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arXiv · 2610.32447

Characterizations of the plane and the catenoid as capillary minimal surfaces

Eungbeom Yeon
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Abstract

In this paper, we prove that an annular capillary minimal surface outside the unit ball in $\mathbb{R}^3$ with one complete embedded end and finite total curvature must be either part of the plane or part of the catenoid. The proofs are based on the use of the Weierstrass data through the Hopf differential and associated holomorphic functions. For a planar end, we control the second fundamental form, while for a catenoidal end, we analyze the Gauss map in more detail using the boundary condition. We also prove nonexistence results for annular free boundary minimal surfaces with one end asymptotic to the end of Enneper's surface, under appropriate conditions. Finally, we establish an exterior analogue of Choe's result on capillary minimal surfaces with vertices.

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Eungbeom Yeon. 2026-10-23. Characterizations of the plane and the catenoid as capillary minimal surfaces. https://arxiv.org/abs/2610.32447

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