Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature

Tom Carroll; Jesse Ratzkin

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arXiv · 1407.0864

Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature

Tom Carroll·Jesse Ratzkin

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Abstract

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $Ω\subset M$ be a suitable domain, and let $λ(Ω)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $Ω$. We prove several bounds for the rate of decrease of $λ(Ω)$ and $Ω$ increases, and a result comparing the rate of decrease of $λ$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest

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BibTeX RIS

Tom Carroll, Jesse Ratzkin. 2014-07-03. Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature. https://arxiv.org/abs/1407.0864

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