Capillary surfaces in a convex cone

Abstract

We show that a compact embedded hypersurface $${S\subset\mathbb{R}^{n+1}}$$ with constant higher-order mean curvature in a convex piecewise smooth cone C which is perpendicular to ∂C is part of a hypersphere. Also we prove that an embedded disk type constant mean curvature surface $${S\subset\mathbb{R}^3}$$ in a convex polyhedral cone C which makes constant contact angles with ∂C is a spherical cap if C has at most five faces. This condition on the number of faces can be dropped if C is a right cone over a regular n-gon and the contact angles are the same on ∂S.