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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01rb68xb89k
 Title: Kahler-Einstein metrics and K-stability Authors: Li, Chi Advisors: Tian, Gang Contributors: Mathematics Department Keywords: Continuity methodFano manifoldFutaki invariantKahler-Einstein metricK-stabilityTest configuration Subjects: Mathematics Issue Date: 2012 Publisher: Princeton, NJ : Princeton University Abstract: In this thesis, we study several problems related to the existence problem of Kahler-Einstein metric on Fano manifold. After introduction in the first chapter, in the second chapter, we review the basic theory both from PDE and variational point of view. Tian's program using finite dimensional approximation is then explained. Futaki invariant is discussed in detail for both its definition and calculation. K-stability is introduced following Tian and Donaldson. In the third chapter, we extend the basic theory to the twisted setting. As an important case, the analytic and algebraic theory are both extended to the conic setting. In the third chapter, we study the continuity method on toric Fano manifolds. We calculate the maximal value of parameter for solvability and study the limit behavior of the solution metrics. As a corollary, we prove Tian's partial C0-estimate on toric Fano manifolds. The log-Futaki invariant is calculated on toric Fano manifolds too. In the fourth chapter, we discuss the recent joint work with Dr. Chenyang Xu. We use Minimal Model Program (MMP) to simplify the degeneration and prove Tian's conjecture which reduce the test for K-stability to special degenerations. In the final chapter, we construct examples of rotationally symmetric solitons. These solitons are local models of special singularities of Kahler-Ricci flow. URI: http://arks.princeton.edu/ark:/88435/dsp01rb68xb89k Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Mathematics

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