The Complex Monge–Ampère Equation in Kähler Geometry

Abstract

We will discuss two main cases where the complex Monge–Ampere equation (CMA) is used in Kaehler 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 Kaehler metrics which is equivalent to solving homogeneous CMA on a manifold with boundary. At first, we will introduce basic notions of Kaehler 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 Kaehler–Einstein metrics and more general metrics of constant scalar curvature.