Free boundary and capillary minimal surfaces in spherical caps I: Low genus

2025/12/14 by Keaton Naff, Naff, Keaton, Jonathan J. Zhu +1 · 2 citations
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2512.12877

openalex publication_date 2025/12/14 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

This is the first of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls B_R⊂\mathbbS³. In this article, we extend our previous half-space intersection properties to warped products, and extend (non-)umbilicity of discs and annuli to capillary minimal surfaces in high codimension. We establish a dual operation relating free boundary and capillary minimal surfaces. These results are discussed in a continuous, unified framework, particularly in relation to uniqueness of minimal annuli.

Citations

• Free boundary and capillary minimal surfaces in spherical caps II: Low energy
• New Homogeneous Solutions for the One-Phase Free Boundary Problem
• New free boundary minimal annuli of revolution in the 3-sphere
• Free boundary CMC annuli in spherical and hyperbolic balls
• Half-space intersection properties for minimal hypersurfaces
• An overdetermined eigenvalue problem and the Critical Catenoid conjecture
• Eigenvalue problems and free boundary minimal surfaces in spherical caps
• Free boundary minimal annuli immersed in the unit ball
• Sufficient symmetry conditions for free boundary minimal annuli to be the critical catenoid
• On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball
• On the intersection of minimal hypersurfaces of S^k
• A two-piece property for free boundary minimal surfaces in the ball
• A partially overdetermined problem in a half ball
• Overdetermined problems and constant mean curvature surfaces in cones
• Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk
• Compactness of minimal hypersurfaces with bounded index
• Uniqueness theorems for free boundary minimal disks in space forms
• Free boundary minimal annuli in convex three-manifolds
• Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
• Constant mean curvature surfaces in warped product manifolds
• The first Steklov eigenvalue, conformal geometry, and minimal surfaces
• Smooth compactness of self-shrinkers

Cited by

• A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere
• Free boundary and capillary minimal surfaces in spherical caps II: Low energy

Related

• Free boundary and capillary minimal surfaces in spherical caps II: Low energy
• Free boundary minimal annuli immersed in the unit ball
• Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk
• Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
• Characterizations of the plane and the catenoid as capillary surfaces