Systoles and minimal surfaces in 3-manifolds with boundary - Longa, Eduardo Rosinato (2021)
The aim if this work is twofold. Firstly, we prove the existence of a local foliation around compact infinitesimally rigid capillary surfaces. We then use this fact to show a rigidity result for infinitesimally rigid capillary surfaces in some Riemannian 3-manifolds with mean convex boundary. We also derive bounds on the genus, number of boundary components and area of any compact two-sided capillary minimal surface with low index under certain assumptions on the curvature of the ambient manifold and of its boundary. Secondly, we prove some sharp systolic inequalities for compact 3-manifolds with boundary. They relate the (relative) homological systoles of the manifold to its scalar curvature and mean curvature of the boundary. In the equality case, the universal cover of the manifold is isometric to a cylinder over a disc of nonnegative constant curvature.