Estimates for eigenvalues of the operator $L_r$

TL;DR

This paper derives sharp estimates for the eigenvalues of a specific elliptic operator on compact submanifolds in space forms, extending understanding of spectral properties in geometric analysis.

Contribution

It provides new sharp eigenvalue estimates for the operator $L_r$ on submanifolds in space forms, a novel result in geometric spectral theory.

Findings

Eigenvalue estimates are sharp.

Results apply to submanifolds in arbitrary codimension.

Advances spectral analysis of elliptic operators in geometry.

Abstract

In this paper, we consider an eigenvalue problem of the elliptic operator $$ L_r={\rm div}(T^r\nabla\cdot )$$ on compact submanifolds in arbitrary codimension of space forms $\mathbb{R}^N(c)$ with $c\geq0$. Our estimates on eigenvalues are sharp.