Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

TL;DR

This paper characterizes the shape of stable hypersurfaces with free boundary in convex cones with homogeneous densities, showing they are spherical intersections centered at the cone's vertex.

Contribution

It proves that under convexity and curvature-dimension conditions, the only stable solutions are spherical caps centered at the vertex, extending classical results to weighted settings.

Findings

Stable hypersurfaces are spherical caps centered at the vertex.

Unique minimizers are intersections of the cone with round spheres.

Results apply to convex cones with homogeneous densities.

Abstract

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.