{"id":{"repo_id":"potsdam-diss","oai_identifier":"oai:kobv.de-opus4-uni-potsdam:6723"},"canonical_url":"https://search.dev.ndltd.org/etd/potsdam-diss/oai:kobv.de-opus4-uni-potsdam:6723","repository":{"repo_id":"potsdam-diss","name":"Universität Potsdam - Diss","base_url":"https://publishup.uni-potsdam.de/opus4-ubp/oai"},"display":{"title":"Stability of Einstein Manifolds","abstract":"This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.","abstract_html":"This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.","abstract_has_math":false,"creators":["Kröncke, Klaus"],"institution":"Universität Potsdam","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Bär, Christian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-12-20","date_published":"2013-12-20","updated_at":"2026-07-24T03:51:46Z","subjects":["Einstein-Mannigfaltigkeiten","Ricci-Fluss","Variationsstabilität","Einstein-Hilbert-Wirkung","Einstein manifolds","Ricci flow","variational stability","Einstein-Hilbert action"],"languages":[],"rights":["Creative Commons - Namensnennung, Nicht kommerziell, Keine Bearbeitung 3.0 Deutschland"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://publishup.uni-potsdam.de/frontdoor/index/index/docId/6723","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bär, Christian"]},{"key":"dc:creator","label":"Author","values":["Kröncke, Klaus"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universität Potsdam"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Potsdam"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Einstein-Mannigfaltigkeiten","Ricci-Fluss","Variationsstabilität","Einstein-Hilbert-Wirkung","Einstein manifolds","Ricci flow","variational stability","Einstein-Hilbert action"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Creative Commons - Namensnennung, Nicht kommerziell, Keine Bearbeitung 3.0 Deutschland"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.","Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Stability of Einstein Manifolds","Stabilität von Einstein-Mannigfaltigkeiten"]}]}],"canonical_facts":{"dc:contributor":["Bär, Christian"],"dc:creator":["Kröncke, Klaus"],"dc:description.abstract":["This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.","Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universität Potsdam"],"dc:rights":["Creative Commons - Namensnennung, Nicht kommerziell, Keine Bearbeitung 3.0 Deutschland"],"dc:subject":["Einstein-Mannigfaltigkeiten","Ricci-Fluss","Variationsstabilität","Einstein-Hilbert-Wirkung","Einstein manifolds","Ricci flow","variational stability","Einstein-Hilbert action"],"dc:title":["Stability of Einstein Manifolds","Stabilität von Einstein-Mannigfaltigkeiten"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Potsdam"]},"updated_at":"2026-07-24T03:51:46Z"}