Free boundary and capillary minimal surfaces in spherical caps I: Low genus

TL;DR

This paper explores the geometry of free boundary and capillary minimal surfaces within spherical caps, extending previous results to warped products and high codimension, and establishing dualities and uniqueness properties.

Contribution

It introduces a dual operation linking free boundary and capillary minimal surfaces and extends geometric properties to higher codimension in spherical caps.

Findings

Extended intersection properties to warped products

Established duality between free boundary and capillary minimal surfaces

Discussed uniqueness of minimal annuli

Abstract

This is the first of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset\mathbb{S}^3$. In this article, we extend our previous half-space intersection properties to warped products, and extend (non-)umbilicity of discs and annuli to capillary minimal surfaces in high codimension. We establish a dual operation relating free boundary and capillary minimal surfaces. These results are discussed in a continuous, unified framework, particularly in relation to uniqueness of minimal annuli.