Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds

TL;DR

This paper establishes new upper bounds for eigenvalues of divergence-type operators and Steklov problems on submanifolds within weighted Riemannian manifolds, extending classical spectral bounds to weighted settings.

Contribution

It introduces Reilly-type upper bounds for eigenvalues in weighted manifolds, generalizing existing results to divergence operators and Steklov problems.

Findings

Derived upper bounds for divergence-type operator eigenvalues

Established bounds for Steklov eigenvalues in weighted manifolds

Extended classical spectral inequalities to weighted geometric contexts

Abstract

We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.