{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/14056"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/14056","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds","abstract":"In this thesis we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle $E$ over an arbitrary K\"{a}hler manifold $(X,omega )$. In particular we show that the flow is determined at infinity by the holomorphic structure of $E$. Namely, if we fix an integrable unitary reference connection $A_{0}$ defining the holomorphic structure, then the Yang-Mills flow with initial condition $A_{0}$, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle $E_{infty } $, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of $(E,A_{0})$. Moreover, $E_{infty }$ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of $1$ and $2$ complex dimensions and proves the general case of a conjecture of Bando and Siu. Chapter 1 is an introduction and a review of the background material. Chapter 2 gives the proof of several critical intermediate results, including the existence of an approximate critical hermitian structure. Chapter 3 concludes the proof of the main theorem.","abstract_html":"In this thesis we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle $E$ over an arbitrary K&quot;{a}hler manifold $(X,omega )$. In particular we show that the flow is determined at infinity by the holomorphic structure of $E$. Namely, if we fix an integrable unitary reference connection <span class=\"etd-inline-math\">A<sub>0</sub></span> defining the holomorphic structure, then the Yang-Mills flow with initial condition <span class=\"etd-inline-math\">A<sub>0</sub></span>, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle <span class=\"etd-inline-math\">E<sub>infty </sub> </span>, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of <span class=\"etd-inline-math\">(E,A<sub>0</sub>)</span>. Moreover, <span class=\"etd-inline-math\">E<sub>infty </sub></span> extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of $1$ and $2$ complex dimensions and proves the general case of a conjecture of Bando and Siu. Chapter 1 is an introduction and a review of the background material. Chapter 2 gives the proof of several critical intermediate results, including the existence of an approximate critical hermitian structure. Chapter 3 concludes the proof of the main theorem.","abstract_has_math":true,"creators":["Sibley, Benjamin Caleb"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Wentworth, Richard A"],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013","date_published":"2013","updated_at":"2026-07-24T03:02:31Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1903/14056","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wentworth, Richard A"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Sibley, Benjamin Caleb"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2013-06-28T06:18:36Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2013-06-28T06:18:36Z"]},{"key":"dc:date.issued","label":"Date","values":["2013"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/14056"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle $E$ over an arbitrary K\"{a}hler manifold $(X,omega )$. In particular we show that the flow is determined at infinity by the holomorphic structure of $E$. Namely, if we fix an integrable unitary reference connection $A_{0}$ defining the holomorphic structure, then the Yang-Mills flow with initial condition $A_{0}$, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle $E_{infty } $, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of $(E,A_{0})$. Moreover, $E_{infty }$ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of $1$ and $2$ complex dimensions and proves the general case of a conjecture of Bando and Siu. Chapter 1 is an introduction and a review of the background material. Chapter 2 gives the proof of several critical intermediate results, including the existence of an approximate critical hermitian structure. Chapter 3 concludes the proof of the main theorem."]},{"key":"dc:title","label":"Title","values":["Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wentworth, Richard A"],"dc:contributor.department":["Mathematics"],"dc:creator":["Sibley, Benjamin Caleb"],"dc:date.accessioned":["2013-06-28T06:18:36Z"],"dc:date.available":["2013-06-28T06:18:36Z"],"dc:date.issued":["2013"],"dc:description.abstract":["In this thesis we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle $E$ over an arbitrary K\"{a}hler manifold $(X,omega )$. In particular we show that the flow is determined at infinity by the holomorphic structure of $E$. Namely, if we fix an integrable unitary reference connection $A_{0}$ defining the holomorphic structure, then the Yang-Mills flow with initial condition $A_{0}$, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle $E_{infty } $, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of $(E,A_{0})$. Moreover, $E_{infty }$ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of $1$ and $2$ complex dimensions and proves the general case of a conjecture of Bando and Siu. Chapter 1 is an introduction and a review of the background material. Chapter 2 gives the proof of several critical intermediate results, including the existence of an approximate critical hermitian structure. Chapter 3 concludes the proof of the main theorem."],"dc:identifier.uri":["http://hdl.handle.net/1903/14056"],"dc:title":["Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T03:02:31Z"}