Area-minimizing Hypersurfaces in Singular Ambient Manifolds

TL;DR

This paper investigates the regularity of area-minimizing hypersurfaces within singular ambient manifolds with nonnegative scalar curvature, establishing codimension bounds for singular sets and demonstrating the sharpness of these bounds.

Contribution

It proves that the singular set of such hypersurfaces has codimension at least 3 and provides an example showing this bound is optimal.

Findings

Singular set has codimension at least 3.

Existence of hypersurfaces with singularities of dimension n-3.

Regularity bounds are sharp.

Abstract

We study area-minimizing hypersurfaces in singular ambient manifolds whose tangent cones have nonnegative scalar curvature on their regular parts. We prove that the singular set of the hypersurface has codimension at least 3 in our settings. We also give an example in which n-dimensional minimizing hypersurface in an ambient space as given above may have singularities of dimension $n-3$. This shows that our regularity result is sharp.