A Viscosity Method in the Min-max Theory of Minimal Surfaces

TL;DR

This paper introduces a viscosity-based penalization method in min-max theory for minimal surfaces, enabling the construction of smooth minimal immersions via a limiting process of critical points.

Contribution

It develops a novel viscosity approach to relax the area functional, allowing min-max construction of minimal surfaces and proving smoothness of the resulting minimal immersions.

Findings

Constructs critical points of a relaxed area functional satisfying Palais-Smale condition.

Establishes varifold convergence of critical points to smooth minimal immersions.

Shows that min-max minimal surfaces are realized by smooth possibly branched immersions.

Abstract

We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $\Sigma$ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^q$ norm of the second fundamental form of the immersion times a "viscosity" parameter. This relaxation of the area functional satisfies the Palais-Smale condition for $q>2$. This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the "viscosity" parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in a recent work in collaboration with A.Pigati that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.