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On the Friedlander–Nadirashvili invariants of surfaces

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Abstract

Let M be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant \(I_1(M)\) using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator \(\Delta _g\) of a Riemannian metric g. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use k-th eigenvalues of \(\Delta _g\) to define the invariants \(I_k(M)\) indexed by positive integers k. In the present paper the values of these invariants on surfaces are investigated. We show that \(I_k(M)=I_k({\mathbb {S}}^2)\) unless M is a non-orientable surface of even genus. For orientable surfaces and \(k=1\) this was earlier shown by Petrides. In fact Friedlander and Nadirashvili suggested that \(I_1(M)=I_1({\mathbb {S}}^2)\) for any surface M different from \({\mathbb {RP}}^2\). We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has \(I_k(M)>I_k({\mathbb {S}}^2)\). We also discuss the connection between the Friedlander–Nadirashvili invariants and the theory of cobordisms, and conjecture that \(I_k(M)\) is a cobordism invariant.

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Acknowledgements

The authors are grateful to Iosif Polterovich for fruitful discussions and for his remarks on the initial draft of the manuscript. The authors would like to thank Alexandre Girouard for outlining the proof of Proposition 4.2 and Bruno Colbois for valuable remarks. The authors are thankful to the reviewer for useful remarks and suggestions. During the preparation of this manuscript the first author was supported by Schulich Fellowship. This research is a part of the second author’s PhD thesis at the Université de Montréal under the supervision of Iosif Polterovich.

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Authors and Affiliations

  1. Mathematics 253-37, Caltech, Pasadena, CA, 91125, USA

    Mikhail Karpukhin & Vladimir Medvedev

  2. Département de Mathématiques et de Statistique, Pavillon André-Aisenstadt, Université de Montréal, Montréal, QC, H3C 3J7, Canada

    Vladimir Medvedev

  3. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation

    Vladimir Medvedev

  4. Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Street, Moscow, 119048, Russian Federation

    Vladimir Medvedev

Authors
  1. Mikhail Karpukhin
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  2. Vladimir Medvedev
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Correspondence to Vladimir Medvedev.

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Communicated by F. C. Marques.

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Karpukhin, M., Medvedev, V. On the Friedlander–Nadirashvili invariants of surfaces. Math. Ann. 379, 1767–1805 (2021). https://doi.org/10.1007/s00208-020-02094-2

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  • DOI: https://doi.org/10.1007/s00208-020-02094-2

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