The Calabi flow on Kähler Surfaces with bounded Sobolev constant (I)

2011 ◽  
Vol 354 (1) ◽  
pp. 227-261 ◽  
Author(s):  
Xiuxiong Chen ◽  
Weiyong He
Keyword(s):  
Calabi Flow ◽  
Kähler Surfaces ◽  
Sobolev Constant

scholarly journals Calabi flow on toric varieties with bounded Sobolev constant, I

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hongnian Huang
Keyword(s):  
Ricci Curvature ◽  
Toric Variety ◽  
Toric Varieties ◽  
Kähler Manifold ◽  
Kähler Metric ◽  
Calabi Flow ◽  
Kahler Manifold ◽  
Sobolev Constant ◽  
Kahler Metric ◽  
Uniformly Bounded

AbstractLet (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

scholarly journals The 2-Dimensional Calabi Flow

2006 ◽  
Vol 181 ◽  
pp. 63-73 ◽  
Author(s):  
Shu-Cheng Chang
Keyword(s):  
Type Estimate ◽  
Uniformization Theorem ◽  
Convergence Of Solutions ◽  
Calabi Flow ◽  
Closed Surfaces ◽  
Long Time Existence ◽  
Long Time ◽  
Sobolev Constant ◽  
Time Existence

AbstractIn this paper, based on a Harnack-type estimate and a local Sobolev constant bounded for the Calabi flow on closed surfaces, we extend author’s previous results and show the long-time existence and convergence of solutions of 2-dimensional Calabi flow on closed surfaces. Then we establish the uniformization theorem for closed surfaces.

scholarly journals Attainability of the best Sobolev constant in a ball

2018 ◽  
Vol 375 (1-2) ◽  
pp. 1-16
Author(s):  
Norisuke Ioku
Keyword(s):  
Sobolev Constant

scholarly journals An obstruction bundle relating Gromov-Witten invariants of curves and Kähler surfaces

2012 ◽  
Vol 134 (2) ◽  
pp. 453-506 ◽  
Author(s):  
Junho Lee ◽  
Thomas H. Parker
Keyword(s):  
Kähler Surfaces

scholarly journals Calabi flow and projective embeddings

2010 ◽  
Vol 84 (3) ◽  
pp. 489-523 ◽  
Author(s):  
Joel Fine ◽  
Kefeng Liu ◽  
Xiaonan Ma
Keyword(s):  
Calabi Flow

scholarly journals Local Solution and Extension to the Calabi Flow

2011 ◽  
Vol 23 (1) ◽  
pp. 270-282 ◽  
Author(s):  
Weiyong He
Keyword(s):  
Local Solution ◽  
Calabi Flow
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