A Constrained Mean Curvature Flow and Alexandrov–Fenchel Inequalities

Abstract

In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in [29] evolve monotonically along the flow, and hence we establish a class of new Alexandrov–Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.