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Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

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Abstract

In the paper we prove the path-connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

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References

  1. Boileau, M., Maillot, S., Porti, J.: Three-Dimensional Orbifolds and their Geometric Structures, Panaromas et Synthèses 15. Soc. Math. de France, Paris (2003)

  2. Brendle, S.: A short proof for the convergence of the Yamabe flow on \(S^{n}\). Pure Appl. Math. Q. 3, 499–512 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brendle, S., Schoen, R.: Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc. 22, 287–307 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carr, R.: Construction of manifolds of positive scalar curvature. Trans. Am. Math. Soc. 307, 63–74 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cerf, J.: Sur les difféomorphismes de la sphère de dimension trois \((\Gamma _{4}=0)\). Lecture Notes on Math, vol. 53. Springer-Verlag, Berlin-New York (1968)

    Book  Google Scholar 

  6. Chen, B.L., Zhu, X.P.: Ricci flow with surgery on four-manifolds with positive isotropic curvature. J. Diff. Geom. 74, 177–264 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Chen, B.L., Tang, S.H., Zhu, X.P.: Complete classification of compact four manifolds with positive isotropic curvature. J. Diff. Geom. 91, 41–80 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Gromov, M., Lawson Jr, H.B.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. 111, 423–434 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gromov, M., Lawson Jr, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58, 83–196 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Hamilton, R.: Four-manifolds with positive isotropic curvatures. Commun. Anal. Geom. 5, 1–92 (1997)

    Article  MATH  Google Scholar 

  12. Hamilton, R.: Four-manifolds with positive curvature operator. J. Diff. Geom. 24, 153–179 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoelzel, S.: Surgery stable curvature conditions. arXiv:1303.6531v1, to appear in Math. Ann

  15. Kato, M.: Topology of Hopf surfaces. J. Math. Soc. Jpn. 27, 222–238 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kneser, H.: Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresbericht der Deut. Math. Verein. 38, 248–260 (1929)

    MATH  Google Scholar 

  17. Kreck, M., Stolz, S.: Nonconnected moduli spaces of positive sectional curvature metrics. J. Am. Math. Soc. 6, 825–850 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuiper, N.: On conformally flat spaces in the large. Ann. Math. 50, 916–924 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marques, F.C.: Deforming three-manifolds with positive scalar curvature. Ann. Math. 176, 815–863 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. McCullough, D.: Isometries of elliptic 3-manifolds. J. Lond. Math. Soc. 65, 167–182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Micallef, M., Moore, J.D.: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. Math. 127, 199–227 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Micallef, M., Wang, M.: Metrics with nonnegative isotropic curvatures. Duke Math. J. 72, 649–672 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Milnor, J.: A unique decomposition theorem for 3-manifolds. Am. J. Math. 84, 1–7 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morgan, J., Tian, G.: Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, American Mathematical Society (2007)

  25. Nguyen, H.: Isotropic curvature and the Ricci flow. Internat. Math. Res. Notices 3, 536–558 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Ohba, K.: On fiber bundles over \(S^{1}\) having small Seifert manifolds as fibers. J. Fac. Sci. Tokyo 37, 659–702 (1990)

    MathSciNet  MATH  Google Scholar 

  27. Ohshika, K.: Discrete Groups. Iwanami Series in Modern Mathematics. American Mathematical Society

  28. Perelman, G.: Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109v1

  29. Rosenberg, J.: Manifolds of positive scalar curvature: a progress report. Surv. Diff. Geom 11, 259–294 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rosenberg, J., Stolz, S.: Metrics of positive scalar curvature and connections with surgery. Surveys on surgery theory. Ann. Math. Stud 149, 353–386 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Ruberman, D.: Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants. Geom. Topology 5, 895–924 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28, 159–183 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  33. Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Steenrod, N.: Topology of Fibre Bundles. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  35. Ue, M.: Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds II. J. Math. Soc. Jpn. 43, 149–183 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Weyl, H.: Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Vierteljahrsschr. Naturforsch. Ges. Zür. 61, 40–72 (1916)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank Professors F. C. Marques, S. H.Tang and X. P. Zhu for many helpful discussions. The authors are grateful to anonymous referees for giving us many valuable suggestions. The first author is partially supported by NSFC 11025107, 11521101 and a Grant (No. 141gzd02) from SYSU.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Bing-Long Chen & Xian-Tao Huang

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  1. Bing-Long Chen
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  2. Xian-Tao Huang
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Correspondence to Bing-Long Chen.

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Chen, BL., Huang, XT. Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds. Math. Ann. 366, 819–851 (2016). https://doi.org/10.1007/s00208-015-1343-4

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  • DOI: https://doi.org/10.1007/s00208-015-1343-4

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