New results on the singularity analysis of the Kaehler-Ricci flow

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In this thesis, we study the singularity development of the Kaehler-Ricci flow on holomorphic fibrations, and classify the singularity models of some classes of fibrations using parabolic rescaling. We first study the collapsing behavior of Calabi-Yau fibrations under the Kaehler-Ricci flow. A compact Kaehler manifold with semi-ample canonical line bundle admits a fibration of Calabi-Yau manifolds with possibly singular fibers. The convergence behavior for this class of manifolds under the Kaehler-Ricci flow was first studied by Song and Tian who establish the metric convergence in the sense of currents. In this thesis, we obtain the optimal collapsing rate of the nonsingular Calabi-Yau fibers, thus improving Song and Tian's work in analytic and geometric aspects. Secondly, we focus on a specific type of holomorphic fibration, namely the CP^1-bundles over Kaehler-Einstein manifolds. Fiber collapsing in the sense of Gromov-Hausdorff convergence was shown to occur in this case by Song, Szekelyhidi and Weinkove. We study the finite-time singularities for these manifolds using parabolic rescaling and dilation procedures adapted from Hamilton and Perelman in their works of the Ricci flow on 3-manifolds. We prove that when the flow metric has cohomogeneity-1 symmetry the collapsing occurs as a Type I singularity and we show that the singularity is modelled by C^n X CP^1.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2012
Issuance monographic
Language English


Associated with Fong, Tsz Ho
Associated with Stanford University, Department of Mathematics
Primary advisor Schoen, Richard (Richard M.)
Thesis advisor Schoen, Richard (Richard M.)
Thesis advisor Brendle, Simon, 1981-
Thesis advisor White, Brian, 1957-
Advisor Brendle, Simon, 1981-
Advisor White, Brian, 1957-


Genre Theses

Bibliographic information

Statement of responsibility Tsz Ho Fong.
Note Submitted to the Department of Mathematics.
Thesis Thesis (Ph.D.)--Stanford University, 2012.
Location electronic resource

Access conditions

© 2012 by Tsz Ho Fong
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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