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On branched minimal immersions of surfaces by first eigenfunctions

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Abstract

It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first nonzero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from \({\mathbb {S}}^2\) differ significantly from the properties of those induced from \({\mathbb {S}}^m\) with \(m>2\). This feature appears to be novel and needs to be taken into account in the existing proofs of the sharp upper bounds for the first nonzero eigenvalue of the Laplacian on the 2-torus and the Klein bottle. In the present paper we address this issue and give a detailed overview of the complete proofs of these upper bounds following the works of Nadirashvili, Jakobson–Nadirashvili–Polterovich, El Soufi–Giacomini–Jazar, Nadirashvili–Sire and Petrides.

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Acknowledgements

The authors are grateful to I. Polterovich and A. Girouard for suggesting this problem. The authors are also grateful to I. Polterovich, A. Girouard, G. Ponsinet, A. Penskoi, G. Kokarev, and Alex Wright for fruitful discussions and especially to A. Girouard for a careful first reading of this manuscript. The authors are grateful to the anonymous referee for providing useful comments and suggestions. This research is part of the third author’s PhD thesis at the Université de Montréal under the supervision of Iosif Polterovich.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Donato Cianci

  2. Department of Mathematics, University of California, 340 Rowland Hall, Irvine, CA, 92697-3875, USA

    Mikhail Karpukhin

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

    Vladimir Medvedev

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  1. Donato Cianci
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  2. Mikhail Karpukhin
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  3. Vladimir Medvedev
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Correspondence to Donato Cianci.

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The first author was supported by a Centre de Recherches Mathématiques-Laval Postdoctoral Fellowship while some of this research was conducted. The second author was partially supported by a Schulich Fellowship at McGill University.

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Cianci, D., Karpukhin, M. & Medvedev, V. On branched minimal immersions of surfaces by first eigenfunctions. Ann Glob Anal Geom 56, 667–690 (2019). https://doi.org/10.1007/s10455-019-09683-8

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  • DOI: https://doi.org/10.1007/s10455-019-09683-8

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