[{"key":"dc.contributor.author","value":"Li, Ka-Fai","language":null},{"key":"dc.date.accessioned","value":"2018-05-24T19:04:29Z","language":null},{"key":"dc.date.available","value":"2018-05-24T19:04:29Z","language":null},{"key":"dc.date.issued","value":"2018","language":"en"},{"key":"dc.identifier.uri","value":"http:\/\/hdl.handle.net\/2429\/66082","language":null},{"key":"dc.description.abstract","value":"We first study the general theory of K\u00e4hler-Ricci flow on non-compact complex\r\nmanifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate,\r\nwe prove a general existence theorem for K\u00e4hler metrics lying in the C^\\infty\r\n_\\loc closure of complete bounded curvature K\u00e4hler metrics that are uniformly equivalent\r\nto a fixed background metric. In particular we do not assume any curvature bounds. Next, we compare the maximal existence time of two complete bounded curvature solutions by using the equivalence of the initial metrics and using this, we also estimate the maximal existence time of a complete bounded curvature solution in terms of the curvature bound of a background metric. We also prove a uniqueness theorem for K\u00e4hler-Ricci flow which slightly improves the result of\r\n[13] in the K\u00e4hler case.\r\nWe apply the above results to study the K\u00e4hler-Ricci flow on some specific non-compact complex manifolds. We first study the K\u00e4hler-Ricci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete non-negatively curved U(n)-invariant K\u00e4hler metric admits a longtime U(n)-invariant solution to the K\u00e4hler-Ricci flow, and the solution converges to the standard Euclidean metric after rescaling.\r\nThen we study the K\u00e4hler-Ricci flow on a quasi-projective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of Lott-Zhang [23], we construct a K\u00e4hler-Ricci flow solution starting from certain smooth K\u00e4hler metrics. In particular, if the metric is the restriction of a smooth K\u00e4hler metric in the ambient space, then the solution instantaneously becomes complete and has cusp singularity at the divisor. We also produce a solution starting from some complete metrics that may not have bounded curvature, and the solution is likewise complete with cusp singularity for positive time. On the other hand, if the initial data has bounded curvature and is asymptotic to the standard cusp model in a certain sense, we find the maximal existence time of the corresponding complete bounded curvature solution to the K\u00e4hler-Ricci flow.","language":"en"},{"key":"dc.language.iso","value":"eng","language":"en"},{"key":"dc.publisher","value":"University of British Columbia","language":"en"},{"key":"dc.rights","value":"Attribution-NonCommercial-NoDerivatives 4.0 International","language":"*"},{"key":"dc.rights.uri","value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","language":"*"},{"key":"dc.title","value":"The K\u00e4hler-Ricci flow on non-compact manifolds","language":"en"},{"key":"dc.type","value":"Text","language":"en"},{"key":"dc.degree.name","value":"Doctor of Philosophy - PhD","language":"en"},{"key":"dc.degree.discipline","value":"Mathematics","language":"en"},{"key":"dc.degree.grantor","value":"University of British Columbia","language":"en"},{"key":"dc.date.graduation","value":"2018-09","language":"en"},{"key":"dc.type.text","value":"Thesis\/Dissertation","language":"en"},{"key":"dc.description.affiliation","value":"Science, Faculty of","language":"en"},{"key":"dc.description.affiliation","value":"Mathematics, Department of","language":"en"},{"key":"dc.degree.campus","value":"UBCV","language":"en"},{"key":"dc.description.scholarlevel","value":"Graduate","language":"en"},{"key":"atmire.cua.enabled","value":"","language":""}]