Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space

TL;DR

This paper establishes stability results for the first eigenvalue of second order divergence-type operators on hypersurfaces in Euclidean space, with applications to $r$-stability and almost-Einstein hypersurfaces.

Contribution

It introduces new stability bounds for the first eigenvalue of these operators and explores their implications for geometric stability and Einstein-like properties.

Findings

Derived upper bounds for the first eigenvalue.

Applied results to $r$-stability of hypersurfaces.

Connected eigenvalue bounds to almost-Einstein conditions.

Abstract

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to $r$-stability as well as to almost-Einstein hypersurfaces.