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An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane

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Abstract

We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace–Beltrami operator on the real projective plane. For a metric of unit area this eigenvalue is not greater than \({20\pi.}\) This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting metric is singular and could be realized as a union of the projective plane and the sphere touching at a point, with standard metrics and the ratio of the areas 3:2. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6.

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Acknowledgments

The authors are very indebted to Mikhail Karpukhin, Iosif Polterovich and the referee for useful remarks and suggestions. The second author is very grateful to the Institut de Mathématiques de Marseille (I2M, UMR 7373) for the hospitality. The second author is very indebted to Pavel Winternitz for fruitful discussions.

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Authors and Affiliations

  1. CNRS, I2M UMR 7353 — Centre de Mathématiques et Informatique, Technopole Chateau-Gombert, 39, rue Frederic Joliot-Curie, 13453, Marseille Cedex 13, France

    Nikolai S. Nadirashvili

  2. Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University, Leninskie Gory, GSP-1, Moscow, Russia, 119991

    Alexei V. Penskoi

  3. Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str., Moscow, Russia, 119048

    Alexei V. Penskoi

  4. Interdisciplinary Scientific Center J.-V. Poncelet (ISCP, UMI 2615), Bolshoy Vlasyevskiy Pereulok 11, Moscow, Russia, 119002

    Alexei V. Penskoi

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  1. Nikolai S. Nadirashvili
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  2. Alexei V. Penskoi
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Correspondence to Alexei V. Penskoi.

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The work of the second author was partially supported by the Simons Foundation and by the Young Russian Mathematics award.

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Nadirashvili, N.S., Penskoi, A.V. An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane. Geom. Funct. Anal. 28, 1368–1393 (2018). https://doi.org/10.1007/s00039-018-0458-7

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