Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

Abstract

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball ${\mathbb{B}}^{n+1}$ and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the $n$-th quermassintegral and non-decreases the $k$-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in ${\mathbb{B}}^{n+1}$. This generalizes the result in \cite{SWX} for convex hypersurfaces with free boundary in ${\mathbb{B}}^{n+1}$.

arXiv commentTo appear in Trans. Amer. Math. Soc

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