Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T05:32:16.863Z Has data issue: false hasContentIssue false
  • English
  • Français

The 2-Dimensional Calabi Flow

Published online by Cambridge University Press:  11 January 2016

Shu-Cheng Chang
Shu-Cheng Chang*
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu Taiwan 30043, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 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.

Keywords

53C2158G03
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 181 , March 2006 , pp. 63 - 73
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[C] Croke, C. B., Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ec. Norm. Super., 13 (1980), 419–435.CrossRefGoogle Scholar
[Ca1] Calabi, E., Extremal Käahler metrics, Seminars on Differential Geometry (Yau, S. T., ed.), Princeton Univ. Press and Univ. of Tokyo Press, Princeton, New York (1982), pp. 259–290.Google Scholar
[Ca2] Calabi, E., Extremal Köahler metrics II, Differential Geometry and Complex Analysis (Chavel, I. and Farkas, H. M., eds.), Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1985), pp. 95–114.Google Scholar
[Ch1] Chang, S.-C., Critical Riemannian 4-manifolds, Math. Z., 214 (1993), 601–625.Google Scholar
[Ch2] Chang, S.-C., Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2, J. of Mathematics of Kyoto University, 40 (2000), no. 2, 363–377.Google Scholar
[Ch3] Chang, S.-C., Global existence and convergence of solutions of the Calabi flow on Einstein 4-manifolds, Nagoya Math. J., 163 (2001), 193–214.CrossRefGoogle Scholar
[Ch4] Chang, S.-C., Recent developments on the Calabi flow, Contemporary Mathematics, 367 (2005), 17–42.Google Scholar
[Chen] Chen, X. X., Calabi flow in Riemann surfaces revisited: a new point of view, IMRN (2001), no. 6, 275–297.Google Scholar
[Chru] Chruásciel, P. T., Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation, Commun. Math. Phys., 137 (1991), 289–313.CrossRefGoogle Scholar
[CW] Chang, S.-C. and Wu, J.-T., On the existence of extremal metrics for L2 -norm of scalar caurvature on closed 3-manifolds, J. of Mathematics of Kyoto University, 39 (1999), no. 3, 435–454.Google Scholar
[G] Gursky, M. J., Compactness of conformal metrics with integral bounds on curvature, Duke Math. J., 72 (1993), no. 2, 339–367.CrossRefGoogle Scholar
[GT] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equation of second order, Springer-Verlag, New York, 1983.Google Scholar
[KW] Kazdan, J. L. and Warner, F. W., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math., 101 (2) (1975), 317–331.Google Scholar