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The minimal length product over homology bases of manifolds

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

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Notes

  1. The notion of almost minimality is that \(Z_2\) is minimal for area among hypersurfaces in its cohomological class up to some chosen small additive constant. Because this constant can be fixed as small as wanted, the proof remains the same if we actually suppose that \(Z_2\) is minimal. We do so and refer the reader to the original proof in [6] for more details.

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Acknowledgements

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). https://doi.org/10.1007/s00208-021-02150-5

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Mathematics Subject Classification

  • 53C23