Envelopes with Prescribed Singularities

Abstract

We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact K\"ahler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on K\"ahler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nystr\"om concerning the Hele-Shaw flow on an arbitrary Reimann surface.