Monge-Ampère Equations, Geodesics and Geometric Invariant Theory

Existence and uniqueness theorems for weak solutions of a complex MongeAmpère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed. 1. A complex Monge-Ampère equation The equation of primary interest in this lecture is the following Dirichlet problem for a completely degenerate complex Monge-Ampère equation: Let M̄ be a complex manifold of dimension m with smooth boundary ∂M̄ , Ω0 a smooth (1, 1)-form with Ω0 = 0. Then for any φ ∈ C(∂M̄), find Φ ∈ PSH(M̄,Ω0) so that (Ω0 + √ −1 2 ∂∂̄Φ) = 0, Φ|∂M̄ = φ. (1) Here the space PSH(M̄,Ω0) is the space of Ω0-plurisubharmonic functions on M̄ defined in section §7, and the equation is to hold in the generalized sense also defined there. We shall be especially interested in general existence and uniqueness theorems for (1), as well as in the construction of explicit solutions when M̄ is of the form M̄ = X × A, where X is a closed Kähler manifold equipped with a positive line bundle L, A = {w ∈ C; 1 ≤ |w| ≤ e} is an annulus in C, and φ is a boundary value function which is invariant with respect to the rotations of A. Our main results [25] are the general existence and uniqueness theorems for the Monge-Ampère equation (1) stated in Theorems 2-5, and, when M̄ = X × A, the explicit construction in Theorem 1, which says in particular that the solutions of (1) can be approximated by geodesic paths of Bergman metrics. Lecture at the Journées Equations aux Dérivées Partielles, Forges-les-Eaux, June 6-10, 2005. Research supported in part by National Science Foundation grants DMS-02-45371 and DMS-01-00410.