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A uniqueness theorem in Kähler geometry

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Abstract

We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F(Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler–Einstein.

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References

  1. Ball J.M.: Differentiablity properties of symmetric and isotropic functions. Duke Math. J. 51, 699–728 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Besse, A.: Einstein manifolds. Ergeb. Math. Grenzgeb. Band 10. Springer-Verlag, Berlin and New York (2009, in press)

  3. Bian, B., Guan, P.: A microscopic convexity principle for fully nonlinear elliptic equations. Invent. Math. (2009, in press)

  4. Caffarelli L., Guan P., Ma X.N.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60, 1769–1791 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Calabi, E.: Extremal Kähler metrics, in seminar on differential geometry. In: Annals of Mathematics Study, vol. 102, pp. 259–290. Princeton University Press, Princeton, NJ (1982)

  6. Chen X.X., Tian G.: Ricci flow on Kähler-Einstein surfaces. Invent. Math. 147, 487–544 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ecker K., Huisken G.: Immersed hypersurfaces with constant Weingarten curvature. Math. Ann. 283, 329–332 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Glaeser G.: Fontions composées différentiables. Ann. Math. 77(1), 193–209 (1963)

    Article  MathSciNet  Google Scholar 

  9. Guan P., Ma X.N: The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation. Invent. Math. 151, 553–577 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Howard A., Smyth B., Wu H.: On compact Kähler manifolds of nonnegative bisectional curvature, I. Acta Math. 147, 51–56 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, Q.: Ph.D. thesis, McGill University (2008)

  12. Wu H.: On compact Kähler manifolds of nonnegative bisectional curvature. II. Acta Math. 147, 57–70 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Yau S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Pengfei Guan & Qun Li

  2. Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China

    Xi Zhang

Authors
  1. Pengfei Guan
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  2. Qun Li
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  3. Xi Zhang
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Corresponding author

Correspondence to Pengfei Guan.

Additional information

P. Guan was supported in part by an NSERC Discovery grant, X. Zhang was supported by NSF in China, No. 10771188.

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Guan, P., Li, Q. & Zhang, X. A uniqueness theorem in Kähler geometry. Math. Ann. 345, 377–393 (2009). https://doi.org/10.1007/s00208-009-0358-0

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  • DOI: https://doi.org/10.1007/s00208-009-0358-0

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