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Global existence and convergence of solutions of the Calabi flow on Einstein 4-manifolds

Published online by Cambridge University Press:  22 January 2016

Shu-Cheng Chang
Shu-Cheng Chang*
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, 30043, R.O.C.scchang@math.nthu.edu.tw
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Abstract

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In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

Keywords

53C2158G03
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 163 , September 2001 , pp. 193 - 214
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[A] Aubin, T., Nonlinear analysis on manifolds, Monge-Ampère Equations, Die Grundlehren der Math. Wissenschaften, Vol. 252, Springer-Verlag, New York, 1982.Google Scholar
[B] Besse, A., Einstein manifolds, Springer-Verlag, New York, 1986.Google Scholar
[C] Croke, C. B., Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ec. Norm. Super., 13 (1980), 419–435.Google Scholar
[Ca] Calabi, E., Extremal Kähler 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
[Ch1] Chang, S.-C., Critical Riemannian 4-manifolds, Math. Z., 214 (1993), 601–625.CrossRefGoogle Scholar
[Ch2] Chang, S.-C., Compactness theorems and the Calabi flow on Kaehler surfaces with stable tangent bundle, to appear.Google Scholar
[Ch3] Chang, S.-C., Compactness theorems of extremal-Kähler manifolds with positive first Chern class, Annals of Global Analysis and Geometry, 17 (1999), 267–287.Google Scholar
[Ch4] Chang, S.-C., The Calabi flow on Einstein manifolds, Lectures on Analysis and Geom etry (Yau, S. T. ed.), International Press, Hong Kong (1997), pp. 29–39.Google Scholar
[Ch5] Chang, S.-C., The Calabi flow on surfaces, to appear.Google Scholar
[Chru] Chrusciel, P. T., Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation, Commun. Math. Phys., 137 (1991), 289–313.Google Scholar
[CW] Chang, S.-C. and Wu, J. T., On the existence of extremal metrics for L2-norm of scalar curvature on closed 3-manifolds, J. of Mathematics Kyoto University, 39–3 (1999), 435–454.Google Scholar
[Chav] Chavel, I., Eigenvalues in Riemannian geometry, Academic Press, New York, 1984.Google Scholar
[CY] Chang, S.-Y. A. and Yang, P., Compactness of isospectral conformal metrics on 3-spheres, Comment. Math. Helvetici, 64 (1989), 363–374.Google Scholar
[O] Obata, M., The conjecture on conformal transformations of Riemannian manifolds, J. Diff. Geo., 6 (1971), 247–258.Google Scholar
[S] Schoen, R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., 20 (1984), 479–495.Google Scholar
[W] Wu, L.-F., The Ricci flow on complete R2 , Communications in Analysis and Geometry, Vol 1, No. 3 (1993), 439–472.Google Scholar