Some characterizations of expanding and steady Ricci solitons

Abstract

In this short note, we deal with complete noncompact expanding and steady Ricci solitons of dimension $n\geq 3.$ More precisely, under an integrability assumption, we obtain a characterization for the generalized cigar Ricci soliton and the Gaussian Ricci soliton.

1. Introduction

A gradient Ricci soliton is a Riemannian manifold $\Sigma$ satisfying

\begin{align*}Ric+\nabla^{2}f=\lambda g,\end{align*}

where Ric denotes the Ricci tensor, $f\;:\;\Sigma\rightarrow\mathbb{R}$ is a smooth function, and $\lambda\in\mathbb{R}.$ A Ricci soliton is called expanding, steady or shrinking if, respectively, $\lambda \lt 0, $ $\lambda = 0$ or $ \lambda \gt 0.$ Ricci flow was introduced by Hamilton in his seminal work [6] to study closed three manifolds with positive Ricci curvature. Ricci solitons generate self-similar solutions to the Ricci flow and often arise as singularity models of the flow; therefore, it is important to study and classify them in order to understand the geometry of singularities.

A standard example of expanding Ricci soliton is given by $(\mathbb{R}^{n},g_0, -\frac{|x|^{2}}{4}),$ where $g_0$ is the Euclidean metric. In fact, note that $Ric+\nabla^{2}f=-\frac{1}{2}.$ We recall that an expanding Ricci soliton is related to the limit solution of Type III singularities of the Ricci flow, see [7]. Besides, the characterization of expanding Ricci soliton has attracted the attention of many researchers, see for instance [2, 3, 8–11].

In the steady case, Hamilton [6] discovered the first example of a complete noncompact steady soliton on $\mathbb{R}^{2}$ called the cigar soliton, where the metric is given by $ds^{2}=\frac{dx^{2}+dy^{2}}{1+x^{2}+y^{2}}$ with potential function $f(x,y)=-\log\!(1+x^{2}+y^{2}),$ $(x,y)\in\mathbb{R}^{2}$ . The cigar has positive Gaussian curvature $R = 4e^{f}$ and linear volume growth, and it is asymptotic to a cylinder of finite circumference at infinity. In the three-dimensional case, the known examples are given by quotients of $\mathbb{R}$ , $\mathbb{R}\times\Sigma^{2}$ , where $\Sigma^{2}$ is the cigar soliton, and the rotationally symmetric one constructed by Bryant [1].

We say that $\Sigma$ is a generalized cigar soliton, if $\Sigma$ is isometric to $M\times\mathbb{R}^{n-2},$ where M is the cigar soliton. Recently, Deruelle [5] obtained the following rigidity result to generalized cigar soliton

Theorem 1. Let $\Sigma$ be a complete nonflat noncompact steady gradient Ricci soliton of dimension $n\geq 3$ such that the sectional curvature is nonnegative and $R\in L^{1}(\Sigma).$ Then the universal covering of $\Sigma$ is isometric to $M\times\mathbb{R}^{n-2}$ , where M is the cigar soliton.

In [2], Catino et al. obtained a suitable Bochner-type formula for the tensor $\left(Ric-\frac{R}{2}\right)e^{-f}$ , where R is the scalar curvature, to guarantee that the condition $R\in L^{1}(\Sigma)$ in the above theorem can be relaxed to $\liminf_{r\rightarrow\infty}\frac{1}{r}\int_{B_r(0)}R=0.$ Besides, using a similar strategy they were able to prove the following rigidity result addressed to expanding Ricci solitons

Theorem 2. Let $\Sigma$ be a complete noncompact expanding gradient Ricci soliton of dimension $n\geq 3$ such that the sectional curvature is nonnegative. If $R\in L^1(\Sigma)$ , then $\Sigma$ is isometric to a quotient of the Gaussian soliton $\mathbb{R}^{n}.$

In this paper, motivated by Deruelle [5] and Catino et al. [2], we obtain rigidity results for steady and expanding Ricci solitons under an assumption that the scalar curvature lies in $L^{p}(\Sigma)$ , with respect to a suitable volume element. We point out that our rigidity results are obtained from a different approach. Now, we can state our first result.

Theorem 3. Let $\Sigma$ be a complete noncompact steady gradient Ricci soliton of dimension $n\geq 3$ such that the sectional curvature is nonnegative. If $Re^{-f}\in L^{p}_{-f}(\Sigma)$ , $p>1$ , then $\Sigma$ is either isometric to a quotient of $\mathbb{R}^{n}$ or $M\times\mathbb{R}^{n-2}$ , where M is the cigar soliton.

We recall that, from [4], a complete three-dimensional noncompact steady gradient Ricci soliton has nonnegative scalar curvature. Thus, we conclude that

Corollary 1. Let $\Sigma$ be a complete three-dimensional noncompact steady gradient Ricci soliton. If $Re^{-f}\in L^{p}_{-f}(\Sigma)$ , $p>1$ , then $\Sigma$ is either isometric to a quotient of $\mathbb{R}^{3}$ or $M\times\mathbb{R}$ , where M is the cigar soliton.

Analogously, we can apply the same ideas of Theorem 3 to guarantee a rigidity result addressed to complete noncompact expanding gradient Ricci soliton as follows.

Theorem 4. Let $\Sigma$ be a complete noncompact expanding gradient Ricci soliton of dimension $n\geq 3$ such that the sectional curvature is nonnegative. If $Re^{-f}\in L^{p}_{-f}(\Sigma)$ , $p>1$ , then $\Sigma$ is isometric to a quotient of the Gaussian soliton $\mathbb{R}^{n}.$

2. Proof of the theorems

Let $\psi$ be a smooth function on $\Sigma$ , let us define the weighted Laplacian on $\Sigma^n$ by

\begin{align*}\Delta_{\psi}\varphi=\Delta\varphi-\langle\nabla\psi,\nabla\varphi\rangle\end{align*}

for all $\varphi\in C^{\infty}(\Sigma^n)$ , where $\langle,\rangle$ denotes the Riemannian metric on $\Sigma.$

In what follows, we denote the space of Lebesgue integrable functions on $\Sigma^n$ by

\begin{align*}L^1(\Sigma^n)=\left\{\varphi\in C^\infty(\Sigma^n)\;:\;\int_{\Sigma^n}|\varphi|d\Sigma\lt +\infty\right\},\end{align*}

where $d\Sigma$ stands for the volume element induced by the metric of $\Sigma^n$ . Furthermore, given a smooth function $\psi\;:\;\Sigma\rightarrow\mathbb{R}$ , we denote by $L^1_{\psi}(\Sigma^n)$ the set of Lebesgue integrable functions on $\Sigma^n$ with respect to the modified volume element

\begin{align*}d\mu=e^{-\psi}d\Sigma.\end{align*}

Given an oriented Riemannian manifold $\Sigma^n$ and $p>1$ , we can consider the following space of integrable functions

\begin{align*}L^p_{\psi}(\Sigma^n)=\{\varphi\in C^\infty(\Sigma^n)\;:\;|\varphi|^p\in L^1_{\psi}(\Sigma^n)\}.\end{align*}

From a straightforward adaptation of [12, Theorem 3], we obtain the following criterion of integrability.

Lemma 1. Let $\Sigma^n$ be an n-dimensional complete oriented Riemannian manifold. If $\varphi\in C^\infty(\Sigma^n)$ is a nonnegative $\psi$ -subharmonic function on $\Sigma^n$ and $\varphi\in L^p_{\psi}(\Sigma^n)$ , for some $p>1$ , then $\varphi$ is constant.

Now, we can prove our main result.

Proof of Theorem 3. Let $k\in\mathbb{R}$ be a constant. Thus, a straightforward calculation shows that

(2.1) \begin{equation}\Delta(Re^{kf})=e^{kf}(\Delta R+2k\langle\nabla f,\nabla R\rangle+kR\Delta f+k^{2}R|\nabla f|^{2}).\end{equation}

Since $\Sigma$ is a steady gradient Ricci soliton, from Lemma $2.3$ of [10], we have

(2.2) \begin{equation}\Delta R=-2|Ric|^{2}+\langle \nabla R, \nabla f\rangle.\end{equation}

Note that

(2.3) \begin{equation} e^{kf}\langle \nabla R, \nabla f\rangle=\langle \nabla (e^{kf}R),\nabla f\rangle -Rke^{kf}|\nabla f|^{2}.\end{equation}

Plugging (2.3) and (2.2) into (2.1) and taking the trace of the steady soliton equation, we conclude that:

\begin{align*}\Delta(Re^{kf})-(2k+1)\langle \nabla (e^{kf}R),\nabla f\rangle=e^{kf}(\!-\!2|Ric|^{2}-\!kR^{2}+R|\nabla f|^{2}(\!-\!k^{2}-\!k)).\end{align*}

Finally, from the definition of weighted Laplacian, we get that

\begin{equation*}\Delta_{(2k+1)f}(Re^{kf})=e^{kf}(\!-\!2|Ric|^{2}-\!kR^{2}+R|\nabla f|^{2}(\!-\!k^{2}-\!k))\end{equation*}

Choosing $k=-1$ , we conclude that

\begin{align*}\Delta_{-f}(Re^{-f})=e^{-f}(\!-\!2|Ric|^{2}+R^{2}).\end{align*}

Since the sectional curvature of $\Sigma$ is nonnegative, we get that $-2|Ric|^{2}+R^{2}\geq 0.$ In fact, given $\lambda_k$ , $k=1,2,...,n$ , the eigenvalue of the Ricci tensor, it is not hard to see that $\sum_{i\neq j}\lambda_i>\lambda_j$ and, therefore, $R\geq 2\lambda_j.$ Thus,

\begin{align*}2|Ric|^{2}=2\sum\lambda_i^{2}\leq R\sum\lambda_i=R^{2}.\end{align*}

From above inequality, we conclude that

\begin{align*}\Delta_{-f}(Re^{-f})=e^{-f}(\!-\!2|Ric|^{2}+R^{2})\geq 0.\end{align*}

On the other hand, since $Re^{-f}$ is a nonnegative function and $Re^{-f}\in L^{p}_{-f}(\Sigma)$ , from Lemma 1, we conclude that $Re^{-f}$ is a constant. If R is constant zero, from [5], $\Sigma$ is isometric to a quotient of $\mathbb{R}^{n}.$ If $Re^{-f}=c,$ where c is a nonzero constant, we get that $\Sigma$ has finite $-f$ -volume and, therefore, $R\in L^{1}(\Sigma).$ From [5], we conclude the desired result.

We recall that a complete three-dimensional steady gradient Ricci soliton has nonnegative sectional curvature. Thus, as a consequence of anterior result, we get that

Corollary 2. Let $\Sigma$ be a complete three-dimensional noncompact steady gradient Ricci soliton. If $Re^{-f}\in L^{p}_{-f}(\Sigma)$ , $p>1$ , then $\Sigma$ is either isometric to a quotient of $\mathbb{R}^{3}$ or $M\times\mathbb{R}$ , where M is the cigar soliton.

Now, we are able to prove our rigidity result, in the expanding case, as follows.

Proof of Theorem 4. In fact, since we are supposing that $Ric+\nabla^{2}f=\lambda g,$ from Lemma 2.3, [10], we conclude that

\begin{equation*}\Delta R=-2|Ric|^{2}+2R\lambda+\langle \nabla R, \nabla f\rangle.\end{equation*}

Thus, following the same steps of the anterior result, we conclude from (2.1) and above equation that

\begin{align*}\Delta(Re^{kf})-(2k+1)\langle \nabla (e^{kf}R),\nabla f\rangle=e^{kf}(\!-\!2|Ric|^{2}+2R\lambda+kR(n\lambda-R)+R|\nabla f|^{2}(\!-\!k^{2}-\!k)).\end{align*}

Again, choosing $k=-1$ , we conclude that

(2.4) \begin{equation}\Delta_{-f}(Re^{-f})=e^{-f}(\!-\!2|Ric|^{2}+R^{2} +R(2-n)\lambda)\end{equation}

Since the sectional curvature is nonnegative, reasoning like the anterior result, we get that $-2|Ric|^{2}+R^{2}\geq 0.$ Taking into account that $\lambda \lt 0,$ we get that

\begin{align*}\Delta_{-f}(Re^{-f})\geq 0.\end{align*}

Finally, from Lemma 1, we get that $Re^{-f}$ is a constant and, therefore, from (2.4) we guarantee that $R=0.$ Since $\Sigma$ has nonnegative sectional curvature, we conclude that $\Sigma$ has sectional curvature equals to zero. Thus, we conclude that $\Sigma$ must be a quotient of the Gaussian soliton $\mathbb{R}^n.$