Hypersurfaces with free boundary in a convex conical domain in a space form

This study presents rigidity results for hypersurfaces with free boundary in a convex conical domain in a space form. Let $C$ be a conical domain with a piecewise linear boundary in a space form. A compact hypersurface with free boundary $M\subset C$ is part of a geodesic sphere if either $C$ is convex and $M$ is an embedded and has constant higher order mean curvature, or $H_l$ does not vanish and the ratio $H_r/H_{l},$ $l \lt r$, is constant on $M$.