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The Complex Monge–Ampère Equation in Kähler Geometry

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Pluripotential Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2075))

Abstract

We will discuss two main cases where the complex Monge–Ampère equation (CMA) is used in Käehler geometry: the Calabi–Yau theorem which boils down to solving nondegenerate CMA on a compact manifold without boundary and Donaldson’s problem of existence of geodesics in Mabuchi’s space of Käehler metrics which is equivalent to solving homogeneous CMA on a manifold with boundary. At first, we will introduce basic notions of Käehler geometry, then derive the equations corresponding to geometric problems, discuss the continuity method which reduces solving such an equation to a priori estimates, and present some of those estimates. We shall also briefly discuss such geometric problems as Käehler–Einstein metrics and more general metrics of constant scalar curvature.

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Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    Zbigniew Błocki

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Błocki, Z. (2013). The Complex Monge–Ampère Equation in Kähler Geometry. In: Pluripotential Theory. Lecture Notes in Mathematics(), vol 2075. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36421-1_2

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