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The complex Monge-Ampere equation in Kähler geometry

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2nd Forum of Polish Mathematicians (2 ; 01.07.2008 ; Częstochowa, Polska)
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We discuss two cases when the complex Monge-Ampere equation appears in Kähler geometry: the Calabi conjecture (with its solution by Yau) and the equation for geodesics in the Mabuchi space of Kähler metrics, introduced independently by Semmes and Donaldson.
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Bibliografia
  • [1] Aubin T., Equations du type de Monge-Ampère sur les variétés Kähleriennes compactes, C.R. Acad. Sci. Paris 283 (1976), 119-121.
  • [2] Błocki Z., Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds, Math. Z. 244 (2003), 153-161.
  • [3] Błocki Z., Uniqueness and stability for the Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), 1697-1702.
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  • [5] Błocki Z., The Monge-Ampère equation on compact Kähler manifolds, unpublihed lecture notes based on the course given at Winter School in Complex Analysis, Toulouse, 2005, available at http://gamma.im.uj.edu.pl/~blocki/
  • [6] Błocki Z., A gradient estimate in the Calabi-Yau theorem, preprint, 2008.
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  • [9] Chen X.X., The space of Kähler metrics, J. Diff. Geom. 56 (2000), 189-234.
  • [10] Demailly J.P., Complex Analytic and Differential Geometry, 1997, available at http://wwwfourier.ujf-grenoble.fr/~demailly/books.html
  • [11] Donaldson S.K., Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, 13-33, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999.
  • [12] Donaldson S.K., Scalar curvature and stability of toric varieties, J. Diff. Geom. 62 (2002), 289-349.
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  • [14] Hörmander L., An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, Amsterdam 1990.
  • [15] Kołodziej S., The complex Monge-Ampère equation, Acta Math. 180 (1998), 69-117.
  • [16] Mabuchi T., Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math. 24 (1987), 227-252.
  • [17] Newlander A., Nirenberg L., Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391-404.
  • [18] Pliś S., A counterexample to the regularity of the degenerate complex Monge-Ampère equation, Ann. Polon. Math. 86 (2005), 171-175.
  • [19] Semmes S., Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495-550.
  • [20] Yau S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Comm. Pure Appl. Math. 31 (1978), 339-411.
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Bibliografia
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bwmeta1.element.baztech-article-BPC6-0014-0001
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