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, Volume 158, Issue 1–2, pp 223–234 | Cite as

Level set flow in 3D steady gradient Ricci solitons

  • Chih-Wei Chen
  • Kuo-Wei Lee
Article
  • 45 Downloads

Abstract

Given a nontrivial 3-dimensional steady gradient Ricci soliton, if the scalar curvature has decay order between \(-b\) and \(-a\) for some \(a\in (0,1], b\ge a\), then the umbilical ratio of the level set of the potential function lies in the class \(O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})\).

Mathematics Subject Classification

Primary 53C44 Secondary 53C25 

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Notes

Acknowledgements

The first author would like to thank Prof. Pengfei Guan for helpful discussions during the workshop for Besson’s 60th birthday. The second author is supported by the MOST research Grant 106-2115-M-018-002-MY2.

References

  1. 1.
    Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow, volume XII of Surv. Diff. Geom. Int. Press. (2008)Google Scholar
  3. 3.
    Cao, H.-D., He, C.: Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three. arXiv: 1412.2714, to appear in Comm. Anal. Geom
  4. 4.
    Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82, 363–382 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, C.-W.: On the regularity of the Ricci flow. National Taiwan University and University of Grenoble, Ph.D. Thesis (2011)Google Scholar
  6. 6.
    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, volume 77 of Graduate Studies in Mathematics. Am. Math. Soc. (2006)Google Scholar
  7. 7.
    Catino, G., Mastrolia, P., Monticelli, D.D.: Classification of expanding and steady Ricci solitons with integral curvature decay. Geom. Topol. 20(5), 2665–2685 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deruelle, A.: Steady gradient soliton with curvature in \({L}^1\). Commun. Anal. Geom. 20(1), 31–53 (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of ancient compact solutions to the Ricci flow on surfaces. J. Differ. Geom. 91(2), 171–214 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deng, Y., Zhu, X.: Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature. Math. Z 279, 211–226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Deng, Y., Zhu, X.: 3d steady Gradient Ricci Solitons with linear curvature decay. arXiv:1612.05713v1 (2016)
  12. 12.
    Guo, H.: Area growth rate of the level surface of the potential function on the \(3\)-dimensional steady gradient Ricci soliton. Proc. Am. Math. Soc. 137(6), 2093–2097 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guo, H.: Remarks on noncompact steady gradient Ricci solitons. Math. Ann. 345, 883–894 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, H.: Evolution equation of the Gauss curvature under hypersurface flows and its applications. Acta Math. Sin. Engl. Ser. 26(7), 1299–1308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84, 463–480 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Munteanu, O., Sung, C.-J., Wang, J.: Poisson equation on complete manifolds. arXiv:1701.02865 (2017)
  17. 17.
    Munteanu, O., Wang, J.: Smooth metric measure spaces with non-negative curvature. Commun. Anal. Geom. 19(3), 451–486 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ni, L., Wallach, N.: On a classification of the gradient Ricci solitons. Math. Res. Lett. 15(5), 941–955 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 (2002)
  21. 21.
    Wu, P.: On the potential function of gradient steady Ricci solitons. J. Geom. Anal. 23(1), 221–228 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational Kaohsiung Normal UniversityKaohsiungTaiwan
  2. 2.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan

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