Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.32447

Characterizations of the plane and the catenoid as capillary minimal surfaces

Eungbeom Yeon
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Symmetries and Higher-Form Connections in Derived Differential Geometry

We introduce a general definition of higher-form connections on principal $\infty$-bundles in differential geometry. These connections are built as infinitesimal parallel transports and thus satisfy the fake-curvature condition. We achieve this by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. That allows us to introduce the Atiyah $L_\infty$-algebroid of a principal $\infty$-bundle and establish its global sections as the $L_\infty$-algebra of the derived higher symmetry group of the bundle. We define the space of $p$-form connections on an $\infty$-bundle as the space of order $p$ splittings of its Atiyah $L_\infty$-algebroid. This can be cast equivalently as lifting the classifying map of a bundle on a manifold to the order $p$ truncation of the de Rham stack of the manifold. We demonstrate that our new concept of derived geometric $p$-form connections recovers the known notion of connections on higher U(1)-bundles defined via Čech-Deligne differential cocycles. We further relate the $L_\infty$-algebras of derived higher symmetries of higher U(1)-bundles and higher Courant algebroids. Some applications in higher gauge theory and in supergravity are mentioned.

math.DG↗
arXiv2026

Height and distance-squared functions on implicit plane curves

We introduce families of Lagrange functions associated with height and distance-squared functions on implicit plane curves. We show that degeneracies of these families characterize geometric properties of the curves, such as inflections and vertices, and give geometric interpretations of the corresponding Lagrange multipliers in terms of contact with lines and circles.

math.DG↗
arXiv2026

Minimal Surfaces in $\mathbb{R}^4$ with Three Planar Ends II: Symmetry and Nonexistence

In this paper, we study complete minimal surfaces in $\mathbb{R}^4$ with three embedded planar ends parallel to those of the union of the Lagrangian catenoid and the plane containing its waist circle. We prove that no such complete embedded minimal surface of genus $g\geq 2$ with finite total curvature admits at least $4(g+1)$ symmetries. This contrasts sharply with the genus-one case, where highly symmetric examples do exist and are uniquely determined up to scaling and rigid motion.

math.DG↗