preprint

Inserted: 24 sep 2024
Last Updated: 24 sep 2024

Year: 2024

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).

Tags: ANGEVA