Short loops in surfaces with a circle boundary component

TL;DR

This paper extends classical systolic bounds to Riemannian 2-tori with boundary, showing the shortest non-contractible loop length can be controlled by the area, even with boundary constraints.

Contribution

It proves a new bound on the shortest non-contractible loop in Riemannian 2-tori with boundary, generalizing Loewner's theorem to surfaces with boundary.

Findings

Shortest non-contractible loop length is bounded by surface area.

Boundary conditions do not prevent systolic bounds.

Results apply to tori embedded in Euclidean space with a circle boundary.

Abstract

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length of the shortest non-contractible loop in $T$ is bounded in terms of the area of $T$.