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 Copy

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

Language

eng

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.

Graduation Semester

2024-05

Type of Resource

Text

Handle URL

https://hdl.handle.net/2142/124563

Copyright and License Information

Copyright 2024 Hanna Kim

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