Maximization of the second conformal eigenvalue of spheres

Petrides, Romain

doi:10.1090/S0002-9939-2014-12095-8

Maximization of the second conformal eigenvalue of spheres
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Proc. Amer. Math. Soc. 142 (2014), 2385-2394 Request permission

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

Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
B. Colbois and J. Dodziuk, Riemannian metrics with large $\lambda _1$, Proc. Amer. Math. Soc. 122 (1994), no. 3, 905–906. MR 1213857, DOI 10.1090/S0002-9939-1994-1213857-9
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, DOI 10.1023/A:1026257431539
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, DOI 10.1007/BF01458460
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
Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993), no. 1, 73–93. MR 1198600
Nikolai Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere, J. Differential Geom. 61 (2002), no. 2, 335–340. MR 1972149
G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356. MR 61749, DOI 10.1512/iumj.1954.3.53017