Sharp systolic inequalities for $3$-manifolds with boundary

TL;DR

This paper establishes precise inequalities connecting the homological systoles of compact 3-manifolds with boundary to their scalar and boundary mean curvatures, characterizing the equality case geometrically.

Contribution

It introduces sharp systolic inequalities for 3-manifolds with boundary, linking topological and geometric properties in a novel way.

Findings

Derived inequalities relating homological systoles to curvature

Characterized the equality case as a cylinder over a disk

Provided geometric conditions for equality in the inequalities

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.