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Mosc. Math. J., 2015, Volume 15, Number 1, Pages 123–140 (Mi mmj553)  

This article is cited in 6 scientific papers (total in 6 papers)

Conformal spectrum and harmonic maps

Nikolai Nadirashvilia, Yannick Sireb

a CNRS, I2M UMR 7353, Centre de Mathématiques et Informatique, Marseille, France
b Université Aix-Marseille, I2M UMR 7353, Marseille, France

Abstract: This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace–Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a rather constructive proof of the existence of a critical metric which is smooth outside of a finite number of conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that there exists a subspace of the eigenspace associated to the first maximized eigenvalue such that the corresponding eigenvector gives a harmonic map from the surface to a Euclidean sphere.

Key words and phrases: eigenvalues, isoperimetric inequalities.

DOI: https://doi.org/10.17323/1609-4514-2015-15-1-123-140

Full text: http://www.mathjournals.org/.../2015-015-001-008.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 35P15
Received: April 2, 2014; in revised form July 3, 2014
Language:

Citation: Nikolai Nadirashvili, Yannick Sire, “Conformal spectrum and harmonic maps”, Mosc. Math. J., 15:1 (2015), 123–140

Citation in format AMSBIB
\Bibitem{NadSir15}
\by Nikolai~Nadirashvili, Yannick~Sire
\paper Conformal spectrum and harmonic maps
\jour Mosc. Math.~J.
\yr 2015
\vol 15
\issue 1
\pages 123--140
\mathnet{http://mi.mathnet.ru/mmj553}
\crossref{https://doi.org/10.17323/1609-4514-2015-15-1-123-140}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3427416}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000354886200008}


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  • http://mi.mathnet.ru/eng/mmj/v15/i1/p123

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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. Nikolai Nadirashvili, Yannick Sire, “Maximization of higher order eigenvalues and applications”, Mosc. Math. J., 15:4 (2015), 767–775  mathnet  crossref  mathscinet
    2. A. Grigor'yan, N. Nadirashvili, Ya. Sire, “A lower bound for the number of negative eigenvalues of Schrödinger operators”, J. Differential Geom., 102:3 (2016), 395–408  crossref  mathscinet  zmath  isi
    3. N. Nadirashvili, Ya. Sire, “Isoperimetric inequality for the third eigenvalue of the Laplace–Beltrami operator on $\mathbb{S}^2$”, J. Differ. Geom., 107:3 (2017), 561–571  crossref  mathscinet  zmath  isi
    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. N. S. Nadirashvili, V A. 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
    6. S. Ariturk, “An annulus and a half-helicoid maximize Laplace eigenvalues”, J. Spectr. Theory, 8:2 (2018), 315–346  crossref  mathscinet  zmath  isi  scopus
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