Vol. 94, No. 1, 1981

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On the spectrum of Cartan-Hadamard manifolds

Mark Allan Pinsky

Vol. 94 (1981), No. 1, 223–230
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 (d1)2k24 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
Authors
Mark Allan Pinsky
Northwestern University
IL
United States
http://www.math.northwestern.edu/people/facultyProfiles/mark.pinsky.html