Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds

TL;DR

This paper extends symmetric rearrangement inequalities from Euclidean space to certain Riemannian manifolds, exploring geometric measure theory tools and conditions needed for these inequalities to hold in curved spaces.

Contribution

It introduces methods to generalize rearrangement inequalities to Riemannian manifolds, including proving a smooth co-area formula and reformulating geometric inequalities in this setting.

Findings

Rearrangement inequalities hold on specific Riemannian manifolds under certain conditions.

The smooth co-area formula is established for these manifolds.

Geometric inequalities are reformulated for curved spaces.

Abstract

This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and P\'{o}lya-Szeg\H{o} inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n = (0,\infty)\times \Sigma^{n-1}$, and test what results carry over from the $\mathbb{R}^n$ setting or what assumptions about $M^n$ need to be added. Of particular interest was proving the smooth co-area formula in the Riemannian manifolds setting and re-formulating particular geometric inequalities.