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Rigidity of shrinking Ricci solitons

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Abstract

We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.

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References

  1. Besse A.L.: Einstein manifolds. Ergebnisse der Mathematik, vol. 10. Springer, Berlin (1987)

    Google Scholar 

  2. Cao H.-D.: Geometry of Ricci solitons. Chin. Ann. Math. 27B(82), 121–142 (2006)

    Google Scholar 

  3. Cao X.: Compact gradient shrinking Ricci solitons with positive curvature operator. J. Geom. Anal. 17, 425–433 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Cao, X., Wang, B., Zhang, Z.: On locally conformally flat gradient shrinking Ricci solitons. arXiv:0807.0588v3

  5. Chow, B., Knopf, D.: The Ricci flow: an introduction. Mathematical Surveys and Monograps. American Mathematical Society, Providence (2004)

  6. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow. Graduates studies in Mathematics. American Mathematical Society, Providence (2006)

  7. Derdzinski, A.: Compact Ricci solitons, preprint

  8. Eminenti M., La Nave G., Mantegazza C.: Ricci solitons-the equation point of view. Manuscripta Math. 127, 345–367 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernández-López M., García-Río E.: A remark on compact Ricci solitons. Math. Ann. 340, 893–896 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gray A.: Einstein-like manifolds which are not Einstein. Geom. Dedic. 7, 259–280 (1978)

    Article  MATH  Google Scholar 

  11. Hamilton R.S.: The formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7–136 (1995) International Press

    MathSciNet  Google Scholar 

  12. Munteanu, O., Sesum, N.: On gradient Ricci solitons. arXiv:0910.1105v1

  13. Naber, A.: Some geometry and analysis on Ricci solitons. arXiv:math/0612532v1

  14. Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. arXiv:0710.5579v2

  15. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:0211159v1

  16. Petersen P., Wylie W.: Rigidity of gradient Ricci solitons. Pac. J. Math 241, 329–345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Petersen P., Wylie W.: On gradient Ricci solitons with symmetry. Proc. Am. Math. Soc 137, 2085–2092 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Petersen, P., Wylie, W. On the classification of gradient Ricci solitons. arXiv:0712.1298v5

  19. Zhang Z.-H. Gradient shrinking solitons with vanishing Weyl tensor. arXiv:0807.1582v1

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Authors and Affiliations

  1. Faculty of Mathematics, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Manuel Fernández-López & Eduardo García-Río

Authors
  1. Manuel Fernández-López
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  2. Eduardo García-Río
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Correspondence to Eduardo García-Río.

Additional information

Supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain).

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Fernández-López, M., García-Río, E. Rigidity of shrinking Ricci solitons. Math. Z. 269, 461–466 (2011). https://doi.org/10.1007/s00209-010-0745-y

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  • DOI: https://doi.org/10.1007/s00209-010-0745-y

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