Characterizations of the plane and the catenoid as capillary minimal surfaces

Abstract

In this paper, we prove that an immersed 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.

arXiv comment16 pages, 1 figure. In response to referee comments, an issue with a reference used in Theorem 1.1 has been corrected, and the proof has been revised accordingly. The statement of Theorem 1.2 has been revised to exclude certain boundary umbilic cases. This revision was posted as arXiv:2610.32447 by mistake. Withdrawal of that duplicate submission has been requested