The minimal length product over homology bases of manifolds

Florent Balacheff*, Steve Karam, Hugo Parlier

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Minkowski’s second theorem can be stated as an inequality for n-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsler manifolds. This includes manifolds for which first Betti number and dimension do no necessarily coincide, a prime example being the case of surfaces. This class of manifolds is described by a non-vanishing condition for the hyperdeterminant reduced modulo 2 of the multilinear map induced by the fundamental class of the manifold on its first Z2-cohomology group using the cup product.

Original languageEnglish
Pages (from-to)825-854
Number of pages30
JournalMathematische Annalen
Volume380
Issue number1-2
DOIs
Publication statusPublished - Jun 2021

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