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

Abstract

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.

arXiv comment22 pages

Cited by in corpus (7)

• The Calabi flow with small initial energy
• Stability of Calabi flow near an extremal metric
• The algebraic structure of geometric flows in two dimensions
• Normalized Ricci flow on Riemann surfaces and determinants of Laplacian
• Pseudo-Calabi Flow
• Combinatorial Calabi flows on surfaces
• Kähler non-collapsing, eigenvalues and the Calabi flow