Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons

Let $\left(\bar{M}^{n+1},\bar{g},f\right)$ be a smooth metric measure space and let $M^{n}$ be a oriented hypersurface in $\bar{M}^{n+1}$. The weighted mean curvature vector $\vec{H}_{f}$ is defined by

$$\vec{H}_{f}=\vec{H}+\left(\bar{\nabla}f\right)^{\perp}$$

and the weighted mean curvature $H_{f}$ is defined by $\vec{H}_{f}=-H_{f}N$, where $\perp$ is the projection to the normal bundle and $N$ is the unit normal vector. A hypersurface $M^{n}$ is said to have constant weighted mean curvature (CWMC) if $H_{f}$ is a constant (see Section 2). These hypersurfaces are also known as $\lambda$-hypersurfaces where $\lambda=H_{f}$. CWMC hypersurfaces can be seen as critical points of the weighted area functional with respect to weighted volume-preserving variations (see [9], [23]). When $H_{f}=0$ they are called $f$-minimal hypersurfaces. For certain choices of $f$, $f$-minimal hypersurfaces are very important singularities of the mean curvature flow in $\mathbb{R}^{n+1}$, namely self-shrinkers (see p.758 and p.768 in [15]), translating solitons (see p.153 and p.154 in [19]) and self-expanders (see p.9016 and p.9017 in [3]). It turns out that $\left(\mathbb{R}^{n+1},\bar{g}_{\text{can}},f\right)$ is a Ricci soliton for each of these choices of $f$ (see Section 2). Thus the study of self-shrinkers, translating solitons and self-expanders in $\mathbb{R}^{n+1}$ is equivalent to the study of $f$-minimal hypersurfaces in the shrinking, steady and expanding Ricci soliton $\left(\mathbb{R}^{n+1},\bar{g}_{\text{can}},f\right)$ respectively.

There has been a great interest in CWMC hypersurfaces in the Gaussian shrinking Ricci soliton $\left(\mathbb{R}^{n+1},\bar{g}_{\text{can}},f\right)$ with $f(x)=\frac{|x|^{2}}{4}$. In [23] Mcgonagle-Ross classified stable CWMC hypersurfaces properly immersed in the Gaussian shrinking Ricci soliton. They also proved that there are no CWMC hypersurfaces properly immersed in this ambient space with index one. In [7] Q.M.Cheng-Ogata-Wei classified complete CWMC hypersurfaces in the Gaussian shrinking Ricci soliton with a certain condition on the norm of the second fundamental form and the mean curvature. In [9] Q.M.Cheng-Wei classified complete CWMC hypersurfaces embedded in the Gaussian shrinking Ricci soliton with polynomial volume growth and $H-H_{f}\geq 0$. In [17] Guang classified compact CWMC surfaces embedded in the Gaussian shrinking Ricci soliton with $H_{f}\geq 0$ and constant norm of the second fundamental form. He also classified complete CWMC hypersurfaces embedded in the Gaussian shrinking Ricci soliton with a certain condition on the norm of the second fundamental form. In [10] Q.M.Cheng-Wei generalized this result to complete CWMC surfaces and removed the condition on $H_{f}$.

There has also been a great interest in f-minimal and CWMC hypersurfaces in the cylinder shrinking Ricci soliton $\left(\mathbb{R}^{n+1-k}\times S_{\sqrt{2\left(k-1\right)}}^{k},\bar{g},f\right)$, where $\bar{g}$ is the product metric and $f\left(x,y\right)=\frac{\left|x\right|^{2}}{4}$ (here $x$ is the position vector in $\mathbb{R}^{n-k+1}$ and $y$ is the position vector in $\mathbb{R}^{k+1}$). In [12] X.Cheng-Mejia-Zhou classified compact f-minimal hypersurfaces in a cylinder shrinking Ricci soliton with a certain condition on the norm of the second fundamental form. They also classified compact f-minimal hypersurfaces in this ambient space with index one. In [14] X.Cheng-Zhou generalized these results to complete hypersurfaces. In [2] Barbosa-Santana-Upadhyay obtained theorems for CWMC hypersurfaces similar to the results of Mcgonagle-Ross [23] but in the cylinder shrinking Ricci soliton ambient space. They also obtained theorems for CWMC hypersurfaces in cylinder shrinking Ricci soliton similar to the results of X.Cheng-Mejia-Zhou [12] and X.Cheng-Zhou [14] about $L_{f}$-stable (which are those that the second variation of weighted area is non-negative) $f$-minimal hypersurfaces.

In this paper we study geometric properties and classification results for CWMC hypersurfaces in shrinking Ricci solitons.

Let $\left(\bar{M}^{n+1},\bar{g},f\right)$ be a gradient Ricci soliton with

$$\bar{R}ic+\bar{\nabla}\bar{\nabla}f=\lambda\bar{g},$$

where $\lambda$ is a constant. We know that

$$\left|\bar{\nabla}f\right|^{2}+\bar{R}-2\lambda f=C,$$

where $\bar{R}$ is the scalar curvature of $\bar{M}^{n+1}$ and $C$ is a constant (see p.85 in [18] and Lemma 1.1 in [4]). Let us define

$$D^{\pm}(c_{1},c_{2})=\left\{x\in\bar{M}^{n+1};f\left(x\right)<\Gamma^{\pm}(x)\right\},$$

where

$$\Gamma^{\pm}(x)=\frac{1}{2\lambda}\left\{\frac{1}{4}\left(\pm c_{1}+\sqrt{c_{1}^{2}+4c_{2}}\right)^{2}+\bar{R}(x)-C\right\}$$

and $c_{1}$ and $c_{2}$ are constants such that $c_{1}^{2}+4c_{2}\geq 0$.

Note that $D^{+}(c_{1},c_{2})$ and $D^{-}(c_{1},c_{2})$ are open subsets of $\bar{M}^{n+1}$. In this paper we will always use the above notations and conventions.

In Theorem 3 in [27], Vieira-Zhou proved that if $M^{n}$ is a complete $f$-minimal hypersurface properly immersed in the cylinder shrinking Ricci soliton then: (a) $M^{n}$ cannot lie inside the closed product $\bar{B}^{n+1-k}_{\sqrt{2\left(n-k\right)}}\left(0\right)\times S_{\sqrt{2(k-1)}}^{k}(0)$, unless $M^{n}=S^{n-k}_{\sqrt{2\left(n-k\right)}}\left(0\right)\times S_{\sqrt{2(k-1)}}^{k}(0)$ (see also the earlier work of Cavalcante-Espinar [6] for CWMC hypersurfaces in the Gaussian shrinking Ricci soliton $\mathbb{R}^{n+1}$); (b) $M^{n}$ cannot lie outside the product $B^{n+1-k}_{\sqrt{2\left(n+1-k\right)}}\left(0\right)\times S_{\sqrt{2(k-1)}}^{k}(0)$. We generalize this result in two different ways: we extend the result to CWMC hypersurfaces and we consider an arbitrary shrinking Ricci soliton ambient space.

We remark that $\bar{\nabla}\bar{\nabla}f$ is a $(0,2)$ tensor in $\bar{M}^{n+1}$, so we can consider its restriction to $M^{n}$ (in this case the trace $tr_{M^{n}}\bar{\nabla}\bar{\nabla}f$ is a function in $M^{n}$). Note that we can state Theorem 1 in a different way using the fact that

$$tr_{M^{n}}\bar{\nabla}\bar{\nabla}f=n\lambda-\bar{R}+\bar{R}ic\left(N,N\right),$$

where $N$ is the unit normal vector of $M^{n}$.

Using the above result together with a classification result for the level sets of the potential function (see Theorem 17) we obtain a new result for the cylinder shrinking Ricci soliton ambient space.

In Theorem 3.5 in [17], Guang proved that if $M^{n}$ is a compact CWMC hypersurface in the Gaussian shrinking Ricci soliton $\mathbb{R}^{n+1}$ satisfying $H_{f}\geq 0$ and

$$|A|^{2}\leq\frac{1}{2}+\frac{H_{f}\left(H_{f}+\sqrt{H_{f}^{2}+2n}\right)}{2n},$$

then $M^{n}=S^{n}_{r}(0)$, where $r=-H_{f}+\sqrt{H_{f}^{2}+2n}$. We generalize this result in two different ways: we assume an upper bound on the mean curvature (this assumption is weaker since $H^{2}\leq n|A|^{2}$) and we consider an arbitrary shrinking Ricci soliton ambient space. Here $A$ is the second fundamental form and $H$ is the mean curvature.

In particular, we obtain a new result for the cylinder shrinking Ricci soliton ambient space.

Note that the assumption on the upper bound of $H$ is sharp because equality holds for $M^{n}=S^{n-k}_{r}(0)\times S^{k}_{\sqrt{2(k-1)}}(0)$, where $r=-H_{f}+\sqrt{H_{f}^{2}+2(n-k)}$.

This is a new result even for $f$-minimal hypersurfaces.

In Theorem 1.2 in [7], Q.M.Cheng-Ogata-Wei proved that if $M^{n}$ is a complete CWMC hypersurface in the Gaussian shrinking Ricci soliton $\mathbb{R}^{n+1}$ with polynomial volume growth and satisfying

$$\left(H-\frac{H_{f}}{2}\right)^{2}\geq\frac{H_{f}^{2}}{4}+\frac{n}{2},$$

then $M^{n}=S^{n}_{r}(0)$, where $r=-H_{f}+\sqrt{H_{f}^{2}+2n}$. We generalize this result to any ambient space which is a smooth metric measure space.

In particular, we recover the result of Q.M.Cheng-Ogata-Wei (see Corollary 26) and we obtain a new result for the cylinder shrinking Ricci soliton ambient space.

This is a new result even for $f$-minimal hypersurfaces.

In the second part of this paper, we obtain some classification theorems for the Gaussian shrinking Ricci soliton ambient space. Le-Sesum [21] proved that if $M^{n}$ is a complete self-shrinker in the Gaussian shrinking Ricci soliton $\mathbb{R}^{n+1}$ with polynomial volume growth and satisfying $|A|^{2}<1/2$, then $M^{n}$ is a hyperplane passing through the origin. Later, Cao-Li [5] extended this result to arbitrary codimension by showing that if $|A|^{2}\leq 1/2$, then $M^{n}$ is a generalized cylinder. Later, Guang [17] extended this result to CWMC hypersurfaces by assuming polynomial volume growth and a condition on the norm of the second fundamental form. Many of the classification results for CWMC hypersurfaces assume that the hypersurface has polynomial volume growth in order to use integration techniques. Recently, there has been some papers trying to avoid this assumption by using the Omori-Yau maximum principles ([7], [8], [25], etc). We can replace the assumption of polynomial volume growth in Guang’s result by an assumption on the integral of the second fundamental form.

We remark that for $M^{n}=S^{n}_{r}(0)$ the assumption on the upper bound is not satisfied unless $r=\sqrt{2n}$ (that is $H_{f}=0$) and for $S^{k}_{r}(0)\times\mathbb{R}^{n-k}$ the assumption on the upper bound is not satisfied unless $r=\sqrt{2k}$ (that is $H_{f}=0$). Therefore the result is only sharp in these cases. It would be interesting to find a sharp result for $H_{f}\neq 0$.

This is a new result even for self-shrinkers.

We remark that Ancari-Miranda [1] proved recently it is possible to obtain a similar result if $|A|^{2}\leq 1/2$ and $H\in L^{q}_{f}(M)$ for some even number $q\geq 2$ which is a weaker assumption when $q$ is even.

For self-shrinker submanifolds $M^{n}$ in $\mathbb{R}^{n+p}$, Q.M.Cheng-Peng [8] proved that if $\sup|A|^{2}<1/2$, then $M^{n}$ is a hyperplane in $\mathbb{R}^{n+1}$. For codimension 1 we generalize Q.M.Cheng-Peng’s result to CWMC hypersurfaces.

This paper is organized as follows. In section 2, we describe our notations and conventions and prove some basic results. In section 3, we prove Theorem 1, Theorem 3, Theorem 7 and related results. In Section 4, we prove Theorem 10 and related results.

In this section we describe our notations and conventions and prove some basic results.

Smooth metric measure spaces. Let $\left(M,g\right)$ be a Riemannian manifold and let $f$ be a smooth function on $M$. The triple $\left(M,g,f\right)$ is called a smooth metric measure space. The measure $e^{-f}dvol$ is called weighted volume. If $u$ and $v$ are functions on $M^{n}$ the $L_{f}^{2}$ inner product $u$ and $v$ is defined by

$$\left\langle u,v\right\rangle_{L_{f}^{2}\left(M\right)}=\int_{M}uve^{-f}$$

and the $L_{f}^{p}$ norm of $u$ is defined by

$$\left|u\right|_{L_{f}^{p}\left(M\right)}=\left(\int_{M}\left|u\right|^{p}e^{-f}\right)^{\frac{1}{p}},$$

where $p\geq 1$. The operator

$$\Delta_{f}=\Delta-\left\langle\nabla f,\nabla\cdot\right\rangle$$

is called weighted Laplacian. It is well known that the weighted Laplacian is a densely defined self-adjoint operator in $L_{f}^{2}$, that is, if $u$ and $v$ are smooth functions on $M$ with compact support we have

$$\int_{M}\left(\Delta_{f}u\right)ve^{-f}=-\int_{M}\left\langle\nabla u,\nabla v\right\rangle e^{-f}.$$

The Bakry-Émery-Ricci curvature is defined by

$$Ric_{f}=Ric+\nabla\nabla f.$$

The triple $\left(M,g,f\right)$ is called a gradient Ricci soliton if

$$Ric_{f}=\lambda g,$$

where $\lambda$ is a constant. If $\lambda$ is positive, zero or negative it is called shrinking, steady or expanding, respectively.

Hypersurfaces in smooth metric measure spaces. Let $\left(\bar{M}^{n+1},\bar{g},f\right)$ be a smooth metric measure space and let $M^{n}$ be a oriented hypersurface of $\bar{M}^{n+1}$. The second fundamental form is defined by

$$A\left(u,v\right)=\langle\bar{\nabla}_{u}v,N\rangle,$$

where $u$ and $v$ are vector fields on $M^{n}$, $\bar{\nabla}$ is the Riemannian connection of $\bar{M}^{n+1}$ and $N$ is the unit normal vector. The mean curvature vector is defined by

$$\vec{H}=(tr_{M^{n}}A)N,$$

The weighted mean curvature vector is defined by

$$\vec{H}_{f}=\vec{H}+\left(\bar{\nabla}f\right)^{\perp}.$$

We remark that

$$\vec{H}_{f}=\sum_{i=1}^{n}\langle\bar{\nabla}_{e_{i}}e_{i},N\rangle N+\langle\bar{\nabla}f,N\rangle N.$$

We see that $\vec{H}_{f}$ does not depend on the choice of the normal vector. The weighted mean curvature is defined by

$$\vec{H}_{f}=-H_{f}N.$$

A hypersurface is said to have constant weighted mean curvature (CWMC) if the weighted mean curvature is constant. Note that $\left(M^{n},g,f\right)$ is also a smooth metric measure space with weighted measure $e^{-f}dvol_{M}$ and weighted Laplacian

$$\Delta_{f}=\Delta-\left\langle\nabla f,\nabla\cdot\right\rangle,$$

where $g=\bar{g}|M$, $\nabla$ is the Riemannian connection of $M^{n}$ and $\Delta$ is the Laplacian of $M^{n}$.

In the rest of the section we prove some results for the level set of the potential function of a shrinking Ricci soliton.

Using the normalization $\lambda=1/2$ and $C=\bar{R}$ (which is used in many papers, see for example pag 2 in [4]) we have:

For 3-dimensional shrinking Ricci solitons we have:

For 4-dimensional shrinking Ricci solitons we have:

Another consequence of Theorem 17 is the following.