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 [Reference Hamilton6] 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 [Reference Lott7]. Besides, the characterization of expanding Ricci soliton has attracted the attention of many researchers, see for instance [Reference Catino2, Reference Chan3, Reference Ma8–Reference Schulze and Simon11].
In the steady case, Hamilton [Reference Hamilton6] 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 [Reference Bryant1].
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 [Reference Deruelle5] 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 [Reference Catino2], 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 [Reference Deruelle5] and Catino et al. [Reference Catino2], 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 [Reference Chen4], 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 [Reference Yau12, 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
\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 [Reference Petersen and Wylie10], we have
\begin{equation}\Delta R=-2|Ric|^{2}+\langle \nabla R, \nabla f\rangle.\end{equation}
Note that
\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 [Reference Deruelle5],
$\Sigma$
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 [Reference Deruelle5], 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, [Reference Petersen and Wylie10], 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
\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
$\mathbb{R}^n.$
Acknowledgments
The author is partially supported by Paraíba State Research Foundation (FAPESQ), Brazil, grant 3025/2021 and CNPq, Brazil, grant 306524/2022-8, respectively.
Data availability
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
