Generic regularity of isoperimetric regions in dimension eight

TL;DR

This paper proves that in eight-dimensional closed Riemannian manifolds, isoperimetric regions typically have smooth, nondegenerate boundaries under generic conditions on the metric and volume, advancing understanding of geometric regularity.

Contribution

It establishes generic regularity results for isoperimetric regions in dimension eight, showing smoothness of boundaries under generic metrics and volumes, which was previously unknown.

Findings

Isoperimetric regions have smooth, nondegenerate boundaries for generic metrics.

Regularity holds for fixed volume and generic metrics, and vice versa.

Results apply specifically to eight-dimensional closed Riemannian manifolds.

Abstract

We establish generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight. In particular, we show that every isoperimetric region has a smooth nondegenerate boundary for a generic choice of smooth metric and enclosed volume, or for a fixed enclosed volume and a generic choice of smooth metric.