Integral formulas for hypersurfaces in cones and related questions
Filomena Pacella, Giulio Tralli
TL;DR
The paper analyzes Minkowski-type integral identities for hypersurfaces within a cone that meet the boundary orthogonally, correcting a second identity previously reported in CP. It provides two proofs of the corrected Minkowski formula and uses these results to establish a corrected rigidity theorem for constant mean curvature hypersurfaces under a star-shaped and integral boundary condition, showing the hypersurface must be a spherical sector relative to the cone. A stability-based eigenvalue criterion is then derived, linking the geometry to the first Neumann eigenvalue $\lambda_1(\Gamma)$ via a Reilly-type averaging argument, which allows relaxation of convexity assumptions in isoperimetric-type problems inside cones. Collectively, the work clarifies the role of boundary terms in Minkowski identities, refines rigidity results for CMC surfaces in cones, and highlights spectral data as a control parameter for stability in geometric variational problems.
Abstract
We discuss the validity of Minkowski integral identities for hypersurfaces inside a cone, intersecting the boundary of the cone orthogonally. In doing so we correct a formula provided in [3]. Then we study rigidity results for constant mean curvature graphs proving the precise statement of a result given in [9] and [10]. Finally we provide an integral estimate for stable constant mean curvature hypersurfaces in cones.