Sharp upper bound for the first eigenvalue

TL;DR

This paper establishes sharp upper bounds for the first eigenvalue of the Laplacian on closed hypersurfaces within certain noncompact symmetric and Riemannian manifolds, extending spectral geometry bounds in curved spaces.

Contribution

It provides the first sharp upper bounds for the first eigenvalue of the Laplacian on hypersurfaces in these specific curved ambient spaces.

Findings

Sharp upper bounds for the first eigenvalue are derived.

Results apply to hypersurfaces in noncompact rank-1 symmetric spaces.

Bounds depend on curvature constraints of the ambient manifold.

Abstract

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq K_{\mathbb{M}} \leq \delta^2$ or $K_{\mathbb{M}} \leq k$ where $k = -\delta^2$ or 0. In this paper we give sharp upperbounds for the first eigenvalue of laplacian of $M$.