Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY OF RICCI FLOWS BASED ON KILLING CONDITIONS

  • Zhao, Peibiao (DEPARTMENT OF APPLIED MATHEMATICS NANJING UNIVERSITY OF SCIENCE AND TECHNOLOGY) ;
  • Cai, Qihui (DEPARTMENT OF APPLIED MATHEMATICS NANJING UNIVERSITY OF SCIENCE AND TECHNOLOGY)
  • Published : 2009.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

C. Guenther studied the stability of DeTurck flows by using maximal regularity theory and center manifolds, but these arguments can not solve the stability of Ricci flows because the Ricci flow equation is not strictly parabolic. Recognizing this deficiency, the present paper considers and obtains the stability of Ricci flows, and of quasi-Ricci flows in view of some Killing conditions.

Keywords

References

  1. M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379-392. https://doi.org/10.4310/jdg/1214429060
  2. P. Butzer and H. Johnen, Lipschitz spaces on compact manifolds, J. Functional Analysis 7 (1971), 242-266. https://doi.org/10.1016/0022-1236(71)90034-6
  3. Q. H. Cai and P. B. Zhao, Stability of quasi-DeTurck flows in Riemannian manifolds of quasi-constant sectional curvatures, Chinese Ann. Math. Ser. A 29 (2008), no. 1, 97-106.
  4. H. D. Cao and B. Chow, Recent developments on the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 59-74. https://doi.org/10.1090/S0273-0979-99-00773-9
  5. B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical Surveys and Monographs, 110. American Mathematical Society, Providence, RI, 2004.
  6. G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Rational Mech. Anal. 101 (1988), no. 2, 115-141. https://doi.org/10.1007/BF00251457
  7. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157-162. https://doi.org/10.4310/jdg/1214509286
  8. G. Dore and A. Favini, On the equivalence of certain interpolation methods, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 4, 1227-1238.
  9. J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
  10. C. Guenther, J. Isenberg, and D. Knopf, Stability of the Ricci flow at Ricci-flat metrics, Comm. Anal. Geom. 10 (2002), no. 4, 741-777. https://doi.org/10.4310/CAG.2002.v10.n4.a4
  11. R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222. https://doi.org/10.1090/S0273-0979-1982-15004-2
  12. R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. https://doi.org/10.4310/jdg/1214436922
  13. R. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153-179. https://doi.org/10.4310/jdg/1214440433
  14. R. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
  15. R. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7-136, Int. Press, Cambridge, MA, 1995
  16. R. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), no. 4, 695-729. https://doi.org/10.4310/CAG.1999.v7.n4.a2
  17. D. Henry, Geometric Theorem of Semilinear Parabolic Equations, Springer Verlag, Berlin, 1981.
  18. J. Isenberg and M. Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Differential Geom. 35 (1992), no. 3, 723-741. https://doi.org/10.4310/jdg/1214448265
  19. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. https://doi.org/10.2307/1994022
  20. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479-495. https://doi.org/10.4310/jdg/1214439291
  21. G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations 8 (1995), no. 4, 753-796.
  22. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition. Johann Ambrosius Barth, Heidelberg, 1995.
  23. K. Yano, Differential Geometry on Complex and Almost Complex Spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49 A Pergamon Press Book. The Macmillan Co., New York 1965.
  24. R. Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc. 338 (1993), no. 2, 871-896. https://doi.org/10.2307/2154433
  25. P. B. Zhao and H. Z. Song, Quasi-Einstein hypersurfaces in a hyperbolic space, Chinese Quart. J. Math. 13 (1998), no. 2, 49-52.
  26. P. B. Zhao and X. P. Yang, On quasi-Einstein field equation, Northeast. Math. J. 21 (2005), no. 4, 411-420.
  27. P. B. Zhao and X. P. Yang, On stationary hypersurfaces in Euclidean spaces, Acta Math. Sci. Ser. B Engl. Ed. 26 (2006), no. 2, 349-357. https://doi.org/10.1016/S0252-9602(06)60057-X

Cited by

  1. On Intrinsic Properties of Ricci Flow Curve vol.41, pp.1, 2017, https://doi.org/10.1007/s40995-017-0213-1