On the spectrum of Cartan-Hadamard manifolds

Mark Allan Pinsky

Vol. 94 (1981), No. 1, 223–230

DOI: 10.2140/pjm.1981.94.223

Abstract

Let M be a simply-connected complete d-dimensional Riemannian manifold of nonpositive sectional curvature K. If K ≦−k² < 0, then the infimum of the L² spectrum of the negative Laplacian is greater than or equal to (d− 1)²k²∕4 with equality in case K →−k² sufficiently fast at infinity. This general result is obtained by analyzing a system of ordinary differential equations. If either d = 2 or the manifold possesses appropriate symmetry, the result is obtained under weaker conditions by analyzing a Riccati equation. Finally the case k = 0 is treated separately.

Mathematical Subject Classification 2000

Primary: 58G25

Secondary: 53C20

Milestones

Received: 23 January 1980

Revised: 25 April 1980

Published: 1 May 1981