Spines of minimal length

TL;DR

This paper investigates minimal spines on Riemannian surfaces, introducing the spine systole, analyzing minimal spines, and classifying them on flat tori and hyperbolic surfaces, advancing understanding of their geometric properties.

Contribution

It introduces the spine systole as a new invariant, classifies minimal spines on flat tori, and studies the finiteness of minimal spines of bounded length on hyperbolic surfaces.

Findings

Spine systole is a proper function on moduli space.

Global minima of the spine systole are extremal surfaces.

Number of minimal spines of bounded length is finite on hyperbolic surfaces.

Abstract

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine systole is a proper function and has its global minima precisely on the extremal surfaces (those containing the biggest possible discs). We also study minimal spines, which are critical points for the length functional. We completely classify minimal spines on flat tori, proving that the number of them is a proper function on moduli space. We also show that the number of minimal spines of uniformly bounded length is finite on hyperbolic surfaces.