Maximization of the second Laplacian Eigenvalue on the Sphere

2020/11/23 by Hanna N. Kim, Kim, Hanna N. · 2 citations
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2011.11494

openalex publication_date 2020/11/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on S^m. For S², 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 S².

Cited by

• Maximizing higher eigenvalues in dimensions three and above
• An improved upper bound for the second eigenvalue on tori

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