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Conformal metric sequences with integral-bounded scalar curvature

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Let (Mg) be a smooth compact Riemiannian manifold without boundary and \(g_k\) be a metric conformal to g. Suppose \(\text{ vol }(M,g_k)+\Vert R_k\Vert _{L^p(M,g_k)}<C\), where \(R_k\) is the scalar curvature and \(p>\frac{n}{2}\). We will use the 3-circles theorem to study the bubble tree convergence of \(g_k\).

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References

  1. Anderson, M.T.: Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102(2), 429–445 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Anderson, M.T., Cheeger, J.: Diffeomorphism finiteness for manifolds with Ricci curvature and \(L^\frac{n}{2}\)-norm of curvature bounded. Geom. Funct. Anal. 1(3), 231–252 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Brendle, S., Marques, F.C.: Blow-up phenomena for the Yamabe equation II. J. Differ. Geom. 81(2), 225–250 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Chang, S.Y.A., Yang, P.: Compactness of isospectral conformal metrics on \({\cal{S}}^{3}\). Comment. Math. Helvetici 64(3), 363–374 (1989)

    MathSciNet  MATH  Google Scholar 

  5. Chang, S.Y.A., Yang, P.: Isospectral conformal metrics on 3-manifolds. J. Am. Math. Soc. 3(1), 117–145 (1990)

    MathSciNet  MATH  Google Scholar 

  6. Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am. J. Math. 92, 61–74 (1970)

    MathSciNet  MATH  Google Scholar 

  7. Cheeger, J.: \(L^p\)-bounds on curvature, elliptic estimates and rectifiability of singular sets. C. R. Math. Acad. Sci. Paris 334(3), 195–198 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Cheeger, J.: Integral bounds on curvature elliptic estimates and rectifiability of singular sets. Geom. Funct. Anal. 13(1), 20–72 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Cheeger, J., Colding, T.H., Tian, G.: On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12(5), 873–914 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Chen, J., Li, Y.: Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows. Adv. Math. 263, 357–388 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Ding, W., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3(3–4), 543–554 (1995)

    MathSciNet  MATH  Google Scholar 

  12. Druet, O.: Compactness for Yamabe metrics in low dimensions. Int. Math. Res. Not. 23, 1143–1191 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Gursky, M.: Compactness of conformal metrics with integral bounds on curvature. Duke Math. J. 72(2), 339–367 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Khuri, M., Marques, F., Schoen, R.: A compactness theorem for the yamabe problem. J. Differ. Geom. 81(1), 143–196 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Li, Y., Zhang, L.: Compactness of solutions to the Yamabe problem II. Calc. Var. PDEs 24(2), 185–237 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Li, Y., Zhang, L.: Compactness of solutions to the Yamabe problem III. J. Funct. Anal. 245(2), 438–474 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Lin, F., Han, Q.: Elliptic Partial Differential Equations. Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York. American Mathematical Society, Providence (1997)

    Google Scholar 

  18. Marques, F.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. J. Differ. Geom. 71(2), 315–346 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Petersen, P., Wei, G.: Relative volume comparison with integral curvature bounds. Geom. Funct. Anal 7(6), 1031–1045 (1997)

    MathSciNet  MATH  Google Scholar 

  20. Petersen, P., Wei, G.: Analysis and geometry on manifolds with integral Ricci curvature bounds. II. Trans. Am. Math. Soc. 353(2), 457–478 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Qing, J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3(1–2), 297–315 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Qing, J., Tian, G.: Bubbling of the heat flows for harmonic maps from surfaces. Commun. Pure. Appl. Math. 50(4), 295–310 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Second Edition. Graduate Texts in Mathematics, vol. 137. Springer, New York (2001)

    MATH  Google Scholar 

  24. Tian, G., Zhang, Z.: Regularity of Kähler–Ricci flows on Fano manifolds. Acta Math. 216(1), 127–176 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Schoen, R., Zhang, D.: Prescribed scalar curvature on the \(n\)-sphere. Calc. Var. PDEs 4(1), 1–25 (1996)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Prof. Hao Yin for bringing Three Circles Theorem to our attention. The authors also thank Prof. Chong Song for helpful suggestions during the preparation of this paper.

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Correspondence to Zhipeng Zhou.

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Li, Y., Zhou, Z. Conformal metric sequences with integral-bounded scalar curvature. Math. Z. 295, 1443–1473 (2020). https://doi.org/10.1007/s00209-020-02533-5

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