A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

TL;DR

This paper establishes a sharp upper bound for the first eigenvalue of the Laplacian on compact hypersurfaces within rank-1 symmetric spaces, linking geometric curvature and second fundamental form.

Contribution

It provides a novel upper bound for the first Laplacian eigenvalue in rank-1 symmetric spaces based on Ricci curvature and second fundamental form.

Findings

Derived an explicit upper bound involving Ricci curvature and second fundamental form.

Applicable to hypersurfaces in simply connected rank-1 symmetric spaces.

Enhances understanding of spectral geometry in curved ambient spaces.

Abstract

Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$.