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.

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball B n + 1 {\mathbb {B}}^{n+1} and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the n n -th quermassintegral and non-decreases the k k -th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in B n + 1 {\mathbb {B}}^{n+1} . This generalizes the result of Scheuer [J. Differential Geom. 120 (2022), pp. 345–373] for convex hypersurfaces with free boundary in B n + 1 {\mathbb {B}}^{n+1} .

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces problem. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.This project is partly supported by SPP 2026 of DFG "Geometry at infinity".