Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

Abstract

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.

Introduction

Let $(M^n,g)$ be a n-dimensional $(n⩾2)$ compact, connected, oriented manifold without boundary, isometrically immersed by X into the $(n+1)$-dimensional Euclidean space ${\mathbb{R}}^{n+1}$. The spectrum of Laplacian of $(M,g)$ is an increasing sequence of real numbers

$$0={\lambda }_0(M)<{\lambda }_1(M)⩽{\lambda }_2(M)⩽\cdots ⩽{\lambda }_k(M)⩽\cdots ⟶+\infty .$$

The eigenvalue 0 (corresponding to constant functions) is simple and ${\lambda }_1(M)$ is the first positive eigenvalue. In [15], Reilly proved the following well-known upper bound for ${\lambda }_1(M)$

$${\lambda }_1(M)⩽\frac{n}{\mathrm{V}(M)}\underset{M}{\int }{H}^{2}d{v}_g,$$

where H is the mean curvature of the immersion. He also proved an analogous inequality involving the higher order mean curvatures. Namely, for $r\in \{1,\cdots ,n\}$

$${\lambda }_1(M){\left(\underset{M}{\int }{H}_{r-1}d{v}_g\right)}^2⩽\mathrm{V}(M)\underset{M}{\int }{H}_r^{2}d{v}_g,$$

where $H_k$ is the k-th mean curvature, defined by the k-th symmetric polynomial of the principal curvatures. Moreover, Reilly studied the equality cases and proved that equality in (1) as in (2) is attained if and only if $X(M)$ is a geodesic sphere.

Later, Alias and Malacarné [2] gave analogues of Reilly inequality of the operators $L_r$ arising in the stability of hypersurfaces with constant higher order mean curvatures, i.e.,

$${\lambda }_1(L_r){\left(\underset{M}{\int }{H}_{s-1}d{v}_g\right)}^2⩽c(r)\left(\underset{M}{\int }{H}_{r}d{v}_g\right)\left(\underset{M}{\int }{H}_s^{2}d{v}_g\right).$$

For $r=s$, they recover the classical inequality proved by Alencar, do Carmo and Rosenberg [1].

On the other hand, a weighted manifold $(\overline{M},\overline{g},{\overline{\mu }}_f)$ is a Riemannian manifold $(\overline{M},\overline{g})$ endowed with a weighted volume form ${\overline{\mu }}_f=e^{-f}d{v}_{\overline{g}}$, where f is a real-valued smooth function on $\overline{M}$ and $d{v}_{\overline{g}}$ is the Riemannian volume form associated with the metric $\overline{g}$. In the present note, we will focus on the case where $(\overline{M},\overline{g})$ is the Euclidean space $({\mathbb{R}}^N,can)$ with its canonical flat metric and we will consider isometric immersions of Riemannian manifolds $(M^n,g)$ into $({\mathbb{R}}^N,can)$. For such an immersion, we define the weighted mean curvature vector ${\mathbf{H}}_f=\mathbf{H}+{(\overline{\mathrm{\nabla }}f)}^{\perp }$, where H is the mean curvature vector of the immersion and ${(\overline{\mathrm{\nabla }}f)}^{\perp }$ is the projection of $\overline{\mathrm{\nabla }}f$ on the normal bundle $T^{\perp }M$.

We can define on M a divergence and a Laplace operator associated with the volume form ${\mu }_f=e^{-f}d{v}_g$ by

$${\mathrm{div}}_{f}Y=\mathrm{div}Y-〈\mathrm{\nabla }f,Y〉\text{and}{\mathrm{\Delta }}_{f}u=-{\mathrm{div}}_f(\mathrm{\nabla }u)=\mathrm{\Delta }u+〈\mathrm{\nabla }f,\mathrm{\nabla }u〉,$$

where ∇ is the gradient on M, that is the projection on TM of $\overline{\mathrm{\nabla }}$. We call them the f-divergence and the f-Laplacian; it is often called Bakry-Émery Laplacian, Witten Laplacian or drifting Laplacian in the literature. It is a classical fact that ${\mathrm{\Delta }}_f$ has a discrete spectrum composed of an infinite sequence of non-negative real numbers. The first eigenvalue is 0, has multiplicity one and corresponds to constant functions.

In [4], Batista, Cavalcante and Pyo proved the following upper bound for the first positive eigenvalue of ${\mathrm{\Delta }}_f$

$${\lambda }_1({\mathrm{\Delta }}_f)⩽\frac{{\int }_M||{\mathbf{H}}_f-\overline{\mathrm{\nabla }}f|{|}^2{\mu }_f}{n{V}_f(M)}=\frac{{\int }_M(||\mathbf{H}|{|}^2+||\mathrm{\nabla }f|{|}^2){\mu }_f}{n{V}_f(M)},$$

where $V_f(M)={\int }_M{\mu }_f$ is the f-volume of M. This inequality is a weighted version of the classical Reilly inequality (1).

More recently, Domingo-Juan and Miquel [6] obtained the same inequality with a more complete characterization of the equality case by the use of mean curvature flow. In [17], the first author extended Batista-Cavalcante-Pyo's result for the operators $L_{T,f}=-{\mathrm{div}}_f(T\mathrm{\nabla })$. Namely, he proved the following Reilly-type upper bound

$${\lambda }_1(L_{T,f}){(\underset{M}{\int }\mathrm{tr}(S){\mu }_f)}^2⩽(\underset{M}{\int }\mathrm{tr}(T){\mu }_f)\underset{M}{\int }(||{\mathbf{H}}_S|{|}^2+||S\mathrm{\nabla }f|{|}^2){\mu }_f,$$

where S and T are two symmetric and divergence-free $(1,1)$-tensors and T is positive definite and ${\mathbf{H}}_S$ and ${\mathbf{H}}_T$ are defined by $\mathrm{tr}(A\circ S)$ and $\mathrm{tr}(A\circ T)$ respectively with A the shape operator of the immersion. For S and T the tensors respectively associated with the higher order mean curvature $H_s$ and $H_r$, this inequality is a weighted extension of (3).

Finally, over the past years, many authors considered geometric problems involving anisotropic mean curvature and higher order mean curvatures (see [9], [10], [13] for instance). The setting is the following. Let $F:{\mathbb{S}}^n⟶{\mathbb{R}}_{+}$ be a smooth function satisfying the following convexity assumption

$$A_F={(\mathrm{\nabla }dF+F\mathrm{Id}{}_{|T_x{\mathbb{S}}^n})}_x>0,$$

for all $x\in {\mathbb{S}}^n$, where ∇dF is the Hessian of F. Here, >0 mean positive definite in the sense of quadratic forms. Now, we consider the following map

$$\begin{array}{cccc}\hfill \varphi :& \hfill {\mathbb{S}}^n\hfill & \hfill ⟶\hfill & {\mathbb{R}}^{n+1}\hfill \\ \hfill & \hfill x\hfill & \hfill ⟼\hfill & F(x)x+{({\mathrm{grad}}_{|S^n}F)}_x.\hfill \end{array}$$

The image $W_F=\varphi ({\mathbb{S}}^n)$ is called the Wulff shape of F and is a smooth hypersurface of ${\mathbb{R}}^{n+1}$. Moreover, from the so-called convexity condition (5), $W_F$ is convex. Note that if F is a positive constant c, the Wulff shape is nothing else but the sphere of radius c. In [8], He obtained upper bounds for the anisotropic operators $L_{r,F}$ which are the anisotropic version of the operators $L_r$. Namely, he proved under the assumption that $H_{r+1}^F>0$,

$${\lambda }_1(L_{r,F}){(\underset{M}{\int }{H}_{s-1}^{F}F(N)d{v}_g)}^2⩽\alpha c(r)(\underset{M}{\int }{H}_r^{F}F(N)d{v}_g)(\underset{M}{\int }{(H_s^F)}^{2}F(N)d{v}_g),$$

where α is constant greater than 1 defined from F and $H_k^F$ is the k-th anisotropic mean curvature. Both definitions is given in the next section. This estimate is the anisotropic generalization of Alias-Malacarné inequality (3).

The main result of the present note is to obtain upper bounds generalizing both inequalities (4) and (6) for a wide class of anisotropic and weighted operators $L_{T,f,F}$ (defined in Section 2). Namely, we have proved the following.

Theorem 1.1

Let $(M^n,g)$ be a closed oriented manifold isometrically immersed into the Euclidean space ${\mathbb{R}}^{n+1}$ endowed with the density $e^{-f}$ and let $F:{\mathbb{S}}^n\to {\mathbb{R}}^{+}$ be a smooth function satisfying the convexity assumption (5). Let S and T be two F-symmetric $(1,1)$-tensors with vanishing F-divergence. Assume moreover that T is positive definite. Then, the first eigenvalue of the operator $L_{T,F,f}$ satisfies

$${\lambda }_1(L_{T,F,f}){(\underset{M}{\int }tr(S){\mu }_{F,f})}^2⩽n\alpha (\underset{M}{\int }tr(T){\mu }_{F,f})\left(\underset{M}{\int }({(H_S^F)}^2+||S({\mathrm{\nabla }}^{F}f)|{|}_{\nu }^2){\mu }_{F,f}\right),$$

where μ is the measure defined by ${\mu }_{F,f}=F(N)e^{-f}d{v}_g$.

Note that the constant α in the statement is the same as Inequality (6) and will be defined below (see relation (9)).