Congruence theorems for capillary hypersurfaces and a substatic Heintze--Karcher inequality

We prove congruence theorems for capillary hypersurfaces of Euclidean space supported on a hyperplane or on a sphere, extending a result of Ros for closed hypersurfaces. We also establish a Heintze–Karcher type inequality for capillary hypersurfaces in substatic manifolds.