Isoperimetric inequalities and monotonicity formulas for submanifolds in warped product manifolds

TL;DR

This paper establishes isoperimetric inequalities and monotonicity formulas for submanifolds in warped product manifolds, providing new volume bounds and insights into minimal surfaces within these geometries.

Contribution

It introduces new linear isoperimetric inequalities and monotonicity formulas for submanifolds in specific warped product manifolds, with equality cases and applications to minimal surfaces.

Findings

Proved linear isoperimetric inequalities in de Sitter-Schwarzschild and Reissner-Nordstrom manifolds

Derived monotonicity formulas for submanifolds with bounded mean curvature

Established volume lower bounds and an isoperimetric inequality for minimal surfaces

Abstract

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as consequences, we give lower bound estimates for the volume of these submanifolds in terms of the warping function. We conclude the paper with an isoperimetric inequality for minimal surfaces.