Upper bounds of second Laplacian eigenvalues on the sphere and the projective space
Kim, Hanna N.
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https://hdl.handle.net/2142/124563
Description
Title
Upper bounds of second Laplacian eigenvalues on the sphere and the projective space
Author(s)
Kim, Hanna N.
Issue Date
2024-04-26
Director of Research (if dissertation) or Advisor (if thesis)
Laugesen, Richard
Doctoral Committee Chair(s)
Bronski, Jared
Committee Member(s)
Albin, Pierre
Hung, Pei-Ken
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Spectral theory, shape optimization
Abstract
In the first part, we prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on $\mathbb{S}^m$. For $\mathbb{S}^{2}$, 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 $\mathbb{S}^2$.
Next, we consider an analogous conjecture for the second non-zero Laplacian eigenvalue on $n$-dimensional real projective space. The sharp result in 2 dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric degenerates to that of the disjoint union of a round projective space and a sphere. That conjecture is open in higher dimensions, but this dissertation proves it up to a constant factor that tends to 1 as the dimension tends to infinity.
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