Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in $\mathbb{R}^n$, spheres, and closed orientable surfaces

Mehdi Eddaoudi

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arxiv 2502.13089 v1 pith:MXZHRSTX submitted 2025-02-18 math.SP math.DG

Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains in mathbb{R}^n, spheres, and closed orientable surfaces

Mehdi Eddaoudi This is my paper

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keywords eigenvaluesboundedboundsconjecturedomainslowerreciprocalsresult

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We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by Wang and Xia, which connects to a conjecture of Ashbaugh and Benguria. For spheres and surfaces, we extend known results on the first and second eigenvalues, and strengthen an analogous conjecture involving the conformal volume of Li and Yau.

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A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues
math.SP 2026-06 unverdicted novelty 9.0

Proves that the ball minimizes the sum of reciprocals of the first m nonzero Neumann eigenvalues among smooth bounded domains of fixed volume in R^m, with equality only for balls.