Compact singular minimal surfaces with boundary

TL;DR

This paper investigates the geometric properties of compact singular minimal surfaces based on boundary information, providing estimates, conditions for rotational symmetry, and non-existence results for certain boundary configurations.

Contribution

It introduces new boundary-based estimates and conditions for minimal surfaces, including criteria for rotational symmetry and non-existence results for specific boundary setups.

Findings

Derived area and height estimates from boundary data.

Identified conditions under which the surface is rotational.

Proved non-existence of surfaces with certain distant boundary curves.

Abstract

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally, we deduce non-existence results when the boundary is formed by two curves that are sufficiently far apart.