Existence and regularity of min-max anisotropic minimal hypersurfaces

TL;DR

This paper proves the existence and regularity of anisotropic minimal hypersurfaces in closed manifolds using min-max methods, confirming a conjecture and establishing smoothness in three dimensions.

Contribution

It introduces a min-max approach for anisotropic minimal hypersurfaces with singular sets of codimension two, confirming a longstanding conjecture by Allard.

Findings

Existence of anisotropic minimal hypersurfaces in closed manifolds.

Regularity results showing smoothness in three-dimensional cases.

A uniform upper bound for density ratios in the anisotropic setting.

Abstract

In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff measure. In particular, in a closed $3$-manifold, we obtain a smooth anisotropic minimal surface. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture by Allard [Invent. Math., 1983].