{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/124563"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/124563","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Upper bounds of second Laplacian eigenvalues on the sphere and the projective space","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-05-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-05-01","abstract_has_math":false,"creators":["Kim, Hanna N."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Laugesen, Richard","Bronski, Jared","Albin, Pierre","Hung, Pei-Ken"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-05","date_published":"2024-05","updated_at":"2026-07-22T22:25:02Z","subjects":["Spectral Theory","Shape Optimization"],"languages":["en","eng"],"rights":["Copyright 2024 Hanna Kim"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/124563","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Laugesen, Richard","Bronski, Jared","Albin, Pierre","Hung, Pei-Ken"]},{"key":"dc:creator","label":"Author","values":["Kim, Hanna N."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-05","2024-04-26"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Spectral Theory","Shape Optimization"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Hanna Kim"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/124563"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-05-01","The student, Hanna Kim, accepted the attached license on 2024-04-23 at 11:26.","The student, Hanna Kim, submitted this Dissertation for approval on 2024-04-23 at 11:33.","This Dissertation was approved for publication on 2024-04-26 at 14:58.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20570 on 2024-09-16 at 00:44:24","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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Upper bounds of second Laplacian eigenvalues on the sphere and the projective space"]}]}],"canonical_facts":{"dc:contributor":["Laugesen, Richard","Bronski, Jared","Albin, Pierre","Hung, Pei-Ken"],"dc:creator":["Kim, Hanna N."],"dc:date":["2024-05","2024-04-26"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-05-01","The student, Hanna Kim, accepted the attached license on 2024-04-23 at 11:26.","The student, Hanna Kim, submitted this Dissertation for approval on 2024-04-23 at 11:33.","This Dissertation was approved for publication on 2024-04-26 at 14:58.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20570 on 2024-09-16 at 00:44:24","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."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/124563"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Hanna Kim"],"dc:subject":["Spectral Theory","Shape Optimization"],"dc:title":["Upper bounds of second Laplacian eigenvalues on the sphere and the projective space"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:02Z"}