Variational properties of the total inverse mean curvature in the plane under boundary constraints

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 HeintzeKarcher 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 HeintzeKarcher 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 halfplane, 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.