On Stable Capillary Hypersurfaces with Planar Boundaries

• Published in: The Journal of geometric analysis Vol. 33; no. 6
• Main Author: Souam, Rabah
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Contact angle; Convex and Discrete Geometry; Differential Geometry; Domains; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Half spaces; Hyperplanes; Hyperspaces; Mathematics; Mathematics and Statistics; Spherical caps; Constant mean curvature hypersurfaces; 53A10; Stability; 49Q10; Capillary hypersurfaces
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01257-2

We study stable immersed capillary hypersurfaces Σ in domains B of R n + 1 bounded by hyperplanes. When B is a half-space, we show Σ is a spherical cap. When B is a domain bounded by k hyperplanes P 1 , … , P k , 2 ≤ k ≤ n + 1 , having independent normals, and Σ has contact angle θ i with P i and does not touch the edges of B , we prove there exists δ > 0 , depending only on P 1 , ⋯ , P k , so that if θ i ∈ ( π 2 - δ , π 2 + δ ) for each i ,  then Σ has to be a piece of a sphere.