Rigidity Characterizations of Conformal Solitons

Abstract

We study the rigidity of conformal solitons, give a sufficient and necessary condition that guarantees that every closed conformal soliton is gradient conformal soliton, and prove that complete conformal solitons with a nonpositive Ricci curvature must be trivial under an integral condition. In particular, by using a p-harmonic map from a complete gradient conformal soliton in a smooth Riemannian manifold, we classify complete noncompact nontrivial gradient conformal solitons under some suitable conditions, and similar results are given for gradient Yamabe solitons and gradient k-Yamabe solitons.

1. Introduction

In this paper, we focus on the rigidity of conformal solitons and gradient conformal solitons under some suitable conditions. Let us recall the concept of conformal solitons, which was introduced by Catino et al. [1]. An n-dimensional complete Riemannian manifold $(M^n,g)$ is called a conformal soliton if there exists a smooth vector field X on $M^n$ satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{L}_{X}g=\psi g\end{array}$$

(1)

for some smooth function $\psi :M^n\to \mathbb{R}$. Here, $L_{X}g$ denotes the Lie derivative of g in the X direction. The soliton is denoted by $(M^n,g,X,\psi )$ for simplicity.

In particular, if X is a gradient vector field, namely, $X:=\nabla f$ for some smooth function f on $M^n$, then the soliton is called a gradient conformal soliton, and Equation (1) becomes

$$\begin{array}{c}\hfill \mathrm{Hess}\left(f\right)=\psi g,\end{array}$$

(2)

where $\mathrm{Hess}\left(f\right)$ denotes the Hessian of the potential function f. When either f is a constant or X is a killing vector field, then the soliton is called trivial.

The conformal solitons include some Yamabe-type solitons. For instance, if $\psi :=2(n-1)({\sigma }_k-\lambda )$ for some constant $\lambda \in \mathbb{R}$, then the solitons (1) and (2), respectively, are called a k-Yamabe soliton and a gradient k-Yamabe soliton. If $\lambda >0$, $\lambda =0$, or $\lambda <0$, then the soliton is called shrinking, steady, or expanding, respectively. Here, the ${\sigma }_k$ curvature of g is defined as

$${\sigma }_k={\sigma }_k\left(g^{-1}A\right)=\sum _{i_1<\cdots <i_k}{\lambda }_{i_1}\cdots \cdots {\lambda }_{i_k} \mathrm{for} 1\le k\le n,$$

where $A={\displaystyle \frac{1}{n-2}}\left(\mathrm{Ric}-{\displaystyle \frac{R_g}{2(n-1)}}g\right)$ is the Schouten tensor, and ${\lambda }_1,\dots ,{\lambda }_n$ denote the eigenvalues of the symmetric endomorphism $g^{-1}A$. Here and below, $\mathrm{Ric}$ and $R_g$, respectively, stand for the Ricci curvature tensor and the scalar curvature of g. Since ${\sigma }_1={\displaystyle \frac{R_g}{2(n-1)}}$, it is easy to see that gradient 1-Yamabe solitons simply correspond to gradient Yamabe solitons. It is well known that Yamabe solitons correspond to self-similar solutions of the Yamabe flow, which was introduced by Hamilton in [2].

In recent years, many authors have been devoted to studying the rigidity of Yamabe-type solitons, and derived many interesting results (see [3,4,5,6,7]). For instance, Hsu [8] proved that any closed gradient Yamabe soliton is trivial. Later, for $k>1$, Catino et al. [1] showed that every closed gradient k-Yamabe soliton with a non-negative Ricci curvature is trivial. Tokura et al. [9] proved that any closed gradient k-Yamabe soliton is trivial and gave a sufficient and necessary condition that guarantees that every closed k-Yamabe soliton is a gradient soliton. The same result was obtained in [10] for closed almost-Yamabe solitons.

Inspired by the above-mentioned results, in Section 2, we present an equivalent characterization of closed conformal solitons.

Theorem 1. 

An n-dimensional closed conformal soliton $(M^n,g,X,\psi )$ is gradient if and only if

$${\int }_{M^n}\mathrm{Ric}(Y,\nabla h)d{V}_g\le 0,$$

where $\nabla h$ and Y denote the Hodge–de Rham decomposition components of X.

Shaikh et al. [11] proved that in a compact gradient Yamabe soliton, the potential function agrees with the Hodge–de Rham potential up to a constant. In Section 2, for compact gradient conformal solitons, we prove the following result:

Theorem 2. 

Let $(M^n,g,\nabla f,\psi )$ be an n-dimensional compact oriented gradient conformal soliton. Then, up to a constant, the potential function f agrees with the Hodge–de Rham potential h.

L. Bo et al. [12] obtained the triviality of complete k-Yamabe solitons with a nonpositive Ricci curvature under an integral condition. In Section 3, we prove the following result for complete conformal solitons:

Theorem 3. 

Let $(M^n,g,X,\psi )$ be an n-dimensional complete conformal soliton with a nonpositive Ricci curvature. If the vector field X satisfies

$${\int }_{M^n}{\left|X\right|}^{2}d{V}_g<\infty ,$$

then $M^n$ is trivial.

Remark 1. 

It is clear that $f(x,y)=x+y-x^2-y^2$ is a smooth function on ${\mathbb{R}}^2$; then, we have $R_g=0$ and $f_{ij}=-2g_{ij}$, where $f_{ij}$ denotes the components of the Hessian of f, and $g_{ij}={\delta }_{ij}$. Letting $\psi :=R_g-2$. Then, $({\mathbb{R}}^2,{\delta }_{ij},\nabla f,\psi )$ is a complete nontrivial gradient Yamabe soliton. It can be checked that $\nabla f=(1-2x,1-2y)$ and ${\int }_{{\mathbb{R}}^2}{|\nabla f|}^{2}dxdy=\infty$. Hence, the example shows that the condition ${\int }_{M^n}{\left|X\right|}^{2}d{V}_g<\infty$ is necessary.

Moreover, for a compact case, we obtain the following conclusion:

Corollary 1. 

If $(M^n,g,X,\psi )$ is an n-dimensional closed conformal soliton satisfying

$${\int }_{M^n}\mathrm{Ric}(X,X)d{V}_g\le 0,$$

then $M^n$ is trivial.

For complete noncompact gradient Yamabe solitons, Li Ma et al. [13] proved the following result:

Theorem 4 

([13]). Let $(M^n,g)$ be an n-dimensional complete noncompact nontrivial gradient Yamabe soliton. If $|R_g-\lambda |\in L^1\left(M\right)$, ${\int }_M\mathrm{Ric}(\nabla f,\nabla f)d{V}_g\le 0$, and the potential function f has at most quadratic growth on M, namely, $\left|f\left(x\right)\right|\le Cd{(x,x_0)}^2$, $|\nabla f|\le C(1+d{(x,x_0)}^2)$ near infinity, where C is a uniform constant, then $M^n$ has constant scalar curvature, and $R_g=\lambda$.

This result was generalized to complete gradient k-Yamabe solitons by Cunha et al. [14], and they reported the following result:

Theorem 5 

([15]). Let $(M^n,g)$ be an n-dimensional complete noncompact nontrivial gradient k-Yamabe soliton. If ${\sigma }_k\ge \lambda$, $|\nabla f|$ has at most linear growth on $M^n$, and ${\int }_{M}h\left(u\right)(\nabla u,\nabla f)d{V}_g\ge 0$ for $h\in C^1\left(\mathbb{R}\right)$, u is a nonconstant solution of $\Delta u+h\left(u\right)=0$, and the energy satisfies ${\int }_{B_R}{|\nabla u|}^{2}d{V}_g=o\left(\log R\right)$, as $R\to \infty$, then the ${\sigma }_k$-curvature is constant and ${\sigma }_k=\lambda$. Here, $B_R:=B(x_0,R)$ denotes a ball with center at fixed point $x_0\in M$ and radius $R>0$.

Motivated by the results above and the recent work of Branding [16] on harmonic maps from gradient Ricci solitons, in Section 4, we consider the relationship between the rigidity of gradient conformal solitons and the p-harmonic map from this soliton into a smooth Riemannian manifold. More precisely, we prove the following:

Theorem 6. 

Let $(M^n,g,\nabla f,\psi )$ be an n-dimensional complete noncompact nontrivial gradient conformal soliton. If $\psi \ge 0$ and $|\nabla f|$ has at most linear growth, u is a nontrivial p-harmonic map from the soliton into a smooth Riemannian manifold, and the energy satisfies ${\int }_{B_R}{\left|du\right|}^{p}d{V}_g=o\left(\log R\right)$, as $R\to \infty$, where $n>p\ge 2$. Then, $M^n$ is either

(1) 

the warped product

$$(\mathbb{R},d{s}^2){\times }_{|\nabla f|}({\widehat{N}}^{n-1},\widehat{g}),$$

where $({\widehat{N}}^{n-1},\widehat{g})$ is an $(n-1)$-dimensional complete Riemannian manifold, or

(2) 

rotationally symmetric and equal to the warped product

$$([0,\infty ),d{s}^2){\times }_{|\nabla f|}({\mathbb{S}}^{n-1},{\overline{g}}_S),$$

where $f=as+b$ depends only on s, $a,b\in \mathbb{R}$, and ${\overline{g}}_S$ is the round metric on ${\mathbb{S}}^{n-1}$.

Similar classification results are easily obtained for gradient k-Yamabe solitons (i.e., $\psi :=2(n-1)({\sigma }_k-\lambda )$) and gradient Yamabe solitons (i.e., $\psi :=R_g-\lambda$).

Corollary 2. 

Let $(M^n,g,\nabla f,\psi )$ be an n-dimensional complete noncompact nontrivial gradient k-Yamabe soliton. If ${\sigma }_k\ge \lambda$, and $|\nabla f|$ has at most linear growth, u is a nontrivial p-harmonic map from the soliton into a smooth Riemannian manifold, and

$${\int }_{B_R}{\left|du\right|}^{p}d{V}_g=o\left(\log R\right),\hspace{1em}\mathrm{as}\hspace{1em}R\to \infty .$$

Here, $n>p\ge 2$. Then, the ${\sigma }_k$ curvature is constant, namely, ${\sigma }_k=\lambda$, and $M^n$ is either

(1) 

the warped product

$$(\mathbb{R},d{s}^2){\times }_{|\nabla f|}({\widehat{N}}^{n-1},\widehat{g}),$$

where $({\widehat{N}}^{n-1},\widehat{g})$ is an $(n-1)$-dimensional complete Riemannian manifold, or

(2) 

rotationally symmetric and equal to the warped product

$$([0,\infty ),d{s}^2){\times }_{|\nabla f|}({\mathbb{S}}^{n-1},{\overline{g}}_S),$$

where $f=as+b$ depends only on s, $a,b\in \mathbb{R}$, and ${\overline{g}}_S$ is the round metric on ${\mathbb{S}}^{n-1}$.

Corollary 3. 

Let $(M^n,g,\nabla f,\psi )$ be an n-dimensional complete noncompact nontrivial gradient Yamabe soliton. If $R_g\ge \lambda$ and $|\nabla f|$ has at most linear growth, u is a nontrivial p-harmonic map from the soliton into a smooth Riemannian manifold and

$${\int }_{B_R}{\left|du\right|}^{p}d{V}_g=o\left(\log R\right),\hspace{1em}\mathrm{as}\hspace{1em}R\to \infty .$$

Here, $n>p\ge 2$. Then, the scalar curvature is constant, namely, $R_g=\lambda$, and $M^n$ is either

(1) 

the warped product

$$(\mathbb{R},d{s}^2){\times }_{|\nabla f|}({\widehat{N}}^{n-1},\widehat{g}),$$

where $({\widehat{N}}^{n-1},\widehat{g})$ is an $(n-1)$-dimensional complete Riemannian manifold, or

(2) 

rotationally symmetric and equal to the warped product

$$([0,\infty ),d{s}^2){\times }_{|\nabla f|}({\mathbb{S}}^{n-1},{\overline{g}}_S),$$

where $f=as+b$ depends only on s, $a,b\in \mathbb{R}$, and ${\overline{g}}_S$ is the round metric on ${\mathbb{S}}^{n-1}$.

2. Compact Conformal Solitons

In this section, we complete the proof of Theorems 1 and 2. We first give some important results, which play a key role in the proof of our theorems. First of all, we introduce a Bochner-type formula, which was given by Petersen and Wylie.

Lemma 1 

([17]). If X is a smooth vector field on an n-dimensional Riemannian manifold $(M^n,g)$, then

$$\begin{array}{c}\hfill \mathrm{div}\left(L_{X}g\right)\left(X\right)={\displaystyle \frac{1}{2}}{\Delta \left|X\right|}^2-{|\nabla X|}^2+\mathrm{Ric}(X,X)+{\nabla }_X\mathrm{div}\left(X\right).\end{array}$$

(3)

In particular, if $X=\nabla h$ is a gradient vector field, where h is a smooth function on $M^n$, then

$$\begin{array}{c}\hfill \mathrm{div}\left(L_{\nabla h}g\right)\left(Y\right)=2\mathrm{Ric}(\nabla h,Y)+2g(\nabla \Delta h,Y)\end{array}$$

(4)

for any smooth vector field Y on $M^n$.

The Hodge–de Rham decomposition theorem (see [18]) shows that on a compact oriented Riemannian manifold $(M^n,g)$, any vector field X can be decomposed as follows:

$$\begin{array}{c}\hfill X=\nabla h+Y,\end{array}$$

(5)

where h and Y, respectively, denote a smooth function and a free divergence vector field on $M^n$, namely, $\mathrm{div}\left(Y\right)=0$. As usual, h is called the Hodge–de Rham potential (see [19] for more details).

Proof of Theorem 1. 

From (1) and (5), we know that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{L}_{Y}g={\displaystyle \frac{1}{2}}{L}_{X}g-{\displaystyle \frac{1}{2}}{L}_{\nabla h}g=\psi g-\mathrm{Hess}\left(h\right).\end{array}$$

(6)

Hence, in order to prove that $(M^n,g)$ admits a gradient conformal soliton structure, it is sufficient and necessary to show that $L_{Y}g=0$.

Taking the trace of (6), we obtain

$$\mathrm{div}\left(Y\right)=\mathrm{div}\left(X\right)-\Delta h=n\psi -\Delta h.$$

Since $\mathrm{div}\left(Y\right)=0$, it follows that

$$\begin{array}{c}\hfill \mathrm{div}\left(X\right)=\Delta h=n\psi .\end{array}$$

(7)

From (6), we directly compute that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{4}}{\int }_{M^n}{\left|L_{Y}g\right|}^{2}d{V}_g& ={\int }_{M^n}(n{\psi }^2-2g(\psi g,\mathrm{Hess}\left(h\right))+{\left|\mathrm{Hess}\left(h\right)\right|}^2)d{V}_g\hfill \\ \hfill & ={\int }_{M^n}(n{\psi }^2-2\psi \Delta h+{\left|\mathrm{Hess}\left(h\right)\right|}^2)d{V}_g\hfill \\ \hfill & ={\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2-n{\psi }^2)d{V}_g.\hfill \end{array}$$

(8)

On the other hand, using (5) and the property of the Ricci curvature tensor, we have

$$\mathrm{Ric}(X,X)=\mathrm{Ric}(Y,Y)+2\mathrm{Ric}(\nabla h,Y)+\mathrm{Ric}(\nabla h,\nabla h).$$

Hence,

$$\begin{array}{c}\hfill 2{\int }_{M^n}\mathrm{Ric}(\nabla h,Y)d{V}_g={\int }_{M^n}(\mathrm{Ric}(X,X)-\mathrm{Ric}(Y,Y)-\mathrm{Ric}(\nabla h,\nabla h))d{V}_g.\end{array}$$

(9)

Now, we are ready to calculate every term on the right-hand side of (9).

Using (1), (3) and (7), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\Delta \left|X\right|}^2-{|\nabla X|}^2+\mathrm{Ric}(X,X)=& \mathrm{div}\left(L_{X}g\right)\left(X\right)-{\nabla }_X\mathrm{div}\left(X\right)\hfill \\ \hfill =& -(n-2)g(\nabla \psi ,X).\hfill \end{array}$$

(10)

From (5), we easily have

$$\begin{array}{c}\hfill {|\nabla X|}^2={\left|\mathrm{Hess}\left(h\right)\right|}^2+2g(\nabla \nabla h,\nabla Y)+{|\nabla Y|}^2.\end{array}$$

(11)

Integrating for (11) and using the formula

$$\Delta \nabla h-\nabla \Delta h=\mathrm{Ric}(\nabla h,·),$$

we deduce that

$$\begin{array}{cc}\hfill {\int }_{M^n}{|\nabla X|}^{2}d{V}_g=& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2+2g(\nabla \nabla h,\nabla Y)+{|\nabla Y|}^2)d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2-2g(\Delta \nabla h,Y)+{|\nabla Y|}^2)d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2-2ng(\nabla \psi ,Y)-2\mathrm{Ric}(\nabla h,Y)+{|\nabla Y|}^2)d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2+2n\psi \mathrm{div}\left(Y\right)-2\mathrm{Ric}(\nabla h,Y)+{|\nabla Y|}^2)d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2-2\mathrm{Ric}(\nabla h,Y)+{|\nabla Y|}^2)d{V}_g.\hfill \end{array}$$

(12)

Since $M^n$ is closed, integrating for (10), using (12) and the divergence theorem, we obtain

$$\begin{array}{cc}\hfill {\int }_{M^n}\mathrm{Ric}(X,X)d{V}_g=& {\int }_{M^n}{\left(\right|\nabla X|}^2-(n-2)g(\nabla \psi ,X))d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2+{|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y)+(n-2)\psi \mathrm{div}\left(X\right))d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2+{|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y)+n(n-2){\psi }^2)d{V}_g.\hfill \end{array}$$

(13)

Taking the divergence of (6), we deduce that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{div}\left(L_{Y}g\right)\left(Y\right)& ={\displaystyle \frac{1}{2}}\mathrm{div}\left(L_{X}g\right)\left(Y\right)-{\displaystyle \frac{1}{2}}\mathrm{div}\left(L_{\nabla h}g\right)\left(Y\right)\hfill \\ \hfill & =\mathrm{div}\left(\psi g\right)\left(Y\right)-\mathrm{div}\left(\mathrm{Hess}\right(h\left)\right)\left(Y\right)\hfill \\ \hfill & =g(\nabla \psi ,Y)-\mathrm{div}\left(\mathrm{Hess}\right(h\left)\right)\left(Y\right).\hfill \end{array}$$

(14)

From (3) and (4), we have

$$\begin{array}{c}\hfill \mathrm{div}\left(L_{Y}g\right)\left(Y\right)={\displaystyle \frac{1}{2}}{\Delta \left|Y\right|}^2-{|\nabla Y|}^2+\mathrm{Ric}(Y,Y)+{\nabla }_Y\mathrm{div}\left(Y\right)\end{array}$$

(15)

and

$$\begin{array}{c}\hfill \mathrm{div}\left(\mathrm{Hess}\right(h\left)\right)\left(Y\right)=\mathrm{Ric}(\nabla h,Y)+g(\nabla \Delta h,Y).\end{array}$$

(16)

Substituting (15) and (16) into (14), we obtain

$$\mathrm{Ric}(Y,Y)={|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y)-{\displaystyle \frac{1}{2}}\Delta {\left|Y\right|}^2-2(n-1)g(\nabla \psi ,Y).$$

Using divergence theorem, we arrive at

$$\begin{array}{cc}\hfill {\int }_{M^n}\mathrm{Ric}(Y,Y)d{V}_g=& {\int }_{M^n}{\left(\right|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y)-2(n-1)g(\nabla \psi ,Y))d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y)+2(n-1)\psi \mathrm{div}\left(Y\right))d{V}_g\hfill \\ \hfill =& {\int }_{M^n}{\left(\right|\nabla Y|}^2-2\mathrm{Ric}(\nabla h,Y))d{V}_g.\hfill \end{array}$$

(17)

From (3) and (4), we have

$$\mathrm{div}\left(L_{\nabla h}g\right)(\nabla h)={\displaystyle \frac{1}{2}}{\Delta |\nabla h|}^2-{\left|\mathrm{Hess}\left(h\right)\right|}^2+g(\nabla \Delta h,\nabla h)+\mathrm{Ric}(\nabla h,\nabla h)$$

and

$$\mathrm{div}\left(L_{\nabla h}g\right)(\nabla h)=2\mathrm{Ric}(\nabla h,\nabla h)+2g(\nabla \Delta h,\nabla h),$$

which imply that

$$\mathrm{Ric}(\nabla h,\nabla h)={\displaystyle \frac{1}{2}}{\Delta |\nabla h|}^2-g(\nabla \Delta h,\nabla h)-{\left|\mathrm{Hess}\left(h\right)\right|}^2.$$

Hence,

$$\begin{array}{cc}\hfill {\int }_{M^n}\mathrm{Ric}(\nabla h,\nabla h)d{V}_g& =-{\int }_{M^n}{(g(\nabla \Delta h,\nabla h)+|\mathrm{Hess}\left(h\right)|}^2)d{V}_g\hfill \\ \hfill & =-{\int }_{M^n}{(ng(\nabla \psi ,\nabla h)+|\mathrm{Hess}\left(h\right)|}^2)d{V}_g\hfill \\ \hfill & ={\int }_{M^n}{(n\psi \Delta h-|\mathrm{Hess}\left(h\right)|}^2)d{V}_g\hfill \\ \hfill & ={\int }_{M^n}(n^2{\psi }^2-{\left|\mathrm{Hess}\left(h\right)\right|}^2)d{V}_g.\hfill \end{array}$$

(18)

Finally, substituting (13), (17) and (18) into (9), we arrive at

$$\begin{array}{c}\hfill {\int }_{M^n}\mathrm{Ric}(\nabla h,Y)d{V}_g={\int }_{M^n}{\left(\right|\mathrm{Hess}\left(h\right)|}^2-n{\psi }^2)d{V}_g.\end{array}$$

(19)

Combining (8) and (19), we have

$$0\le {\displaystyle \frac{1}{4}}{\int }_{M^n}{\left|L_{Y}g\right|}^{2}d{V}_g={\int }_{M^n}\mathrm{Ric}(\nabla h,Y)d{V}_g.$$

Under the assumption ${\int }_{M^n}\mathrm{Ric}(\nabla h,Y)d{V}_g\le 0$, we obtain

$${\int }_{M^n}{\left|L_{Y}g\right|}^{2}d{V}_g=0,$$

which shows that $L_{Y}g=0$.

This completes the proof of Theorem 1. □

Proof of Theorem 2. 

Taking the trace of (1) and (2), we have

$$\begin{array}{c}\hfill \mathrm{div}\left(X\right)=\Delta f=n\psi .\end{array}$$

(20)

Taking the divergence of (5), we have

$$\mathrm{div}\left(X\right)=\Delta h.$$

Hence,

$$\begin{array}{c}\hfill \Delta h=n\psi .\end{array}$$

(21)

Combining (20) and (21), we obtain

$$\Delta (f-h)=0,$$

which shows that $f=h+C$ for some constant C.

We complete the proof of Theorem 2. □

3. Complete Conformal Solitons

In this section, we complete the proof of Theorem 3 and Corollary 1. We give a triviality result for complete conformal solitons with a nonpositive Ricci curvature under an integral condition.

Proof of Theorem 3. 

Taking the trace of (1), we have

$$\begin{array}{c}\hfill \mathrm{div}\left(X\right)=n\psi .\end{array}$$

Hence,

$${\nabla }_X\mathrm{div}\left(X\right)=ng(X,\nabla \psi )$$

and

$$\mathrm{div}\left(L_{X}g\right)\left(X\right)=2g(X,\nabla \psi ).$$

Further, using (3), we obtain

$$\begin{array}{c}\hfill {|\nabla X|}^2={\displaystyle \frac{1}{2}}\Delta {\left|X\right|}^2+\mathrm{Ric}(X,X)+(n-2)g(X,\nabla \psi ).\end{array}$$

(22)

Taking a cut-off function $\varphi$ on $M^n$ such that $0\le \varphi \le 1$ and $\varphi =1$ on $B_R\left(x_0\right)$, $Supp\left(\varphi \right)\subset B_{2R}\left(x_0\right)$, ${|\nabla \varphi |}^2\le {\displaystyle \frac{C}{R^2}}$, $\Delta \varphi \le {\displaystyle \frac{C}{R}}$.

Multiplying (22) with ${\varphi }^2$ and using the divergence theorem, we obtain

$$\begin{array}{cc}\hfill {\int }_{B_{2R}}{|\nabla X|}^2{\varphi }^{2}d{V}_g=& {\displaystyle \frac{1}{2}}{\int }_{B_{2R}}{\varphi }^2\Delta {\left|X\right|}^{2}d{V}_g+{\int }_{B_{2R}}\mathrm{Ric}(X,X){\varphi }^{2}d{V}_g\hfill \\ \hfill & +(n-2){\int }_{B_{2R}}{\varphi }^{2}g(X,\nabla \psi )d{V}_g\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2\Delta {\varphi }^{2}d{V}_g+{\int }_{B_{2R}}\mathrm{Ric}(X,X){\varphi }^{2}d{V}_g\hfill \\ \hfill & -(n-2){\int }_{B_{2R}}({\varphi }^2\psi \mathrm{div}\left(X\right)+\psi g(X,\nabla {\varphi }^2))d{V}_g\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2\Delta {\varphi }^{2}d{V}_g+{\int }_{B_{2R}}\mathrm{Ric}(X,X){\varphi }^{2}d{V}_g\hfill \\ \hfill & -2(n-2){\int }_{B_{2R}\setminus B_R}\varphi \psi g(X,\nabla \varphi )d{V}_g\hfill \\ \hfill & -n(n-2){\int }_{B_{2R}}{\psi }^2{\varphi }^{2}d{V}_g.\hfill \end{array}$$

(23)

Using the Cauchy–Schwarz inequality, it follows from (23) that

$$\begin{array}{cc}\hfill 0\le & {\displaystyle \frac{1}{2}}n(n-2){\int }_{B_{2R}}{\psi }^2{\varphi }^{2}d{V}_g\hfill \\ \hfill \le & {\int }_{B_{2R}}{|\nabla X|}^2{\varphi }^{2}d{V}_g+{\displaystyle \frac{1}{2}}n(n-2){\int }_{B_{2R}}{\psi }^2{\varphi }^{2}d{V}_g\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2\Delta {\varphi }^{2}d{V}_g+{\int }_{B_{2R}}\mathrm{Ric}(X,X){\varphi }^{2}d{V}_g\hfill \\ \hfill & +C{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2{|\nabla \varphi |}^{2}d{V}_g\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2\Delta {\varphi }^{2}d{V}_g+C{\int }_{B_{2R}\setminus B_R}{\left|X\right|}^2{|\nabla \varphi |}^{2}d{V}_g\hfill \\ \hfill \le & C{\int }_{B_{2R}\setminus B_R}{\displaystyle \frac{{\left|X\right|}^2}{R^2}}d{V}_g\hfill \\ \hfill \le & {\displaystyle \frac{C}{R^2}}{\int }_{M^n}{\left|X\right|}^{2}d{V}_g\to 0,\hspace{1em}\mathrm{as}\hspace{1em}R\to \infty ,\hfill \end{array}$$

where C is a positive constant. It shows that $\psi =0$ and $L_{X}g=0$. Hence, $M^n$ is trivial.

This completes the proof of Theorem 3. □

From the proof of Theorem 3, we easily obtain Corollary 1 and only need to take $\varphi :=1$.

4. Complete Noncompact Gradient Conformal Solitons

In this section, we study the relationship between the rigidity of gradient conformal solitons and the p-harmonic map from this soliton into a smooth Riemannian manifold. By the property of the stress p-energy tensor of this map, choosing a suitable cut-off function and integrating by parts, the classification results for complete gradient conformal solitons are derived.

We next introduce some basic notations and definitions. Let $(M,g)$ and $(N,h)$ be two smooth Riemannian manifolds. For a smooth map $u:M\to N$, the energy functional is defined by

$$E_p\left(u\right)={\displaystyle \frac{1}{p}}{\int }_M{\left|du\right|}^{p}d{V}_g.$$

It is clear that the critical points of the functional are p-harmonic maps. If $du=0$, then the map is trivial; namely, it is a constant map. In particular when $p=2$, this map is called a harmonic map. It is characterized by the vanishing of the p-tension field, which is given by

$${\tau }_p\left(u\right)={\mathrm{tr}}_g(\overline{\nabla }{\left(\right|du|}^{p-2}du\left)\right),$$

where $\overline{\nabla }$ denotes the induced connection on the vector bundle ${\varphi }^{-1}TN$.

The stress p-energy tensor of the p-harmonic map u is defined as

$$\begin{array}{c}\hfill S_p\left(u\right)(X,Y)={\displaystyle \frac{1}{p}}{\left|du\right|}^{p}g(X,Y)-{\left|du\right|}^{p-2}\langle du\left(X\right),du\left(Y\right)\rangle ,\end{array}$$

(24)

where $X,Y$ are smooth vector fields on M.

It is well known that the stress p-energy tensor satisfies the equation

$$\begin{array}{c}\hfill \mathrm{div}{S}_p\left(u\right)=-\langle {\tau }_p\left(u\right),du\rangle ,\end{array}$$

which implies that if u is a p-harmonic map, then $\mathrm{div}{S}_p\left(u\right)=0$ (see [20] and the references therein).

We now introduce an important result that plays a crucial role in the proof of our classification results.

Lemma 2 

([21]). An n-dimensional complete nontrivial gradient conformal soliton $(M^n,g,\nabla f,\psi )$ is either compact and rotationally symmetric or the warped product

$$(\mathbb{R},d{s}^2){\times }_{|\nabla f|}({\widehat{N}}^{n-1},\widehat{g}),$$

where the scalar curvature ${\widehat{R}}_{\widehat{g}}$ of ${\widehat{N}}^{n-1}$ satisfies

$${|\nabla f|}^2{R}_g={\widehat{R}}_{\widehat{g}}-(n-1)(n-2){\psi }^2-2(n-1)g(\nabla f,\nabla \psi ),$$

or rotationally symmetric and equal to the warped product

$$([0,\infty ),d{s}^2){\times }_{|\nabla f|}({\mathbb{S}}^{n-1},{\overline{g}}_S),$$

where ${\overline{g}}_S$ is the round metric on ${\mathbb{S}}^{n-1}$. Furthermore, the potential function f depends only on s.

Proof of Theorem 6. 

Since u is a p-harmonic map, then $\mathrm{div}{S}_p\left(u\right)=0$. Hence,

$$\begin{array}{c}\hfill \langle \nabla f,\mathrm{div}{S}_p\left(u\right)\rangle =0.\end{array}$$

(25)

Let ${\left\{e_i\right\}}_{i=1}^n$ be an orthonormal basis of the tangent bundle $TM$ of $M^n$ such that ${\nabla }_{e_i}{e}_j=0$ at a fixed point $x_0\in M^n$, where $i,j=1,\cdots ,n$.

Multiplying (25) with ${\eta }^2$ and integrating by parts, where $\eta$ is a function with compact support, we obtain

$$\begin{array}{cc}\hfill 0=& {\int }_{M^n}{\eta }^2\langle \nabla f,\mathrm{div}{S}_p\left(u\right)\rangle d{V}_g\hfill \\ \hfill =& \sum _{i,j=1}^n{\int }_{M^n}\langle {\eta }^2{e}_i\left(f\right),e_j\left(S_p\left(u\right)(e_i,e_j)\right)\rangle d{V}_g\hfill \\ \hfill =& -\sum _{i,j=1}^n{\int }_{M^n}{\eta }^2\langle e_i{e}_j\left(f\right),S_p\left(u\right)(e_i,e_j)\rangle d{V}_g\hfill \\ \hfill & -\sum _{i,j=1}^n{\int }_{M^n}\langle e_i\left({\eta }^2\right)e_j\left(f\right),S_p\left(u\right)(e_i,e_j)\rangle d{V}_g\hfill \\ \hfill =& -\sum _{i,j=1}^n{\int }_{M^n}{\eta }^2\langle e_i{e}_j\left(f\right),{\displaystyle \frac{1}{p}}{\left|du\right|}^p{g}_{ij}-{\left|du\right|}^{p-2}\langle du\left(e_i\right),du\left(e_j\right)\rangle \rangle d{V}_g\hfill \\ \hfill & -{\int }_{M^n}{S}_p\left(u\right)(\nabla f,\nabla {\eta }^2)d{V}_g\hfill \\ \hfill =& -{\displaystyle \frac{1}{p}}{\int }_{M^n}{\eta }^2{\left|du\right|}^p\Delta fd{V}_g-{\int }_{M^n}{S}_p\left(u\right)(\nabla f,\nabla {\eta }^2)d{V}_g\hfill \\ \hfill & +\sum _{i,j=1}^n{\int }_{M^n}{\eta }^2{\left|du\right|}^{p-2}{f}_{ij}\langle du\left(e_i\right),du\left(e_j\right)\rangle d{V}_g,\hfill \end{array}$$

which can be rewritten as

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{p}}{\int }_{M^n}{\eta }^2{\left|du\right|}^p\Delta fd{V}_g=& \sum _{i,j=1}^n{\int }_{M^n}{\eta }^2{\left|du\right|}^{p-2}{f}_{ij}\langle du\left(e_i\right),du\left(e_j\right)\rangle d{V}_g\hfill \\ \hfill & -{\int }_{M^n}{S}_p\left(u\right)(\nabla f,\nabla {\eta }^2)d{V}_g.\hfill \end{array}$$

(26)

Substituting (2) into (26), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{n-p}{p}}{\int }_{M^n}{\eta }^2\psi {\left|du\right|}^{p}d{V}_g=-{\int }_{M^n}{S}_p\left(u\right)(\nabla f,\nabla {\eta }^2)d{V}_g.\end{array}$$

(27)

Now, we take a cut-off function $\eta$ on $M^n$ such that

$$\begin{array}{c}\hfill \eta \left(r\right)=\left\{\begin{array}{cc}1\hfill & r\le R,\hfill \\ 2-{\displaystyle \frac{\log r}{\log R}}\hfill & r\in [R,R^2],\hfill \\ 0\hfill & r\ge R^2.\hfill \end{array}\right.\end{array}$$

where $r\left(x\right)$ is the distance function from x to $x_0$ (see [22]).

Since $|\nabla f|$ has at most linear growth, namely, $|\nabla f|\le Cr$ for some constant $C>0$, using (24), we obtain

$$\begin{array}{cc}\hfill |{\int }_{M^n}{S}_p\left(u\right)(\nabla f,\nabla {\eta }^2)d{V}_g|\le & C{\int }_{M^n}|\nabla {\eta }^2{\left|\right|\nabla f\left|\right|du|}^{p}d{V}_g\hfill \\ \hfill & \le 2C{\int }_{B_R\setminus B_{\sqrt{R}}}{|\nabla \eta ||\nabla f|\left|du\right|}^{p}d{V}_g\hfill \\ \hfill & \le {\displaystyle \frac{4C}{\log R}}{\int }_{B_R\setminus B_{\sqrt{R}}}{\displaystyle \frac{{\left|du\right|}^{p}r\left(x\right)}{r\left(x\right)}}d{V}_g\hfill \\ \hfill & \le {\displaystyle \frac{4C}{\log R}}{\int }_{B_R}{\left|du\right|}^{p}d{V}_g.\hfill \end{array}$$

(28)

From the assumed conditions, combining (27) and (28), we have

$${\int }_{M^n}{\left|du\right|}^p\psi d{V}_g\to 0,\hspace{1em}\mathrm{as}\hspace{1em}R\to \infty ,$$

which shows that $\psi =0$. Hence, $\nabla \nabla f=0$.

According to Lemma 2, we next divide the arguments into three cases.

Case 1. 

Since $M^n$ is compact, by the standard maximum principle, we find that f is constant. Hence, the case cannot happen.

Case 2. 

$M^n$ is the warped product

$$(\mathbb{R},d{s}^2){\times }_{|\nabla f|}({\widehat{N}}^{n-1},\widehat{g}).$$

Since $\nabla \nabla f=0$, we know ${\nabla |\nabla f|}^2=0$, which shows that $\nabla f$ is a constant vector field. Hence, $f\left(s\right)=as+b$.

Case 3. 

$M^n$ is rotationally symmetric and equal to the warped product

$$([0,\infty ),d{s}^2){\times }_{|\nabla f|}({\mathbb{S}}^{n-1},{\overline{g}}_S).$$

Similarly, $f\left(s\right)=as+b$.

This completes the proof of Theorem 6. □

Letting $\psi :=2(n-1)({\sigma }_k-\lambda )$ and $\psi :=R_g-\lambda$, respectively, we obtain Corollaries 2 and 3.

5. Conclusions

In this paper, we studied the rigidity of conformal solitons (which include Yamabe solitons and k-Yamabe solitons) as well as gradient conformal solitons and gave an equivalent characterization for closed conformal solitons to be gradient conformal solitons. This result extends that of Tokura et al. [9] for closed k-Yamabe solitons, and we proved that complete conformal solitons with a nonpositive Ricci curvature must be trivial under an integral condition. This result generalizes the result of L. Bo et al. [12] for complete k-Yamabe solitons. In particular, by using a p-harmonic map from a complete noncompact gradient conformal soliton into a smooth Riemannian manifold, we classified complete noncompact nontrivial gradient conformal solitons under some suitable conditions. Similar results were given for gradient Yamabe solitons and gradient k-Yamabe solitons.