Calabi flow in Riemann surfaces revisited: a new point of view

By Xiuxiong Chen. 2000

87 citations recorded by the source corpus.

In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we prove the long term existence and convergence of the Calabi flow. Thus give a new proof to Chruscial's theorem. The set of simple ideas of global integral estimates and concentration compactness should have further implications in other heat flow problems.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5f9392c11a8e359c7f108f87a489142cedfa5495

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Duong H. Phong
2. Jacob Sturm
3. Xiuxiong Chen
4. S. Brendle
5. Weiyong He
6. G'abor Sz'ekelyhidi
7. Robert J. Berman
8. B. Weinkove

In Geometry and Topology

1. Duong H. Phong
2. Jacob Sturm
3. Xiuxiong Chen
4. G'abor Sz'ekelyhidi
5. Robert J. Berman
6. Weiyong He
7. B. Weinkove
8. S. Brendle

In Nuclear and High Energy Physics

1. S. Brendle
2. Weiyong He
3. Michael Struwe
4. Huabin Ge
5. Jacob Sturm
6. Duong H. Phong
7. Xiuxiong Chen

Citations in the corpus

• On the Calabi Flow (2006, methodology): “In this paper, we follow the path of [16] to tackle the long time existence under some suitable curvature assumption.”
• Curvature flows on surfaces (2002, background): “Finally, Chen observes that the Calabi flow also decreases the Mabuchi energy, introduced in [3].”
• Long time existence of Minimizing Movement solutions of Calabi flow (2012, unknown): “The rigorous proof requires the theory of almost smooth geodesics of Chen-Tian [11] discussed in §3.3.”
• Combinatorial Calabi flows on surfaces (2012, background): “Moreover, these two flows converge to constant scalar curvature metric (see [4], [5], [6], [7], [9], [24], and [25]).”
• Lectures on Stability and Constant Scalar Curvature (2008, background): “The main interest lies in its formulation, otherwise it is a generalization, requiring no new estimates, of results in Chen [26].”
• Global existence and convergence for a higher order flow in conformal geometry (2003, methodology): “The approach used in [6] and [24] is based on integral estimates and”