Anisotropic min-max via phase transitions

TL;DR

This paper introduces a PDE-based method to construct anisotropic min-max hypersurfaces on Riemannian manifolds, extending Allen-Cahn techniques to anisotropic surface energies and proving their convergence to stationary varifolds.

Contribution

It develops an anisotropic Allen-Cahn framework, establishes gradient bounds, and constructs anisotropic min-max hypersurfaces, extending prior isotropic results to anisotropic settings.

Findings

Established a Modica-type gradient bound for anisotropic Allen-Cahn critical points.

Proved energy concentration along stationary anisotropic varifolds.

Constructed possibly singular anisotropic min-max hypersurfaces.

Abstract

We develop a PDE-based approach to the min-max construction of nontrivial integer rectifiable varifolds that are stationary with respect to anisotropic surface energies on closed Riemannian manifolds, in codimension one. Specifically, we study the anisotropic analogue of the Allen-Cahn energy and establish a Modica-type gradient bound for its critical points. Using this in conjunction with certain estimates for stable solutions, we then prove that the energy densities of stable or bounded-Morse-index critical points of its rescalings concentrate along an integer rectifiable varifold that is stationary for the underlying anisotropic integrand. As a consequence, we construct a (possibly singular) anisotropic min-max hypersurface via Allen-Cahn, obtaining an analogue of the result of Hutchinson-Tonegawa in the anisotropic setting.