On The Existence of Min-Max Minimal Surface of Genus $g\geq 2$

TL;DR

This paper develops a min-max theory for genus g≥2 minimal surfaces, showing the existence of branched minimal surfaces with bubble trees using variational methods, extending previous theories to higher genus cases.

Contribution

It introduces a new min-max framework for minimal surfaces of higher genus, utilizing direct variational methods and bubble tree limits, expanding prior work to all genera.

Findings

Existence of genus g≥2 minimal surfaces via min-max methods

Achievement of min-max value by bubble tree limits

Extension of min-max theory to all genera

Abstract

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-$g$ minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding-Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory developed by Colding-Minicozzi and the author to all genera.