Maximization of the second Laplacian eigenvalue on the sphere

Kim, Hanna

doi:10.1090/proc/15908

Maximization of the second Laplacian eigenvalue on the sphere
HTML articles powered by AMS MathViewer

by Hanna N. Kim

Proc. Amer. Math. Soc. 150 (2022), 3501-3512

DOI: https://doi.org/10.1090/proc/15908

Published electronically: April 15, 2022

PDF | Request permission

Abstract:

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on $S^m$. For $S^{2}$, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides’ work and recent developments on the hyperbolic center of mass and provides also a simpler proof for $S^2$.

References

Olivier Druet, On the second conformal eigenvalue of the standard sphere, Asian J. Math. 22 (2018), no. 6, 1047–1073. MR 3919551, DOI 10.4310/AJM.2018.v22.n6.a4
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
P. Freitas and R. S. Laugesen, Two balls maximize the third Neumann eigenvalue in hyperbolic space, Ann. Sc. Norm. Super. Pisa Cl. Sci., To appear, arXiv:2009.09980.
Alexandre Girouard and Richard S. Laugesen, Robin spectrum: two disks maximize the third eigenvalue, Indiana Univ. Math. J. 70 (2021), no. 6, 2711–2742. MR 4359924, DOI 10.1512/iumj.2021.70.8721
Alexandre Girouard, Mikhail Karpukhin, and Jean Lagacé, Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems, Geom. Funct. Anal. 31 (2021), no. 3, 513–561. MR 4311579, DOI 10.1007/s00039-021-00573-5
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
Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
Chiu-Yen Kao, Rongjie Lai, and Braxton Osting, Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces, ESAIM Control Optim. Calc. Var. 23 (2017), no. 2, 685–720. MR 3608099, DOI 10.1051/cocv/2016008
Mikhail Karpukhin, Index of minimal spheres and isoperimetric eigenvalue inequalities, Invent. Math. 223 (2021), no. 1, 335–377. MR 4199444, DOI 10.1007/s00222-020-00992-5
Mikhail Karpukhin, Nikolai Nadirashvili, Alexei V. Penskoi, and Iosif Polterovich, An isoperimetric inequality for Laplace eigenvalues on the sphere, J. Differential Geom. 118 (2021), no. 2, 313–333. MR 4278696, DOI 10.4310/jdg/1622743142
M. Karpukhin and D. L. Stern, Min-max harmonic maps and a new characterization of conformal eigenvalues, 2020, arXiv:2004.04086.
M. Karpukhin, N. Nadirashvili, A. Penskoi, and I. Polterovich, Conformally maximal metrics for Laplace eigenvalues on surfaces, Surv. Differ. Geom. 24 (2019), 205–256.
R. S. Laugesen, Well-posedness of Hersch-Szegő’s center of mass by hyperbolic energy minimization, Ann. Math. Qué. 45 (2021), no. 2, 363–390 (English, with English and French summaries). MR 4308185, DOI 10.1007/s40316-020-00151-5
Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
Nikolai Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere, J. Differential Geom. 61 (2002), no. 2, 335–340. MR 1972149
Nikolai S. Nadirashvili and Alexei V. Penskoi, An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane, Geom. Funct. Anal. 28 (2018), no. 5, 1368–1393. MR 3856795, DOI 10.1007/s00039-018-0458-7
Nikolai Nadirashvili and Yannick Sire, Isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on $\Bbb {S}^2$, J. Differential Geom. 107 (2017), no. 3, 561–571. MR 3715349, DOI 10.4310/jdg/1508551225
Romain Petrides, Maximization of the second conformal eigenvalue of spheres, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2385–2394. MR 3195761, DOI 10.1090/S0002-9939-2014-12095-8
Romain Petrides, On the existence of metrics which maximize Laplace eigenvalues on surfaces, Int. Math. Res. Not. IMRN 14 (2018), 4261–4355. MR 3830572, DOI 10.1093/imrn/rnx004
Manfred Stoll, Harmonic and subharmonic function theory on the hyperbolic ball, London Mathematical Society Lecture Note Series, vol. 431, Cambridge University Press, Cambridge, 2016. MR 3495548, DOI 10.1017/CBO9781316341063
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
H. F. Weinberger, An isoperimetric inequality for the $N$-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633–636. MR 79286, DOI 10.1512/iumj.1956.5.55021