Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds

Abstract

Let (M m , g) be a compact Riemannian manifold isometrically immersed in a simply connected space form (euclidean space, sphere or hyperbolic space). The purpose of this paper is to give optimal upper bounds for the first nonzero eigenvalue of the Laplacian of (M m , g) in terms of r-th mean curvatures and scalar curvature. As consequences, we obtain some rigidity results. In particular, we prove that if (M n , g) is a compact hypersurface of positive scalar curvature immersed in R n+1 and if g is a Yamabe metric, then (M n , g) is a standard sphere.

Extrinsic radius pinching in space forms of nonnegative sectional curvature

“…Indeed, Reilly gave in [20] a sharper upper bound for the first eigenvalue of the Laplacian for hypersurfaces of R n+1 . The analogue of this upper bound was proved by Grosjean for hypersurfaces of S n+1 and H n+1 (see [12]). …”

Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds

“…[1][2][3][4]15]). However, to the authors' knowledge, there are no known estimates for the higher order eigenvalues of L r .…”

“…It should be mentioned that the problem we study include the eigenvalue problem of the linearized operator L r of the r -th mean curvature of a hypersurface which is closely related to the stability problem of hypersurfaces of constant r -th mean curvature in a space form. The first non-zero eigenvalue of L r has been studied recently (see e.g., [1][2][3][4]9,15] and the reference therein). To the authors' knowledge, very little is known about its higher order eigenvalues.…”

r-Minimal submanifolds in space forms

“…Now we will give the relations between the rth mean curvature functions, rth mean curvature vector fields and the tensors T r , some of these relations are given in [11]. When the codimension is 1, the related results can be found in [1-3, 5-7, 10, 21, 23].…”