17 dec 2025 -- 14:30   [open in google calendar]

Aula Fermi, Scuola Normale Superiore

Abstract.

In the Euclidean space, Aleksandrov's theorem asserts that smooth, closed, constant mean curvature hypersurfaces are round spheres. An effective proof of this result is provided by the characterization of the equality case in the so-called Heintze-Karcher inequality, which is the relevant geometric inequality associated with the minimization of the total inverse mean curvature under a volume constraint. We show that the symmetry between Aleksandrov's theorem and the Heintze-Karcher inequality breaks down when boundary conditions are imposed. Precisely, we deal with the variational behavior of the total inverse mean curvature for smooth curves in the half-plane, prescribing both the enclosed volume and a boundary condition. We characterize the existence of equilibrium configurations, and we discuss various notions of stability. As an application, we establish a local minimization property. This talk is based on a joint work with J. Pozuelo and G. Vianello.