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 \cite{WWX2022} 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.

arXiv commentFinal version, to appear in Int. Math. Res. Not. IMRN

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• A Minkowski-type inequality for capillary hypersurfaces in a half-space
• A fully nonlinear locally constrained curvature flow for capillary hypersurface
• Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II
• Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow
• A mean curvature type flow with capillary boundary in a horoball in hyperbolic space
• A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane