On the Second Eigenvalue of the Laplace Operator Penalized by Curvature
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Abstract
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.
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1994
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