Min-max free boundary minimal surface with genus at least one

TL;DR

This paper develops a min-max theory for free boundary minimal surfaces of genus at least one, demonstrating that the area width can be realized by a bubble tree limit of various minimal surfaces.

Contribution

It introduces a new min-max framework for higher genus free boundary minimal surfaces and describes the structure of the limiting bubble tree.

Findings

Width achieved by bubble tree limit of minimal surfaces.

Includes branched genus g free boundary minimal surfaces with nodes.

Possibly finitely many minimal spheres and disks in the limit.

Abstract

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary components. We show that the width for the area functional can be achieved by a bubble tree limit consisting of branched genus $g$ free boundary minimal surfaces with nodes, and possibly finitely many branched minimal spheres and free boundary minimal disks.