{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22489"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22489","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Harmonic forms under metric and topological perturbations","abstract":"We construct examples illustrating various aspects of Hodge theory on Riemannian manifolds. We consider the relationship between the harmonic forms on a product space and the harmonic forms on the factors as well as the harmonic forms on a connected sum and the harmonic forms on the summands. Algebraic topology provides relations between the cohomology groups of these spaces via the Kunneth formula and the connected sum formula. We consider the problem of describing these relationships in terms of DeRham cohomology and in terms of Hodge theory. For a connected sum, we use a localization method to produce DeRham representatives on the connected sum from DeRham representatives on the summands. We give an example of a Riemannian connected sum, the double torus, where it is impossible to extend a harmonic form on one summand to the entire connected sum.","abstract_html":"We construct examples illustrating various aspects of Hodge theory on Riemannian manifolds. We consider the relationship between the harmonic forms on a product space and the harmonic forms on the factors as well as the harmonic forms on a connected sum and the harmonic forms on the summands. Algebraic topology provides relations between the cohomology groups of these spaces via the Kunneth formula and the connected sum formula. We consider the problem of describing these relationships in terms of DeRham cohomology and in terms of Hodge theory. For a connected sum, we use a localization method to produce DeRham representatives on the connected sum from DeRham representatives on the summands. We give an example of a Riemannian connected sum, the double torus, where it is impossible to extend a harmonic form on one summand to the entire connected sum.","abstract_has_math":false,"creators":["Kerofsky, Louis Joseph"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Tondeur, Philippe"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:41:29Z","date_published":"2011-05-07T13:41:29Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Kerofsky, Louis Joseph"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624385","(UMI)AAI9624385"],"render_values":[{"text":"AAI9624385","href":null,"code":true},{"text":"(UMI)AAI9624385","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22489","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tondeur, Philippe"]},{"key":"dc:creator","label":"Author","values":["Kerofsky, Louis Joseph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:41:29Z","10000-01-01","1995"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Kerofsky, Louis Joseph"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624385","(UMI)AAI9624385","http://hdl.handle.net/2142/22489"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We construct examples illustrating various aspects of Hodge theory on Riemannian manifolds. We consider the relationship between the harmonic forms on a product space and the harmonic forms on the factors as well as the harmonic forms on a connected sum and the harmonic forms on the summands. Algebraic topology provides relations between the cohomology groups of these spaces via the Kunneth formula and the connected sum formula. We consider the problem of describing these relationships in terms of DeRham cohomology and in terms of Hodge theory. For a connected sum, we use a localization method to produce DeRham representatives on the connected sum from DeRham representatives on the summands. We give an example of a Riemannian connected sum, the double torus, where it is impossible to extend a harmonic form on one summand to the entire connected sum.","Studying Hodge theory on warped products, we give an explicit solution for the harmonic one forms on the torus embedded in three space with its induced metric. We consider examples of harmonic forms on the same smooth manifold equipped with different metrics. In general, a localized change in metric alters the harmonic representative of a fixed cohomology class globally. We give a surprising example of two metrics on the torus which differ only inside of a ball such that the harmonic forms representing the DeRham class of dx in each metric are identical outside the ball. We utilize the heat equation determined by the Laplacian to examine these examples more carefully.","Finally, possible applications of the methods introduced in this work are discussed. Areas of application include spectral theory, Dehn surgery, the metric dependence of harmonic forms, and the study of warped product spaces.","Made available in DSpace on 2011-05-07T13:41:29Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624385.pdf: 2282430 bytes, checksum: 1d0e93334be85a499c9057592958d81f (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:57:58Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Harmonic forms under metric and topological perturbations"]}]}],"canonical_facts":{"dc:contributor":["Tondeur, Philippe"],"dc:creator":["Kerofsky, Louis Joseph"],"dc:date":["2011-05-07T13:41:29Z","10000-01-01","1995"],"dc:description":["We construct examples illustrating various aspects of Hodge theory on Riemannian manifolds. We consider the relationship between the harmonic forms on a product space and the harmonic forms on the factors as well as the harmonic forms on a connected sum and the harmonic forms on the summands. Algebraic topology provides relations between the cohomology groups of these spaces via the Kunneth formula and the connected sum formula. We consider the problem of describing these relationships in terms of DeRham cohomology and in terms of Hodge theory. For a connected sum, we use a localization method to produce DeRham representatives on the connected sum from DeRham representatives on the summands. We give an example of a Riemannian connected sum, the double torus, where it is impossible to extend a harmonic form on one summand to the entire connected sum.","Studying Hodge theory on warped products, we give an explicit solution for the harmonic one forms on the torus embedded in three space with its induced metric. We consider examples of harmonic forms on the same smooth manifold equipped with different metrics. In general, a localized change in metric alters the harmonic representative of a fixed cohomology class globally. We give a surprising example of two metrics on the torus which differ only inside of a ball such that the harmonic forms representing the DeRham class of dx in each metric are identical outside the ball. We utilize the heat equation determined by the Laplacian to examine these examples more carefully.","Finally, possible applications of the methods introduced in this work are discussed. Areas of application include spectral theory, Dehn surgery, the metric dependence of harmonic forms, and the study of warped product spaces.","Made available in DSpace on 2011-05-07T13:41:29Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624385.pdf: 2282430 bytes, checksum: 1d0e93334be85a499c9057592958d81f (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:57:58Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9624385","(UMI)AAI9624385","http://hdl.handle.net/2142/22489"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Kerofsky, Louis Joseph"],"dc:subject":["Mathematics"],"dc:title":["Harmonic forms under metric and topological perturbations"],"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":"2026-07-22T22:25:20Z"}