Sharp Systolic Inequalities for 3-Manifolds with Boundary

Abstract

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 disk of nonnegative constant curvature.