Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: March 20, 2014
MathSciNet review: 3195761
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish an upper bound on the second eigenvalue of $ n$-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.

References [Enhancements On Off] (What's this?)

  • [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • [2] B. Colbois and J. Dodziuk, Riemannian metrics with large $ \lambda _1$, Proc. Amer. Math. Soc. 122 (1994), no. 3, 905-906. MR 1213857 (95a:58130),
  • [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 (2005f:58051),
  • [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 (87j:53088),
  • [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 (2011a:58057)
  • [6] Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645-A1648 (French). MR 0292357 (45 #1444)
  • [7] Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993), no. 1, 73-93. MR 1198600 (94d:58153)
  • [8] Nikolai Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere, J. Differential Geom. 61 (2002), no. 2, 335-340. MR 1972149 (2004c:58066)
  • [9] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356. MR 0061749 (15,877c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35P15, 58C40, 58J50

Retrieve articles in all journals with MSC (2010): 35P15, 58C40, 58J50

Additional Information

Romain Petrides
Affiliation: UMPA-ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France

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.

American Mathematical Society