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On conformal solutions of the Yamabe flow

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The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.

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References

  1. Barros A, Ribeiro E. Jr: characterizations for compact almost Ricci solitons. Proc. Amer. Math. Soc. 140, 1033–1040 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. A, Barros, R. Batista, and E. Ribeiro Jr, Rigidity of gradient almost Ricci soliton, To appear in Illinois J. of Math. (2012).

  3. A. Barros, R. Batista, and E. Ribeiro Jr, Compact nontrivial almost Ricci solitons with constant scalar curvature are gradient, arXiv:1209.2720v1 [math.DG] (2012).

  4. R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Pure and applied mathematics (Academic Press), 15, New York: Academic Press, (1964).

  5. Bourguignon J., Ezin J.: curvature functions in a conformal class of metrics and conformal transformations. Trans. of the Amer. Math. Soc. 301, 723–736 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brendle S.: of the Yamabe flow for arbitrary initial energy. J. Diff. Geom. 69, 217–278 (2005)

    MathSciNet  MATH  Google Scholar 

  7. S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math. 170 (2007), no. 3, 541–576.

  8. Caminha A.: The geometry of closed conformal vector fields on Riemann spaces. Bull. Braz. Math. Soc. 42, 277–300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chow B.: Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure Appl. Math. 45, 1003–1014 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Chow, P. Lu and L. Ni Hamilton’s Ricci flow, Graduate Studies in Mathematics 77, American Mathematical Society, Providence, RI; Science Press, New York, (2006).

  11. Deshmukh S.: spheres by conformal vector fields. Ann. Univ. Ferrara 56, 231–236 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. G. De Rham, Sur la réducibilité d’un espace de Riemann, Comment. Math. Helv. 26 (1952), 328–344.

    Google Scholar 

  13. L. Di cerbo and M. Disconzi, Yamabe Solitons, Determinant of the Laplacian and the Uniformization Theorem for Riemann Surfaces, Letters in Mathematical Physics. 83 (2008) 13–18.

  14. Hamilton R.: Ricci flow on surfaces. Contemporary Mathematics. 71, 237–261 (1988)

    Article  MathSciNet  Google Scholar 

  15. O’Neill B.: Semi-Riemannian Geometry with Applications to General Relativity. Academic Press, New York. (1983)

    Google Scholar 

  16. S. Pigola et al., Ricci Almost Solitons, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. X (2011), 757–799.

  17. P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons, Pacific J. of Math. 241–2 (2009), 329–345.

    Google Scholar 

  18. Sasaki S., Goto M.: theorems on holonomy groups of Riemannian manifolds. Trans. Amer. Math. Soc. 80, 148–158 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schwetlick H., Struwe M.: Convergence of the Yamabe flow for large energies. J. reine angew. Math. 562, 59–100 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Tanno S., Weber W.: Closed conformal vector fields. J. Diff. Geom. 3, 361–366 (1969)

    MathSciNet  MATH  Google Scholar 

  21. Tashiro Y.: Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251–275 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  22. K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, Inc., New York, (1970).

  23. Ye R.: Global existence and convergence of Yamabe flow. J. Diff. Geom. 39, 35–50 (1994)

    MATH  Google Scholar 

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Correspondence to Ezequiel Barbosa.

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Ezequiel Barbosa: Partially supported by CNPq-BR. Ernani Ribeiro Jr: Partially supported by FUNCAP-BR.

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Barbosa, E., Ribeiro, E. On conformal solutions of the Yamabe flow. Arch. Math. 101, 79–89 (2013). https://doi.org/10.1007/s00013-013-0533-0

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