Dimension Bound of Singular Set of One-Phase Free Boundary Problems in Spaces with Two-Sided Ricci Bound

TL;DR

This paper establishes a sharp upper bound of n-5 on the Hausdorff dimension of the singular set in one-phase free boundary problems within noncollapsed Ricci-bounded manifolds.

Contribution

It provides the first dimension bound for singular sets in such free boundary problems under two-sided Ricci curvature bounds.

Findings

Hausdorff dimension of singular set is at most n-5

Bound is sharp in the context of Ricci-bounded manifolds

Results apply to noncollapsed limits of manifolds with Ricci bounds

Abstract

In this article, we show that for one-phase free boundary problems in noncollapsed limits of $n$-dimensional manifolds with two-sided Ricci curvature bounds, the Hausdorff dimension of the singular set of the free boundary can be bounded by $n-5$, which is sharp in this context.