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Annals of Global Analysis and Geometry

, Volume 56, Issue 4, pp 667–690 | Cite as

On branched minimal immersions of surfaces by first eigenfunctions

  • Donato CianciEmail author
  • Mikhail Karpukhin
  • Vladimir Medvedev
Article

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.

Keywords

Spectral theory Branched minimal immersions Maximal metrics Eigenvalue bounds 

Notes

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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of CaliforniaIrvineUSA
  3. 3.Département de Mathématiques et de Statistique, Pavillon André-AisenstadtUniversité de MontréalMontrealCanada

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