Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds

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 Pólya-Szegő 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.

This work was conducted as part of an undergraduate research program (URSS) at the University of Warwick, under the supervision of Dr Maxwell Stolarski. It applies geometric measure theory to study how symmetric rearrangement inequalities in ${\mathbb{R}}^n$ generalize to a certain type of Riemannian manifold with symmetry. The work is a preliminary investigation into these geometric inequalities