\in M$ , where $V_{x}(r):=$(cf.[V],Theorem 2and [Gri], Corollary2). Later Li and Tam proved the same result under the condition(cf.[LTl], Corollary 2.3). Here we note that
$S_{x}(r)$induced by$g$$\frac{d}{dr}V_{x}(r)$$1/ \frac{d}{dr}V_{x}(r)\not\in L^{1}(\mathrm{R}_{+})$

is the area of the geodesic sphere

for almost all $r>0$ . In view of Lemma in$5.\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{x}$, Li and Tarn’s

assertion is sharper than Varopoulos’ one. The idea of proof of Theorem 1.2 allows us togive an alternative and elementary proof of their result by showing the following.
Theorem 2.1.

Suppose $(M, g)$ admits anon-constant continuous subharmonic function

bounded from above.

Then$1/ \frac{d}{dr}V_{x}(r)\in L^{1}(\mathrm{R}_{+})$

for any point x $\in M$ .

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Here acontinuous function u is said to be subharmonic if D is any relatively

Remark 3.

compact open subset in $M$ ,class$C^{2}$, then$u$satisfies$\Delta_{g}v\equiv 0$$\Delta_{g}u\geq 0$

.

on

$D$

, and $u\leq v$ on$\overline{D}\backslash D$, then $u\leq v$ on $D$ . If

$u$

is of

It is known that apositive Green’s function producessmooth non-constant bounded subharmonic functions (cf. [CTW], Theorem 1.4).
By Theorem 2.1, we get the following in view of Remark 3.

Corollary 2.2.

If $(M, g)$ admits apoint $x\in M$ such that

$1/ \frac{d}{dr}V_{x}(r)\not\in L^{1}(\mathrm{R}_{+})$, then$(M, g)$ is parabolic.

Remark 4. If $(M, g)$ is rotationally symmetric at apoint $x_{*}\in M$ , letting
function from $x_{*}\in M$ , then we can see that (i)on$M\backslash \{x_{*}\}$

and $\lim_{r_{*}(x)arrow 0}w(x)=-\infty$ ,be the distance$r_{*}$$w(x):= \int_{1}^{r_{*}(x)}dr/\frac{d}{dr}V_{x_{*}}(r)$is harmonic$( \mathrm{i}\mathrm{i})\int_{1}^{+\infty}rdr/V_{x_{*}}(r)\geq\int_{1}^{+\infty}dr/\frac{d}{dr}V_{x_{*}}(r)$the radial curvature is non-positive (cf.[GWl] and Lemma in$5.\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{x}$if). In particular$u:=\exp w\geq 0$ defines acontinuous bounded subharmonic function on $M$ if and onlyif

$1/ \frac{d}{dr}V_{x_{*}}(r)\in L^{1}(\mathrm{R}_{+})$. If the Ricci curvature of $(M, g)$ is non-negative, then $(M,g)$

is parabolic if and only if$r/V_{x}(r)\not\in L^{1}(\mathrm{R}_{+})$

for some point $x\in M$ (cf.[V], Theorem 2

and [LT2], Theorem 1.9). However the integrability of $r/V_{x}(r)$ does not always imply theexistence of Green’s function on $(M, g)$ generally as observed in [V].

3. The parabolicity of manifold is related to Liouville theorem for harmonic maps. First

we state the following.
Theorem 3.1.
$i\mathrm{a}n$

Let

$f$

: $(M, g)arrow(N, h)$ be asmooth map from $(M, g)$ to aRiemannia

manifold $N$ provided with asmooth function

such that$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\varphi)\geq\chi h$and$|\nabla\varphi|\leq C$$\varphi$

and acontinuous function $\chi>0$

for aconstant $C>0$ on N.

Suppose$f$is har-monic, and the energy density $\mathrm{e}(f):=(1/2)|df|^{2}$ of $f$ satisfies the following condition $(*)$: $\int_{B_{x}(r)}\mathrm{e}(f)dv_{g}=o$$( \int_{1}^{r}dt/\frac{d}{dt}V_{x}(t))$

for some point $x\in M$ . Then$f$is aconstant map.

This can be induced by applying the case $C_{1}=0$ in Theorem 1.1 to $E(r)= \int_{B_{x}(r)}|\nabla u|^{2}dv_{g}$for $u:=f^{*}\varphi$ and $\mathrm{I}(r)=V_{x}(r)$ . As acorollary we obtain the following Liouville theorem

for harmonic maps to amanifold of asymptotically non-positive curvature.
Corollary 3.2.

Let

$f$

: $(M, g)arrow(N, h)$ be aharmonic map to acomplete Riemannia

179
manifold $(N, h)$ with apole $y\in N$ whose radial curvature

on N. Suppose the energy density
Then

$f$

$\mathrm{e}(f)$

of

$f$

$R_{N}$

satisfies $R_{N}\leq 1/4(1+r_{y})^{2}$satisfies the condition $(*)$ in Theorem 3.1.

is aconstant map.

Remark 5.

It is known that if $(M, g)$ is parabolic, then any harmonic map of finite energy

from $(M, g)$ to any Hadamard manifold $(N, h)$ , i.e., $N$ is simply connected and $R_{N}\leq 0$

on $N$ , is constant (cf.[CTW],Proposition 2.1 and Theorem 3.2). On the other hand there
exists anon-degenerate harmonic map from atwo dimensional Euclidean spaceflat metric to ahyperbolic plane of constantmap on$B_{x}(r)\subset \mathrm{R}^{2}$$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}-1$$\mathrm{R}^{2}$with(cf.[CT]). The energy of such awith $x=(0,0)\in \mathrm{R}^{2}$ diverges not slower than $\log r$ by Theorem 3.1.

However the analyticity of map yields the following Liouville theorem of holomorphic ones

to amanifold of negative curvature bounded away from zero.

Theorem 3.3.

Let

$f$

: $(M, \omega_{M})arrow(N,\omega_{N})$ be aholomorphic map from acomplete

K\"ahler manifold $(M, \omega_{M})$ of dimension $m=\dim_{\mathbb{C}}M$ to aKiihler manifold $(N, \omega_{N})$ . If$(M, \omega_{M})$admits apoint $x\in M$ such thatsmooth 1-form0such that

Remark 6.

$\omega_{N}=d\theta$

The target manifold$1/ \frac{d}{dr}V_{x}(r)\not\in L^{1}(\mathrm{R}_{+})$, and $(N,\omega_{N})$ admits aand $C:= \sup_{N}|\theta|_{g}<+\infty$ , then$(N, \omega_{N})$$f$is aconstant map.

in Theorem 3.3 should be non-compact. Manykinds of hyperbolic Kahler manifold admit such aKahler metric (cf. [Gro]).

4. In Theorem 1.2 we can relax the non-negativity condition ofTheorem 4.1. Let
$k$

$u$

for the case $p=2$ .

be asmooth non-constant solution satisfying the inequality

for locally integrable function

then

$k$

$1/ \frac{d}{dr}\mathrm{I}_{x}^{2}(u, r)\in L^{1}(\mathrm{R}_{+})$$k$$u\Delta_{g}u\geq$

on M. If $k_{-}:= \max\{-k, 0\}\in L^{1}(M)$ and $\int_{M}kdv_{g}\geq 0$ ,for any $x\in M$ .

According to the same spirit as Theorem 4.1 we can show the following which is ageneralization of Theorem 2.1 in [LY] and Theorem 1in [Y].

Theorem 4.3. Let$\geq k$$u$

be asmooth non-constant solution satisfying the inequality$\Delta_{g}\log u$

on $M_{+}:=\{u>0\}$ . If either $k\equiv 0$ or $k_{-}\in L^{1}(M)$ and $\int_{M}kdv_{g}>0$ , then

$1/ \frac{d}{dr}\mathrm{I}_{x}^{p}(u, r)\in L^{1}(\mathrm{R}_{+})$

for any $p>0$ and $x\in M$ .

180
Remark 7. Under the same situation Li and Yau showedwhich follows from

$1/ \frac{d}{dr}\mathrm{I}_{x}^{p}(u, r)\in L^{1}(\mathrm{R}_{+})$$\lim_{rarrow+}\inf_{\infty}\mathrm{I}_{x}^{p}(u, r)/r^{2}>0$

in [LY],in view of Lemma in 5. Appendix.

The condition $\int_{M}kdv_{g}>0$ follows from the non-parabolicity of $(M, g)$ .
Corollary 4.3.

on$M_{+}$

Let$u$

be asmooth non-negative solution of the inequality $\Delta_{g}\log u\geq k$. If $k_{-}\in L^{1}(M)$ , $(M, g)$ is not parabolic, andsome $p>0$ and apoint $x\in M$ , then
Let

$f$

$u$

$1/ \max\{1, \frac{d}{dr}\mathrm{I}_{x}^{p}(u, r)\}\not\in L^{1}(\mathrm{R}_{+})$

forshould be identically zero.

: $(M, \omega_{M})arrow(N, \omega_{N})$ be aholomorphic map from anon-compact complete Kahler

manifoldto acomplex Hermitian manifold$(M, \omega_{M})$curvature is non-positive. Letting$\mathrm{e}(f)$$(N, \omega_{N})$be the energy density

whose holomorphic sectional$\mathrm{e}(f)$

of

$f$

and

$R_{M}$

the point

wise lower bound of the Ricci curvature of $M$ , one can show the following inequality$\Delta_{g}\log \mathrm{e}(f)\geq 2R_{M}$where$\mathrm{e}(f)\neq 0$,(see [R], Proposition 4). Acomplex differential geometric interpretation ofTheorem 4.2 and Corollary 4.3 is the following Liouville theorem for holomorphic maps.

Theorem 4.4.

Let

: $(M, \omega_{M})arrow(N,\omega N)$ be aholomorphic map from anon-compact$f$complete Kiihler manifold $(M, \omega_{M})$ to acomplex Hermitian manifold$(N, \omega_{N})$whose holo

morphic sectional curvature is non-positive, and $R_{M,-}$ the negative part of $R_{M}$ . Suppose(i)$\mathrm{R}\mathrm{m},-\in L^{1}(M)$energy density, (ii) either $(M, \omega_{M})$ is not parabolic or $\int_{M}R_{M}dv_{M}>0$ , and (iii) the$\mathrm{e}(f)$ofapoint $x\in M$ . Then
Remark 8.

In case$f$$f$satisfies$1/ \max\{1, \frac{d}{dr}\mathrm{I}_{x}^{p}(\mathrm{e}(f), r)\}\not\in L^{1}(\mathrm{R}_{+})$

for some $p>0$ andis aconstant map.

$\mathrm{R}\mathrm{m},-\equiv 0$

and $p=1$ the condition (ii) can be dropped as shown in[SY], which deals with harmonic maps of finite energy (see also [LY], theorem 3.1).

Let

$f$

: $(M, \omega M)$

manifold$arrow(N, \omega_{N})$$(M, \omega_{M})$be aholomorphic map from anon-compact complete Kahlerof dimension $m$ to acomplex hermitian manifold$(N,\omega_{N})$of the same

dimension whose Ricci curvature is non-positive. Letting $uf$ denote the ratio $f^{*}V_{N}/V_{M}$ ofthe volume forms$V_{M}$

relative to$\omega M$

and$V_{N}$

relative to$\omega_{N}$curvature of $(M, \omega M)$ , one can see the following inequality$\Delta_{g}\log u_{f}\geq S_{M}$respectively and$S_{M}$

the scalar

181
where $u_{f}\neq 0$ (see [LY], the proof of Theorem 3.5). Hence we can also show the following

theorem (cf.[MY], \S 1 and [LY], Theorem 3.5 and Corollary 3.6).

Theorem 4.5. Let$f$: $(M, \omega_{M})arrow(N, \omega_{N})$ be aholomorphic map from anon-compact

complete Kihler manifold $(M, \omega_{M})$ of dimension $m$ to acomplex hermitian manifold$(N, \omega N)$of the same dimension whose Ricci curvature is non-positive and $S_{M,-}$ the nega-tive part of$S_{M}$

of M. Suppose (i)$\int_{M}S_{M}dv_{M}>0$$x\in M$

. Then$f$, and (iii)$\mathrm{S}\mathrm{m},-\in L^{1}(M)$, (ii) either$(M, \omega_{M})$$1/ \max\{1, \frac{d}{dr}\mathrm{I}_{x}^{p}(u_{f}, r)\}\not\in L^{1}(\mathrm{R}_{+})$

is not parabolic orfor some $p>0$ and apoint

degenerates everywhere on $M$ .

5. Appendix

Let $v(r)>0$ be an absolutely continuous function on $[0, +\infty)$ such that

Lemma.
$\frac{d}{dr}v(r)>0$

for almost all $r\in[0, +\infty)$ .

Then

$v$

satisfies the following integral inequality:$\int_{2}^{r}\frac{tdt}{v(t)}\leq 4\int_{1}^{r}\frac{dt}{\frac{d}{dt}v(t)}$for any $r>2$ .

If $v(r)/r$ is non-decreasing (in particular $v(r)$ is convex and $v(0)=0$ ), then $r/v(r)\in$$L^{1}(\mathrm{R}_{+})$

Proof.

if and only if$1/ \frac{d}{dr}v(r)\in L^{1}(\mathrm{R}_{+})$.

By integration by parts and Schwarz’s inequality we get the following:$\int_{1}^{r}\frac{t-1}{v(t)}dt\leq 2\int_{1}^{r}\frac{dt}{\frac{d}{dt}v(t)}$which implies the desired inequality.

for any $r>1$ ,$\square$

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[LY]
[MY]

[R]
[S]
[V]

[Y]

$\mathrm{S}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{m},\mathrm{K}$$L^{p}$

Address: Department of Mathematics, Graduate School of Science, Machikaneyama-cho
1-16, Toyonaka-shi Osaka, 560-0043, Japan-mail address:kensho@math.wani.osaka-u.ac.jp$E$