Reciprocal sums of Neumann eigenvalues in non-Euclidean space forms

Abstract

Let $M_{\kappa }^n$ be the simply connected space form of dimension $n\ge 2$ and constant sectional curvature $\kappa \in \{-1,1\}$. For every bounded connected smooth domain $\Omega \subset M_{\kappa }^n$, assume in the case $\kappa =1$ that $\Omega$ is contained in an open hemisphere, and let $B_{\Omega }$ be a geodesic ball with $|B_{\Omega }|=|\Omega |$. We prove $$\sum _{j=1}^n\frac{1}{{\mu }_j(\Omega )}\ge \frac{n}{{\mu }_1(B_{\Omega })},$$ where ${\mu }_j(\Omega )$ are the positive Neumann eigenvalues of $\Omega$. Equality holds if and only if $\Omega$ is a geodesic ball. This proves a conjecture proposed by Xia and Wang [Math. Ann. 385, 2023, 863-879].

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