Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

TL;DR

This paper proves that compact, free-boundary, anisotropic minimal hypersurfaces within convex solid cones are necessarily parts of Wulff-shapes, extending the understanding of geometric variational problems with boundary conditions.

Contribution

It establishes that such hypersurfaces minimizing anisotropic area under volume constraints are characterized as segments of Wulff-shapes within convex cones.

Findings

Hypersurfaces are contained in Wulff-shapes.

Results extend classical minimal surface theory to anisotropic, free-boundary settings.

Provides geometric characterization of area-minimizing hypersurfaces in convex cones.

Abstract

We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $\Sigma$ immersed in $\mathcal{C}$ with free boundary in $\partial\mathcal{C}\setminus\{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape.