(Georgia Institute of Technology, 1994-03)
Harrell, Evans M.; Michel, Patricia L.
We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace-Beltrami
operators on manifolds. We significantly improve earlier bounds of Yang and Yau,
Li, and Harrell.
(Georgia Institute of Technology, 1994)
Harrell, Evans M.
Consider the operator - ∇^2 - q(κ), where - ∇^2 is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S^2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue λ [1] is uniquely maximized, among manifolds of fixed area, by the true sphere.