Upper bounds of second Laplacian eigenvalues on the sphere and the projective space

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Kim, Hanna N.

Contributors dc:contributor

• Laugesen, Richard
• Bronski, Jared
• Albin, Pierre
• Hung, Pei-Ken

• Spectral Theory
• Shape Optimization

Statement dc:rights

• Copyright 2024 Hanna Kim

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F124563