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On the Friedlander–Nadirashvili invariants of surfaces

Karpukhin, Mikhail and Medvedev, Vladimir (2021) On the Friedlander–Nadirashvili invariants of surfaces. Mathematische Annalen, 379 (3-4). pp. 1767-1805. ISSN 0025-5831. doi:10.1007/s00208-020-02094-2.

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Let M be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant I₁ (M) using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator Δ_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 Δ_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) = Ik(S²) 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 I₁(M) = I₁ (S²) for any surface M different from RP². 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(S²). 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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Additional Information:© 2020 Springer-Verlag GmbH Germany, part of Springer Nature. Received 10 February 2019; Revised 04 June 2020; Accepted 06 October 2020; Published 19 November 2020. 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. Communicated by F. C. Marques.
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Schullich FellowshipUNSPECIFIED
Université de MontréalUNSPECIFIED
Issue or Number:3-4
Record Number:CaltechAUTHORS:20201119-140226430
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Official Citation:Karpukhin, M., Medvedev, V. On the Friedlander–Nadirashvili invariants of surfaces. Math. Ann. 379, 1767–1805 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106741
Deposited By: Tony Diaz
Deposited On:19 Nov 2020 22:20
Last Modified:06 Aug 2021 21:37

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