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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 ℝ3 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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Cite

BibTeXRIS
BibTeX
@misc{yeon2026characterizations,
  title = {{Characterizations of the plane and the catenoid as capillary minimal surfaces}},
  author = {Eungbeom Yeon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32447},
  eprint = {2610.32447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32447},
}
RIS
TY  - UNPB
TI  - Characterizations of the plane and the catenoid as capillary minimal surfaces
AU  - Eungbeom Yeon
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.32447
UR  - https://arxiv.org/abs/2610.32447
AB  - In this paper, we prove that an annular capillary minimal surface outside the unit ball in 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.
ER  -