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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 6(414), Pages 107–168 (Mi umn9565)  

This article is cited in 9 scientific papers (total in 9 papers)

Extremal metrics for eigenvalues of the Laplace–Beltrami operator on surfaces

A. V. Penskoiabc

a M. V. Lomonosov Moscow State
b National Research University "Higher School of Economics"
c Independent University of Moscow

Abstract: Known results on geometric optimisation of eigenvalues of the Laplace operator are briefly reviewed, and a more detailed survey of recent results in the theory of extremal metrics on surfaces is presented.
Bibliography: 78 titles.

Keywords: geometric optimisation of eigenvalues, extremal metrics, minimal surfaces in spheres.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 2010-220-01-077
НШ-4995.2012.1
Simons Foundation


DOI: https://doi.org/10.4213/rm9565

Full text: PDF file (963 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2013, 68:6, 1073–1130

Bibliographic databases:

UDC: 514.76+517.9
MSC: 58J50, 53C42
Received: 24.10.2013

Citation: A. V. Penskoi, “Extremal metrics for eigenvalues of the Laplace–Beltrami operator on surfaces”, Uspekhi Mat. Nauk, 68:6(414) (2013), 107–168; Russian Math. Surveys, 68:6 (2013), 1073–1130

Citation in format AMSBIB
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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. N. Kuznetsov, A. Nazarov, “Sharp constants in the Poincaré, Steklov and related inequalities (a survey)”, Mathematika, 61:2 (2015), 328–344  crossref  mathscinet  zmath  isi  elib  scopus
    2. Broderick Causley, “Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces”, SIGMA, 12 (2016), 009, 11 pp.  mathnet  crossref
    3. M. A. Karpukhin, “Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces”, Int. Math. Res. Not. IMRN, 2016, no. 20, 6200–6209  crossref  mathscinet  isi  scopus
    4. 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
    5. J. Hirsch, E. Mäder-Baumdicker, “Note on Willmore minimizing Klein bottles in Euclidean space”, Adv. Math., 319 (2017), 67–75  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. M. S. Ermentai, “Ob odnom semeistve minimalnykh izotropnykh torov i butylok Kleina v $\mathbb{C}P^3$”, Sib. elektron. matem. izv., 16 (2019), 955–958  mathnet  crossref
    8. 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
    9. 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  isi
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