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 The KählerRicci flow on noncompact manifolds
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The KählerRicci flow on noncompact manifolds Li, KaFai
Abstract
We first study the general theory of KählerRicci flow on noncompact complex manifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate, we prove a general existence theorem for Kähler metrics lying in the C^\infty _\loc closure of complete bounded curvature Kähler metrics that are uniformly equivalent to 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 KählerRicci flow which slightly improves the result of [13] in the Kähler case. We apply the above results to study the KählerRicci flow on some specific noncompact complex manifolds. We first study the KählerRicci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete nonnegatively curved U(n)invariant Kähler metric admits a longtime U(n)invariant solution to the KählerRicci flow, and the solution converges to the standard Euclidean metric after rescaling. Then we study the KählerRicci flow on a quasiprojective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of LottZhang [23], we construct a KählerRicci flow solution starting from certain smooth Kähler metrics. In particular, if the metric is the restriction of a smooth Kähler 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 KählerRicci flow.
Item Metadata
Title 
The KählerRicci flow on noncompact manifolds

Creator  
Publisher 
University of British Columbia

Date Issued 
2018

Description 
We first study the general theory of KählerRicci flow on noncompact complex
manifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate,
we prove a general existence theorem for Kähler metrics lying in the C^\infty
_\loc closure of complete bounded curvature Kähler metrics that are uniformly equivalent
to 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 KählerRicci flow which slightly improves the result of
[13] in the Kähler case.
We apply the above results to study the KählerRicci flow on some specific noncompact complex manifolds. We first study the KählerRicci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete nonnegatively curved U(n)invariant Kähler metric admits a longtime U(n)invariant solution to the KählerRicci flow, and the solution converges to the standard Euclidean metric after rescaling.
Then we study the KählerRicci flow on a quasiprojective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of LottZhang [23], we construct a KählerRicci flow solution starting from certain smooth Kähler metrics. In particular, if the metric is the restriction of a smooth Kähler 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 KählerRicci flow.

Genre  
Type  
Language 
eng

Date Available 
20180524

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0367001

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201809

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International