Abstract
In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, Y., Ma, L.: The Maximum Principle and the Yamabe Flow, Partial Differential Equations and Their Applications, pp. 211–224. World Scientific, Singapore (1999)
Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)
Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217–278 (2005)
Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)
Cheng, L., Zhu, A.: Yamabe flow and ADM mass on asymptotically flat manifolds. J. Math. Phys. 56, 101507 (2015). https://doi.org/10.1063/1.4934725
Chow, B.: Yamabe flow on locally conformally flat manifolds. Commun. Pure Appl. Math. XLV, 1003–1014 (1992)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence (2006)
Chrusciel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212, 231–264 (2003)
Dai, X., Ma, L.: Mass under Ricci flow. Commun. Math. Phys. 274, 65–80 (2007)
Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346–369 (2013)
Ecker, K., Knopf, D., Ni, L., Topping, P.: Local monotonicity and mean value formulas for evolving Riemannian manifolds. J. Reine Angew. Math. 616, 89–130 (2008). https://doi.org/10.1515/CRELLE.2008.019
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Commun. Pure Appl. Math. 33, 199–211 (1980)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)
Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 2, 255–306 (1982)
Hamilton, R.S.: Formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995)
Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics, vol. 5. New York University, Courant Institute of Mathematical Sciences, New York (1999). x+309 pp. ISBN: 0-9658703-4-0; 0-8218-2700-6
Kazdan, J.L.: Deformation to positive scalar curvature on complete manifolds. Math. Ann. 261(2), 227–234 (1982)
Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Zzv. 16, 151–164 (1981)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type, AMS Translation of Monograph, vol. 23. American Mathematical Society, Providence (1968)
Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)
Ma, L.: Gap theorems for locally conformally flat manifolds. J. Differ. Equ. 260(2), 1414–1429 (2016)
Ma, L.: Expanding Ricci solitons with pinched Ricci curvature. Kodai Math. J. 34, 140–143 (2011)
Ma, L., Cheng, L.: Yamabe flow and Myers type theorem on complete manifolds. J. Geom. Anal. 24(1), 246–270 (2014). https://doi.org/10.1007/s12220-012-9336-y
Ma, L., Cheng, L., Zhu, A.: Extending Yamabe flow on complete Riemannian manifolds. Bull. Sci. Math. 136, 882–891 (2012)
Moss, W.F., Piepenbrink, J.: Positive solutions of elliptic equation. Pac. J. Math. 75, 219–226 (1978)
Ni, L.: The Poisson equation and Hermitian–Einstein metrics on holomorphic vector boundle over complete non-compact Kaehler manifolds. Indiana Univ. Math. J. 51(3), 679–704 (2002)
Schulz, M.: Instantaneously complete Yamabe flow on hyperbolic space. arxiv:1612.02745v1
Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for ’large’ energies. J. Reine Angew. Math. 562, 59–100 (2003)
Souplet, P., Zhang, Q.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045–1053 (2006)
Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92, 47–71 (1988)
Trudinger, N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. XXI, 205–226 (1968)
Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57(2), 273–299 (2001)
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)
Ye, R.: Global existence and convergence of the Yamabe flow. J. Differ. Geom. 39, 35–50 (1994)
Acknowledgements
The author is very grateful to the unknown referees for helpful suggestions, in particular for pointing out the reference [32]. Part of the revision had been done when the author visited the Department of Mathematics at Stanford University in June and July of 2018 and the author would like to thank Prof. R. Schoen for the invitation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Struwe.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research is partially supported by the National Natural Science Foundation of China Nos. 11771124 and 11271111.
