Maximization of the second conformal eigenvalue of spheres
Author:
Romain Petrides
Journal:
Proc. Amer. Math. Soc. 142 (2014), 2385-2394
MSC (2010):
Primary 35P15, 58C40, 58J50
DOI:
https://doi.org/10.1090/S0002-9939-2014-12095-8
Published electronically:
March 20, 2014
MathSciNet review:
3195761
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we establish an upper bound on the second eigenvalue of 
-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
- [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
- [2] B. Colbois and J. Dodziuk, Riemannian metrics with large 𝜆₁, Proc. Amer. Math. Soc. 122 (1994), no. 3, 905–906. MR 1213857, https://doi.org/10.1090/S0002-9939-1994-1213857-9
- [3] Bruno Colbois and Ahmad El Soufi, Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’, Ann. Global Anal. Geom. 24 (2003), no. 4, 337–349. MR 2015867, https://doi.org/10.1023/A:1026257431539
- [4] A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du laplacien et volume conforme, Math. Ann. 275 (1986), no. 2, 257–267 (French). MR 854009, https://doi.org/10.1007/BF01458460
- [5] Alexandre Girouard, Nikolai Nadirashvili, and Iosif Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains, J. Differential Geom. 83 (2009), no. 3, 637–661. MR 2581359
- [6] Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
- [7] Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993), no. 1, 73–93. MR 1198600
- [8] Nikolai Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere, J. Differential Geom. 61 (2002), no. 2, 335–340. MR 1972149
- [9] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356. MR 61749, https://doi.org/10.1512/iumj.1954.3.53017
, Proc. Amer. Math. Soc. 122 (1994), no. 3, 905-906. Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35P15, 58C40, 58J50
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Additional Information
Romain Petrides
Affiliation:
UMPA-ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France
Email:
romain.petrides@ens-lyon.fr
DOI:
https://doi.org/10.1090/S0002-9939-2014-12095-8
Received by editor(s):
July 3, 2012
Published electronically:
March 20, 2014
Communicated by:
Michael Wolf
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
