Abstract
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional E. We show that it has the longtime existence and subconverges to spherical caps. As an application, we solve an isoperimetric problem for hypersurfaces with capillary boundary.
Highlights
We are interested in a mean curvature type flow in the unit ball Bn+1 ⊂ Rn+1 with capillary boundary
The mean curvature flow plays an important role in geometric analysis and has been extensively studied
Mean curvature type flows with a constraint play an important role in the study of isoperimetric problems
Summary
We are interested in a mean curvature type flow in the unit ball Bn+1 ⊂ Rn+1 with capillary boundary. This flow is volume preserving and area decreasing by the Minkowski formulas They obtained the longtime existence of this flow and proved that it smoothly converges to a round sphere if the initial hypersurface is star-shaped. As a result, this yields a flow proof of classical AlexandrovFenchel inequalities of quermassintegrals in convex geometry. We would like to mention the recent articles [33,36] for a mean curvature type flow and a fully nonlinear inverse curvature type flow respectively in the unit ball with free boundary, where new geometric inequalities were proved as applications. In the last Section, we establish a priori estimates and prove the main theorem
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