Torsional Rigidity on Compact Riemannian Manifolds with lower Ricci Curvature Bounds

TL;DR

This paper establishes a reverse Hölder inequality for the fundamental eigenfunction and an isoperimetric inequality for torsional rigidity on domains within compact Riemannian manifolds with lower Ricci curvature bounds.

Contribution

It introduces new inequalities linking eigenfunctions and torsional rigidity to curvature bounds, advancing geometric analysis on Riemannian manifolds.

Findings

Proved a reverse Hölder inequality for eigenfunctions.

Established an isoperimetric inequality for torsional rigidity.

Connected curvature bounds with spectral and geometric properties.

Abstract

In this article we prove a reverse H\"older inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.