Calculus of Variations and Geometric Measure Theory
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B. Martelli - M. Novaga - A. Pluda - S. Riolo

Spines of minimal length

created by novaga on 07 Nov 2015
modified by pluda on 15 Mar 2019


Published Paper

Inserted: 7 nov 2015
Last Updated: 15 mar 2019

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Volume: XVII
Pages: 1067-1090
Year: 2017
Doi: 10.2422/2036-2145.201511_003

ArXiv: 1511.02367 PDF


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.


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