Envelopes with Prescribed Singularities

Abstract

We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler 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-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.

arXiv commentv2: Improved main results, minor typos corrected; 25 pages

References in corpus (7)

• The Space of Kähler metrics (II)
• Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity
• On the singularity type of full mass currents in big cohomology classes
• On the $C^{1,1}$ regularity of geodesics in the space of Kähler metrics
• The Monge-Ampère equation for non-integrable almost complex structures
• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• Regularity of envelopes in Kähler classes

Cited by in corpus (4)

• $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds
• Pluricomplex Green's functions and Fano manifolds
• $C^{1,1}$ regularity of geodesics of singular Kähler metrics
• Geodesics in the space of $m$-subharmonic functions with bounded energy