{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71237"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71237","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Volume and Energy Stability for Immersions (Harmonic, Minimal)","abstract":"Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.","abstract_html":"Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.","abstract_has_math":false,"creators":["Hvidsten, Michael David"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:14Z","date_published":"2014-12-16T06:18:14Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8600213"],"render_values":[{"text":"(UMI)AAI8600213","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71237","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Hvidsten, Michael David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:14Z","10000-01-01","1985"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71237","(UMI)AAI8600213"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.","An interesting question is whether the stability of f coincides for volume and energy. The second variation formulas for volume and energy show us that for variations of f in normal directions, energy stability implies volume stability. Thus, one suspects that there are energy stable harmonic immersions that are volume unstable minimal immersions.","We find a minimal (harmonic) immersion f that is energy stable but not volume stable by looking at maps f: M (--->) N where N is a flat Riemannian manifold, M is a compact manifold without boundary, and dim N = dim M + 1, dim M (GREATERTHEQ) 2. We show that f is a minimal, volume stable immersion if f is totally geodesic. On the other hand, for f: M (--->) N with f harmonic and N flat, we get that f is automatically energy stable.","For oriented surfaces M of genus g immersed in the torus T('3), we show that for g = 0 there are no minimal immersions of M in T('3). For g = 1, we get that M must be totally geodesic and thus a sub-torus. For g (GREATERTHEQ) 2, we show that f cannot be totally geodesic. Thus, a minimal (harmonic) surface in T('3) of genus g (GREATERTHEQ) 2 must be area unstable, but energy stable.","One such minimally immersed surface of genus 9 can be constructed from Schwarz's tetrahedral surface. This surface is a minimal surface immersed in R('3) that is triply periodic, but not oriented. By taking an 8-fold covering on this surface, and then dividing out by the periodic action, we get a minimal surface in T('3) of genus 9 that is orientable. This surface will then be area unstable, but energy stable. Several other examples of surfaces with this stability behavior for energy and volume are also discussed.","Made available in DSpace on 2014-12-16T06:18:14Z (GMT). No. of bitstreams: 1 8600213.pdf: 1754606 bytes, checksum: bbe369b1357e65973fa86d99a7b49c7f (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71403 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","69 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."]},{"key":"dc:title","label":"Title","values":["Volume and Energy Stability for Immersions (Harmonic, Minimal)"]}]}],"canonical_facts":{"dc:creator":["Hvidsten, Michael David"],"dc:date":["2014-12-16T06:18:14Z","10000-01-01","1985"],"dc:description":["Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.","An interesting question is whether the stability of f coincides for volume and energy. The second variation formulas for volume and energy show us that for variations of f in normal directions, energy stability implies volume stability. Thus, one suspects that there are energy stable harmonic immersions that are volume unstable minimal immersions.","We find a minimal (harmonic) immersion f that is energy stable but not volume stable by looking at maps f: M (--->) N where N is a flat Riemannian manifold, M is a compact manifold without boundary, and dim N = dim M + 1, dim M (GREATERTHEQ) 2. We show that f is a minimal, volume stable immersion if f is totally geodesic. On the other hand, for f: M (--->) N with f harmonic and N flat, we get that f is automatically energy stable.","For oriented surfaces M of genus g immersed in the torus T('3), we show that for g = 0 there are no minimal immersions of M in T('3). For g = 1, we get that M must be totally geodesic and thus a sub-torus. For g (GREATERTHEQ) 2, we show that f cannot be totally geodesic. Thus, a minimal (harmonic) surface in T('3) of genus g (GREATERTHEQ) 2 must be area unstable, but energy stable.","One such minimally immersed surface of genus 9 can be constructed from Schwarz's tetrahedral surface. This surface is a minimal surface immersed in R('3) that is triply periodic, but not oriented. By taking an 8-fold covering on this surface, and then dividing out by the periodic action, we get a minimal surface in T('3) of genus 9 that is orientable. This surface will then be area unstable, but energy stable. Several other examples of surfaces with this stability behavior for energy and volume are also discussed.","Made available in DSpace on 2014-12-16T06:18:14Z (GMT). No. of bitstreams: 1 8600213.pdf: 1754606 bytes, checksum: bbe369b1357e65973fa86d99a7b49c7f (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71403 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","69 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."],"dc:identifier":["http://hdl.handle.net/2142/71237","(UMI)AAI8600213"],"dc:subject":["Mathematics"],"dc:title":["Volume and Energy Stability for Immersions (Harmonic, Minimal)"],"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":"2026-07-22T22:26:04Z"}