An upper bound for the first nonzero Neumann eigenvalue

TL;DR

This paper establishes an upper bound for the first nonzero Neumann eigenvalue of a domain in a Riemannian manifold with curvature constraints, relating it to a model space form's eigenvalue scaled by a constant.

Contribution

It provides a new upper bound for Neumann eigenvalues in curved spaces, extending classical results to more general Riemannian manifolds with curvature bounds.

Findings

The upper bound depends on volume, diameter, and dimension.

Equality holds for geodesic balls in space forms.

The bound generalizes known Euclidean results.

Abstract

Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with sectional curvature $K_{\mathbb{M}} \leq k$ and Ricci curvature $\text{Ric}_{\mathbb{M}} \geq (n-1)K$, where $k,K \in \mathbb{R}$. Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$. Here $B_{k}(R)$ is a geodesic ball of radius $R > 0$ in the simply connected space form $\mathbb{M}_{k}$ such that vol$(\Omega)$ = vol$(B_{k}(R))$, and $\mathcal{C}$ is a constant which depends on the volume, diameter of $\Omega$ and the dimension of $\mathbb{M}$.