{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/110435"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/110435","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds","abstract":"We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic-manifolds, and formulate several questions and conjectures. Using similar decomposition principles, we also obtain results on eigenvalues of infinite volume geometrically finite hyperbolic manifolds.","abstract_html":"We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic-manifolds, and formulate several questions and conjectures. Using similar decomposition principles, we also obtain results on eigenvalues of infinite volume geometrically finite hyperbolic manifolds.","abstract_has_math":false,"creators":["Han, Xiaolong"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dunfield, Nathan","Laugesen, Richard","Hirani, Anil","Albin, Pierre"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-09-17T01:10:40Z","date_published":"2021-09-17T01:10:40Z","updated_at":"2026-07-22T22:24:50Z","subjects":["norms of cohomology","hyperbolic manifolds","eigenvalues"],"languages":["en"],"rights":["Copyright 2021 Xiaolong Han"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/110435","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dunfield, Nathan","Laugesen, Richard","Hirani, Anil","Albin, Pierre"]},{"key":"dc:creator","label":"Author","values":["Han, Xiaolong"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2021-09-17T01:10:40Z","2021-04-19","2021-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["norms of cohomology","hyperbolic manifolds","eigenvalues"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Xiaolong Han"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/110435"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic-manifolds, and formulate several questions and conjectures. Using similar decomposition principles, we also obtain results on eigenvalues of infinite volume geometrically finite hyperbolic manifolds.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2021-09-16 without embargo terms","The student, Xiaolong Han, accepted the attached license on 2021-04-04 at 10:29.","The student, Xiaolong Han, submitted this Dissertation for approval on 2021-04-04 at 10:38.","This Dissertation was approved for publication on 2021-04-19 at 11:54.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16234 on 2021-09-16 at 16:40:28","Made available in DSpace on 2021-09-17T01:10:40Z (GMT). No. of bitstreams: 2 HAN-DISSERTATION-2021.pdf: 638951 bytes, checksum: 83a2ae4bf78f27e3f69538486f64d805 (MD5) LICENSE.txt: 4209 bytes, checksum: 3c416f1bcbd6d4025877a73392b225af (MD5) Previous issue date: 2021-04-19"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds"]}]}],"canonical_facts":{"dc:contributor":["Dunfield, Nathan","Laugesen, Richard","Hirani, Anil","Albin, Pierre"],"dc:creator":["Han, Xiaolong"],"dc:date":["2021-09-17T01:10:40Z","2021-04-19","2021-05"],"dc:description":["We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic-manifolds, and formulate several questions and conjectures. Using similar decomposition principles, we also obtain results on eigenvalues of infinite volume geometrically finite hyperbolic manifolds.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2021-09-16 without embargo terms","The student, Xiaolong Han, accepted the attached license on 2021-04-04 at 10:29.","The student, Xiaolong Han, submitted this Dissertation for approval on 2021-04-04 at 10:38.","This Dissertation was approved for publication on 2021-04-19 at 11:54.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16234 on 2021-09-16 at 16:40:28","Made available in DSpace on 2021-09-17T01:10:40Z (GMT). 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