Geometric inequalities involving three quantities in warped product manifolds

TL;DR

This paper establishes sharp geometric inequalities involving three quantities for hypersurfaces in warped product manifolds, with applications to eigenvalue bounds and curvature integrals, extending previous results in the field.

Contribution

It introduces two new families of inequalities relating geometric quantities in warped product manifolds, generalizing and extending prior work on eigenvalues and curvature integrals.

Findings

Sharp inequalities for boundary momentum, area, and volume in warped products.

Applications to eigenvalue bounds for Steklov and Wentzell problems.

Extension of previous curvature integral inequalities.

Abstract

In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family concerns the $k$-th boundary momentum, area, and weighted volume, and has applications to Weinstock-type inequalities for Steklov or Wentzell eigenvalues on star-shaped mean convex domains. This generalizes the main results of [12]. The second family involves a weighted $k$-th mean curvature integral and two distinct quermassintegrals and extends the authors' recent work [33] with G. Wheeler and V.-M. Wheeler.