# Characterizations of the plane and the catenoid as capillary surfaces

## TL;DR

This paper characterizes certain capillary minimal surfaces outside the unit ball, showing they must be planes or catenoids under specific conditions, and rules out some configurations based on flux and asymptotic behavior.

## Contribution

It provides new classification results for capillary minimal surfaces with finite total curvature outside convex domains, extending understanding of their geometric properties.

## Key findings

- Capillary minimal surfaces outside the unit ball with one embedded end are either planes or catenoids.
- Such surfaces cannot exist with certain asymptotic behaviors if flux conditions are met.
- Capillary minimal surfaces outside convex domains bounded by spheres are necessarily parts of planes.

## Abstract

In this paper we prove that a capillary minimal surface outside the unit ball in $\mathbb {R}^3$ with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary minimal surface outside the unit ball with one end asymptotic to the end of the Enneper's surface and finite total curvature cannot exist if the flux vector vanishes on the first homology calss of the surface. Furthermore, we prove that a capillary minimal surface outside the convex domain bounded by several spheres with one embedded end and finite total curvature must be part of the plane.