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A Remark on Covering of Compact Kähler Manifolds and Applications

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Ukrainian Mathematical Journal Aims and scope

Recently, Kolodziej proved that, on a compact Kähler manifold M, the solutions to the complex Monge–Ampère equation with right-hand side in Lp, p > 1, are Hölder continuous with exponent depending on M and ‖f‖p (see [Math. Ann., 342, 379 (2008)]). Then, by using the regularization techniques from [J. Algebr. Geom., 1, 361 (1992)], the authors of [J. Eur. Math. Soc., 16, 619–647 (2014)] determined the optimal exponent of the solutions. We construct a cover of the compact Kähler manifold M that depends only on the curvature of M. Then, as an application, based on the arguments from [Math. Ann., 342, 379 (2008)], we show that the solutions are Hölder continuous with exponent that depends just on the function f on the right-hand side and the upper bound of the curvature of M.

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References

  1. E. Bedford and B. A. Taylor, “The Dirichlet problem for the complex Monge–Ampère operator,” Invent. Math., 37, 1–44 (1976).

    Article  MathSciNet  Google Scholar 

  2. E. Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions,” Acta Math., 149, 1–40 (1982).

    Article  MathSciNet  Google Scholar 

  3. U. Cegrell, “The general definition of the complex Monge–Ampère operator,” Ann. Inst. Fourier (Grenoble), 54, 159–179 (2004).

    Article  MathSciNet  Google Scholar 

  4. J.-P. Demailly, Complex Analytic and Differential Geometry, https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf.

  5. J.-P. Demailly, “Regularization of closed positive currents and intersection theory,” J. Algebraic Geom., 1, 361–409 (1992).

    MathSciNet  MATH  Google Scholar 

  6. J.-P. Demailly, S. Dinew, V. Guedj, H. H. Pham, S. Kolodziej, and A. Zeriahi, “Hölder continuous solutions to Monge–Ampère equation,” J. Eur. Math. Soc. (JEMS), 16, 619–647 (2014).

    Article  MathSciNet  Google Scholar 

  7. T. C. Dinh, V. A. Nguyen, and N. Sibony, “Exponential estimates for plurisubharmonic functions and stochastic dynamics,” J. Different, Geom., 84, 465–488 (2010).

  8. V. Guedj, S. Kolodziej, and A. Zeriahi, “H¨older continuous solutions to the complex Monge–Ampère equations,” Bull. Lond. Math. Soc., 40, 1070–1080 (2008).

    Article  MathSciNet  Google Scholar 

  9. S. Kolodziej, “The Monge–Amp`ere equation,” Acta Math., 180, 69–117 (1998).

    Article  MathSciNet  Google Scholar 

  10. S. Kolodziej, “The Monge–Amp`ere equation on compact Kähler manifolds,” Indiana Univ. Math. J., 52, 667–686 (2003).

    Article  MathSciNet  Google Scholar 

  11. S. Kolodziej, “The complex Monge–Amp`ere equation and pluripotential theory,” Mem. Amer. Math. Soc. (2005).

  12. S. Kolodziej, “Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in Lp: the case of compact Kähler manifolds,” Math. Ann., 342, 379–386 (2008).

    Article  MathSciNet  Google Scholar 

  13. L. M. Hai, P. H. Hiep, and H. N. Quy, “Local property of the class Eχ,loc,” J. Math. Anal. Appl., 302, 440–445 (2013).

  14. P. H. Hiep, “Hölder continuity of solutions to the complex Monge–Ampère operator on compact Kähler manifolds,” Ann. Inst. Fourier (Grenoble), 60, 1857–1869 (2010).

    Article  MathSciNet  Google Scholar 

  15. H. Hein, “Gravitational instantons from rational elliptic surfaces,” J. Amer. Math. Soc., 25, No. 2, 355–393 (2012).

    Article  MathSciNet  Google Scholar 

  16. V. V. Hung and H. N. Quy, “Convergence in capacity on smooth hypersurfaces of compact Kähler manifolds,” Ann. Polon. Math., 103, 175–187 (2012).

    Article  MathSciNet  Google Scholar 

  17. G. Tian and S. T. Yau, “Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry,” in: Advanced Series in Mathematical Physics: Volume 1, Mathematical Aspects of String Theory, Proc. of the Conf. on Mathematical Aspects of String Theory, San Diego, CA, USA (1987), pp. 574–628.

  18. G. Tian and S. T. Yau, “Complete Kähler manifolds with zero Ricci curvature, I,” J. Amer. Math. Soc., 3, No. 3, 579–609 (1990).

    MathSciNet  MATH  Google Scholar 

  19. G. Tian and S. T. Yau, “Complete K¨ahler manifolds with zero Ricci curvature, II,” Invent. Math., 106, No. 1, 27–60 (1991).

    Article  MathSciNet  Google Scholar 

  20. S. T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation,” Comm. Pure Appl. Math., 31, 339–411 (1978).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Tay Bac University, Son La and University of Danang, University of Science and Education, Danang, Vietnam

    V. V. Hung & H. N. Quy

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  1. V. V. Hung
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  2. H. N. Quy
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Correspondence to V. V. Hung.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 1, pp. 138–148, January, 2021. Ukrainian DOI: 10.37863/umzh.v73i1.6038.

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Hung, V.V., Quy, H.N. A Remark on Covering of Compact Kähler Manifolds and Applications. Ukr Math J 73, 156–169 (2021). https://doi.org/10.1007/s11253-021-01915-0

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  • DOI: https://doi.org/10.1007/s11253-021-01915-0

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