The space of partially free boundary minimal half disks

TL;DR

This paper develops a degree theory for partially free boundary minimal half disks in a 3D half ball, establishing their space as a Banach manifold and analyzing associated elliptic operators with mixed boundary conditions.

Contribution

It introduces a novel degree theory framework for these minimal surfaces and proves their space forms a Banach manifold, advancing understanding of free boundary minimal surfaces.

Findings

The space of such surfaces is a Banach manifold.

The projection map is a Fredholm map of index zero.

Established a well-defined mod-2 degree for the projection map.

Abstract

This paper forms part of our ongoing works on the existence of complete non-compact free boundary minimal planes in an asymptotically flat three-dimensional Riemannian manifold with boundary. We set up the degree theory for the space of properly embedded partially free boundary minimal half disks in a three-dimensional half ball. We prove that the space of such surfaces is a Banach manifold. The projection map which projects partially free boundary minimal half disks into their Dirichlet boundaries is a Fredholm map of index zero and has a well-defined mod-2 degree under suitable assumptions. The new major analytic difficulty is to analyze the properties of a second order elliptic operator with mix boundary conditions on a domain with corners.