Holomorphic Extension and the Complex Monge-Ampère Equation
University of Michigan, Ann Arbor
Some time ago, Boutet de Monvel proved a Paley-Wiener type theorem for compact real analytic manifolds M, relating the exponential rate of convergence of the eigenfunction expansion of a real analytic function f with respect to an elliptic operator P with the radius of holomorphic extension of said function into the complex domain, i.e., into a neighborhood of M in its complexification. We discuss what happens when one wants to study this extension globally. This leads to holomorphic differential geometric properties of global complexifications, and especially exhaustions by solutions of the homogeneous complex Monge-Ampère equation. We also discuss another approach to these questions using the Toeplitz operators of Boutet de Monvel and Guillemin. This is joint work with Zhou Zhang and with Victor Guillemin.