The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

TL;DR

This paper establishes the existence of smooth, closed surfaces with prescribed constant anisotropic mean curvature in 3D Riemannian manifolds, partially resolving a longstanding conjecture and expanding understanding of anisotropic minimal surfaces.

Contribution

It proves the existence of anisotropic min-max surfaces with prescribed mean curvature, including singularity analysis, in closed 3D manifolds, addressing a conjecture by Allard.

Findings

Existence of smooth anisotropic min-max surfaces with at most one singular point.

Construction of surfaces with any prescribed constant anisotropic mean curvature.

Partial resolution of Allard's conjecture in three dimensions.

Abstract

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.