Sharp local estimates for the Szeg\"o-Weinberger profile in Riemannian manifolds

TL;DR

This paper establishes sharp local asymptotic bounds for the Szeg"o-Weinberger profile, specifically the first nontrivial Neumann eigenvalue, in geodesic balls within Riemannian manifolds, highlighting the influence of scalar curvature.

Contribution

It provides the first sharp asymptotic estimates for the Neumann eigenvalue profile in Riemannian manifolds, extending previous Dirichlet eigenvalue results and addressing new challenges.

Findings

Derived sharp asymptotic bounds in terms of scalar curvature

Established a local comparison principle based on scalar curvature

Identified difficulties due to degeneracy of eigenvalues in Euclidean balls

Abstract

We study the local Szeg\"o-Weinberger profile in a geodesic ball $B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$. This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $\mu_2$ of the Laplace-Beltrami Operator $\Delta_g$ on $\M$ among subdomains of $B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of $\Delta_g$, but additional difficulties arise due to the fact that $\mu_2$ is degenerate in the unit ball in $\R^N$ and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.