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Mat. Sb., 2013, Volume 204, Number 12, Pages 31–48 (Mi msb8227)  

This article is cited in 8 scientific papers (total in 8 papers)

Nonmaximality of known extremal metrics on torus and Klein bottle

M. A. Karpukhinab

a M. V. Lomonosov Moscow State University
b Independent University of Moscow

Abstract: The El Soufi-Ilias theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the properties of these metrics and prove that none of them is maximal.
Bibliography: 24 titles.

Keywords: extremal metrics, bipolar surface, Otsuki tori, Lawson tau-surfaces.

Funding Agency Grant Number
Dobrushin Foundation
Simons Foundation


DOI: https://doi.org/10.4213/sm8227

Full text: PDF file (516 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2013, 204:12, 1728–1744

Bibliographic databases:

UDC: 514.764.21
MSC: 58E11
Received: 27.02.2013 and 10.06.2013

Citation: M. A. Karpukhin, “Nonmaximality of known extremal metrics on torus and Klein bottle”, Mat. Sb., 204:12 (2013), 31–48; Sb. Math., 204:12 (2013), 1728–1744

Citation in format AMSBIB
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  • https://doi.org/10.4213/sm8227
  • http://mi.mathnet.ru/eng/msb/v204/i12/p31

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Penskoi, “Extremal metrics for eigenvalues of the Laplace–Beltrami operator on surfaces”, Russian Math. Surveys, 68:6 (2013), 1073–1130  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. V. Penskoi, “Generalized Lawson Tori and Klein Bottles”, J. Geom. Anal., 25:4 (2015), 2645–2666  crossref  mathscinet  zmath  isi  elib  scopus
    3. M. Karpukhin, “Spectral properties of a family of minimal tori of revolution in five-dimensional sphere”, Can. Math. Bull., 58:2 (2015), 285–296  crossref  mathscinet  zmath  isi  scopus
    4. Broderick Causley, “Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces”, SIGMA, 12 (2016), 009, 11 pp.  mathnet  crossref
    5. Ch.-Y. Kao, R. Lai, B. Osting, “Maximization of Laplace–Beltrami eigenvalues on closed Riemannian surfaces”, ESAIM Control Optim. Calc. Var., 23:2 (2017), 685–720  crossref  mathscinet  zmath  isi  scopus
    6. N. S. Nadirashvili, A. V. Penskoi, “An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane”, Geom. Funct. Anal., 28:5 (2018), 1368–1393  crossref  mathscinet  zmath  isi  scopus
    7. Alexei V. Penskoi, “Isoperimetric Inequalities for Higher Eigenvalues of the Laplace–Beltrami Operator on Surfaces”, Proc. Steklov Inst. Math., 305 (2019), 270–286  mathnet  crossref  crossref  isi  elib
    8. Cianci D., Karpukhin M., Medvedev V., “On Branched Minimal Immersions of Surfaces By First Eigenfunctions”, Ann. Glob. Anal. Geom., 56:4 (2019), 667–690  crossref  mathscinet  zmath  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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