On the dimension of the singular set of perimeter minimizers in spaces with a two-sided Ricci curvature bound

TL;DR

This paper establishes that in non-collapsed Ricci limit spaces with two-sided Ricci curvature bounds, the singular set of perimeter minimizers has Hausdorff dimension at most N-5, and this bound is proven to be optimal.

Contribution

The paper proves a sharp upper bound on the Hausdorff dimension of the singular set in Ricci limit spaces, extending geometric measure theory in this context.

Findings

Hausdorff dimension of singular set ≤ N-5

Bound is sharp and cannot be improved

Results apply to non-collapsed Ricci limit spaces

Abstract

We show that the Hausdorff dimension of the singular set of perimeter minimizers in non-collapsed Ricci limit spaces with a two-sided Ricci curvature bound is at most $N-5$, where $N$ is the dimension of the ambient space. The estimate is sharp.