Advertisement

Some differentiable sphere theorems

  • Qing Cui
  • Linlin SunEmail author
Article
  • 17 Downloads

Abstract

In this paper, we obtain several new intrinsic and extrinsic differentiable sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected \(n(\ge 4)\)-dimensional Riemannian manifold M is diffeomorphic to \(\mathbb {S}^n\) if one of the following conditions holds pointwisely:
$$\begin{aligned} (i)\ R_0>\left( 1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right) K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left( 1-\frac{6(\sqrt{10}-3)}{n-1}\right) K_{max}. \end{aligned}$$
Here \(K_{max}\), \(Ric^{[k]}\) and \(R_0\) stand for the maximal sectional curvature, the k-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when M is a closed simply connected \(n(\ge 4)\)-dimensional submanifold immersed in \(\bar{M}\). We prove that M is diffeomorphic to \(\mathbb {S}^n\) if it satisfies some curvature pinching conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature \(\bar{K}_{max}\) and the squared norm of mean curvature vector \(\left|H\right|^2\). More precisely, we show that M is diffeomorphic to \(\mathbb {S}^n\) if one of the following conditions holds:
  1. (1)

    \(R_0\ge \left( 1-\frac{2}{n(n-1)}\right) \bar{K}_{max} +\frac{n(n-2)}{(n-1)^2}\left|H\right|^2\), and strict inequality is achieved at some point;

     
  2. (2)

    \(\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar{K}_{max}+\frac{n^2}{8}\left|H\right|^2,\) and strict inequality is achieved at some point;

     
  3. (3)

    \(\dfrac{Ric^{[2]}}{2} \ge \frac{n(n-3)}{n-2}\left( \bar{K}_{max}+\left|H\right|^2\right) ,\) and strict inequality is achieved at some point.

     
It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for \(n=4\).

Mathematics Subject Classification

53C20 53C40 

Notes

Acknowledgements

We would like to thank Dr. Jun Sun for useful discussions and suggestions. We also want to thank the referee for his/her careful reading and useful comments.

References

  1. 1.
    Andrews, B., Baker, C.: Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85(3), 357–395 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Berger, M.: Les variétés Riemanniennes \((1/4)\)-pincées. Ann. Scuola Norm. Sup. Pisa (3) 14, 161–170 (1960)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brendle, S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145(3), 585–601 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brendle, S.: Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151(1), 1–21 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brendle, S.: Ricci Flow and the Sphere Theorem, Graduate Studies in Mathematics, vol. 111. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  7. 7.
    Brendle, S., Schoen, R.: Classification of manifolds with weakly \(1/4\)-pinched curvatures. Acta Math. 200(1), 1–13 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brendle, S., Schoen, R.: Manifolds with \(1/4\)-pinched curvature are space forms. J. Amer. Math. Soc. 22(1), 287–307 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. (2) 96, 413–443 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Costa, E., Ribeiro Jr., E.: Four-dimensional compact manifolds with nonnegative biorthogonal curvature. Mich. Math. J. 63(4), 747–761 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ejiri, N.: Compact minimal submanifolds of a sphere with positive Ricci curvature. J. Math. Soc. Jpn. 31(2), 251–256 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. (2) 90, 75–90 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gu, J., Xu, H.: The sphere theorems for manifolds with positive scalar curvature. J. Differ. Geom. 92(3), 507–545 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gu, J., Xu, H., Zhao, E.: A sharp differentiable pinching theorem for manifolds with positive scalar curvature, preprint (2017)Google Scholar
  15. 15.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84(3), 463–480 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Karcher, H.: A short proof of Berger’s curvature tensor estimates. Proc. Amer. Math. Soc. 26, 642–644 (1970)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Klingenberg, W.: über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv. 35, 47–54 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lawson Jr., H., Simons, J.: On stable currents and their application to global problems in real and complex geometry. Ann. Math. (2) 98, 427–450 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Liu, K., Xu, H., Ye, F., Zhao, E.: Mean curvature flow of higher codimension in hyperbolic spaces. Comm. Anal. Geom. 21(3), 651–669 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Micallef, M., Moore, J.: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. Math. (2) 127(1), 199–227 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Micallef, M., Wang, M.: Metrics with nonnegative isotropic curvature. Duke Math. J. 72(3), 649–672 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Myers, S.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Perelman, G.: Proof of the soul conjecture of Cheeger and Gromoll. J. Differ. Geom. 40(1), 209–212 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rauch, H.: A contribution to differential geometry in the large. Ann. Math. (2) 54, 38–55 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Seshadri, H.: Manifolds with nonnegative isotropic curvature. Comm. Anal. Geom. 17(4), 621–635 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Xin, Y.: An application of integral currents to the vanishing theorems. Sci. Sin. Ser. A 27(3), 233–241 (1984)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Xu, H., Gu, J.: An optimal differentiable sphere theorem for complete manifolds. Math. Res. Lett. 17(6), 1111–1124 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Xu, H., Gu, J.: Geometric, topological and differentiable rigidity of submanifolds in space forms. Geom. Funct. Anal. 23(5), 1684–1703 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Xu, H., Gu, J.: Rigidity of Einstein manifolds with positive scalar curvature. Math. Ann. 358(1–2), 169–193 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Xu, H., Tian, L.: A differentiable sphere theorem inspired by rigidity of minimal submanifolds. Pac. J. Math. 254(2), 499–510 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Yau, S.: Selected expository works of Shing-Tung Yau with commentary. Vol. I. In: Li, Lizhen Ji Peter, Liu, Kefeng, Schoen, Richard (eds.) Advanced Lectures in Mathematics (ALM), vol. 28. International Press and Higher Education Press, Somerville and Beijing (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduChina
  2. 2.School of Mathematics and Statistics and Computational Science Hubei Key LaboratoryWuhan UniversityWuhanChina

Personalised recommendations