Envelopes with Prescribed Singularities

TL;DR

This paper establishes the optimal regularity of certain complex geometric envelopes with prescribed singularities on compact Kähler manifolds, with implications for pluricomplex Green's functions and Hele-Shaw flow.

Contribution

It proves the $C^{1,1}$ regularity of envelopes with prescribed singularities on compact Kähler manifolds and extends results to manifolds with boundary, answering a question on Hele-Shaw flow.

Findings

Optimal $C^{1,1}$ regularity on Zariski open sets

Regularity of pluricomplex Green's functions

Positive answer to Hele-Shaw flow question

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.