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

Jean-François Grosjean

Vol. 206 (2002), No. 1, 93–112

DOI: 10.2140/pjm.2002.206.93

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 (Mⁿ,g) is a compact hypersurface of positive scalar curvature immersed in ℝⁿ⁺¹ and if g is a Yamabe metric, then (Mⁿ,g) is a standard sphere.

Milestones

Received: 28 December 2000

Published: 1 September 2002