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On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality

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Abstract

We show that a problem asked by Yau (Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319, 1992) cannot be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng (Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. 17, pp. 517–542, 2010).

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References

  1. Cao, H.-D.: Existence of gradient Kähler–Ricci solitons. Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), vol. 1–16. A K Peters, Wellesley (1996)

  2. Cao H.-D.: Limits of solutions to the Kähler–Ricci flow. J. Differ. Geom. 65, 257–272 (1997)

    Google Scholar 

  3. Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow. Surveys in differential geometry. vol. XII. Geometric flows, pp. 47–112, Surv. Differ. Geom., vol. 12. Int. Press, Somerville (2008)

  4. Chau A., Tam L.-F.: On the complex structure of Kähler manifolds with nonegative curvature. J. Differ. Geom. 73(3), 491–530 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Chau A., Tam L.-F.: Non-negatively curved Kähler manifolds with average quadratic curvature decay. Commun. Anal. Geom. 15(1), 121–146 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Chau A., Tam L.-F.: On the Steinness of a class of Kähler manifolds. J. Differ. Geom. 79(2), 167–183 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Chen B.-L., Zhu X.-P.: On complete noncompact Kähler manifolds with positive bisectional curvature. Math. Ann. 327(1), 1–23 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen B.-L., Zhu X.-P.: Volume growth and curvature decay of positively curved Kähler manifolds. Q. J. Pure Appl. Math. 1(1), 68–108 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Chen B.-L., Tang S.-H., Zhu X.-P.: A uniformization theorem for complete non-compact Kähler surfaces with positive bisectional curvature. J. Differ. Geom. 67(3), 519–570 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Klembeck Paul F.: A complete Kähler metric of positive curvature on \({\mathbb{C}^{n}}\). Proc. Am. Math. Soc. 64(2), 313–316 (1977)

    MATH  Google Scholar 

  11. Greene, R.E., Wu, H.: Analysis on noncompact Kähler manifolds. Several complex variables (Proc. Sympos. Pure Math., vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975), pp. 69–100. American Mathematical Society, Providence (1977)

  12. Ni L.: A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature. J. Am. Math. Soc. 17(4), 909–946 (2004)

    Article  MATH  Google Scholar 

  13. Ni L.: Ancient solutions to Kähler–Ricci flow. Math. Res. Lett. 12(5–6), 633–653 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Ni, L.: Monotonicity and holomorphic functions. Geometry and analysis, vol. I. Advanced Lecture in Mathematics, vol. 17, pp. 447–457. Higher Education Press and International Press, Beijing (2010)

  15. Ni L., Tam L.-F.: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature. J. Differ. Geom. 64(3), 457–524 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Petrunin, A.M.: An upper bound for the curvature integral. (Russian) Algebra i Analiz 20(2), 134–148 (2008); translation in St. Petersburg Math. J. 20(2), 255–265 (2009)

  17. Siu Y.-T.: Pseudoconvexity and the problem of Levi. Bull. Am. Math. Soc. 84(4), 481–512 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, H.-H., Zheng, F.: Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. I. Advanced Lecture in Mathematics, vol. 17, pp. 517–542. Higher Education Press and International Press, Beijing (2010)

  19. Yau, S.-T.: Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319. International Press, Hong Kong (1992)

  20. Yau, S.-T.: A review of complex differential geometry. Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, pp. 619–625, Part II. American Mathematical Society, Providence (1991)

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  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA

    Bo Yang

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  1. Bo Yang
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Correspondence to Bo Yang.

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Yang, B. On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality. Math. Ann. 355, 765–781 (2013). https://doi.org/10.1007/s00208-012-0803-3

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  • DOI: https://doi.org/10.1007/s00208-012-0803-3

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