Some Isoperimetric Inequalities and Eigenvalue Estimates in Weighted Manifolds

TL;DR

This paper establishes new inequalities relating weighted mean curvature and eigenvalues of the drift Laplacian for submanifolds in weighted manifolds, advancing geometric analysis in this context.

Contribution

It introduces general inequalities involving weighted mean curvature and provides an extrinsic upper bound for the first non-zero eigenvalue of the drift Laplacian.

Findings

Derived a relative linear isoperimetric inequality for submanifolds

Established an upper bound for the first non-zero eigenvalue of the drift Laplacian

Connected geometric curvature properties with spectral estimates

Abstract

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.