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The minimal length product over homology bases of manifolds


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 \(\mathbb {Z}_2\)-cohomology group using the cup product.

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We are grateful to A. Abdesselam, I. Babenko, B. Kahn and J. Milnor for valuable exchanges. We are also indebted to J. Gutt whose proof of a symplectic analog of Minkowski’s first theorem (see Lemma 3.10) inspired a mechanism used in the proof of Theorem 2.5. Finally we would like to thank the referee for valuable comments.

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Correspondence to Florent Balacheff.

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Communicated by F. C. Marques.

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Florent Balacheff acknowledges support from the European Social Fund and the Agencia Estatal de Investigación through the Ramón y Cajal grant RYC-2016-19334 “Local and global systolic geometry and topology”, as well as from the grant ANR-12-BS01-0009-02. Steve Karam acknowledges support from grant ANR CEMPI (ANR-11-LABX-0007-01). Hugo Parlier acknowledges support from the ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033.

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Balacheff, F., Karam, S. & Parlier, H. The minimal length product over homology bases of manifolds. Math. Ann. 380, 825–854 (2021).

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Mathematics Subject Classification

  • 53C23