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The Fourth-Order Q-Curvature Flow on Closed 3-Manifolds

Published online by Cambridge University Press:  11 January 2016

Shu-Cheng Chang  and
Chin-Tung Wu
Shu-Cheng Chang
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu Taiwan 30013 R.O.C., scchang@math.nthu.edu.tw
Chin-Tung Wu
Affiliation:
Department of Mathematics Education, National Pingtung University of Education, Pingtung 90003, Taiwan R.O.C., ctwu@mail.npue.edu.tw
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Abstract

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Let the Paneitz operator P0 be strictly positive on a closed 3-manifold M with a fixed conformal class. It is proved that the solution of a fourth-order Q-curvature flow exists on M for all time and converges smoothly to a metric of constant Q-curvature.

Keywords

53C2158G03
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 185 , 2007 , pp. 1 - 15
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[A] Aubin, T., Nonlinear analysis on manifolds, Monge-Ampére Equations, Die Grundlehren der Math. Wissenschaften, Vol. 252, Springer-Verlag, New York, 1982.Google Scholar
[B] Branson, T., Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293–345.CrossRefGoogle Scholar
[BCY] Branson, T., Chang, S.-Y. A. and Yang, P., Estimates and extremals for zeta functions determinants on four-manifolds, Comm. Math. Phys., 149 (1992), 241–262.CrossRefGoogle Scholar
[Br] Brendle, S., Global existence and convergence for a higher-order flow in conformal geometry, Ann. of Math. (2), 158 (2003), no. 1, 323–343.CrossRefGoogle 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
[CCC] Chang, S.-C., Cheng, J.-H. and Chiu, H.-L., The Q-curvature flow on CR 3-manifolds, preprint.Google Scholar
[Ch] Chang, S. -C., Recent developments on the Calabi flow, Contemporary Mathematics, Vol. 367, AMS (2005), pp. 17–42.CrossRefGoogle Scholar
[Chru] Chruściel, 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
[CW1] Chang, S.-C. and Wu, C. -T., On the existence of extremal metrics for L2-norm of scalar caurvature on closed 3-manifolds, J. of Math. Kyoto Univ., 39 (1999), no. 3, 435–454.Google Scholar
[CW2] Chang, S.-C. and Wu, C. -T., Extremal metrics for quadratic functional of scalar curvature on closed 3-manifolds, Annals of Global Analysis and Geometry, 25 (2004), no. 1, 11–25.CrossRefGoogle Scholar
[CW3] Chang, S.-C. and Wu, C. -T., Extremal metrics for quadratic functional of scalar curvature on complete noncompact 3-manifolds, Indiana Univ. Math. J., 53 (2004), 243–268.CrossRefGoogle Scholar
[CGY] Chang, S.-Y. A., Gursky, M. and Yang, P., An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. Math., 155 (2002), 711–789.CrossRefGoogle Scholar
[CY] Chang, S.-Y. A., and Yang, P., Isospectral conformal metrics on 3-manifolds, J. Ameri. Math. Soc., 3 (1990), 117–145.CrossRefGoogle Scholar
[FH] Fefferman, C. and Hirachi, K., Ambient metric construction of Q-curvature in con-formal and CR geometries, to appear in Math. Res. Lett.Google Scholar
[Gu] Gursky, M. J., Compactness of conformal metrics with integral bounds on curvature, Duke Math. J., 72 (1993), no. 2, 339–367.CrossRefGoogle Scholar
[GV] Gursky, M. J. and Viaclovsky, J. A., A new variational characterization of three-dimensional space forms, Invent. Math., 145 (2001), 251–278.CrossRefGoogle Scholar
[H] Hirachi, K., Scalar pseudo-hermitian invariants and the Szegö kernel on 3-dimensional CR manifolds, Lecture Notes in Pure and Appl. Math. 143, Dekker (1992), pp. 67–76.Google Scholar
[HY] Hang, F. and Yang, P., The Sobolev inequality for Paneitz operator on three manifolds, Calc. of Variations, 21 (2004), 57–83.Google Scholar
[P] Paneitz, S., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint (1983).Google Scholar
[S] Simon, L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525–571.CrossRefGoogle Scholar
[XY] Xu, X. and Yang, P., On a fourth order equation in 3-D, ESAIM: Cont. Optim. Calc. Var., 8 (2002), 1029–1042.Google Scholar
[YZ] Yang, P. and Zhu, M., On the Paneitz energy on standard 3-spheres, ESAIM: Cont. Optim. Calc. Var., 10 (2004), 211–223.Google Scholar