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.

On the "Equation missing" Regularity of Geodesics in the Space of Kähler Metrics

“…Note that if ϕ in Theorem 1.1 is actually smooth near the boundary, one can simply take ϕ ε = max{ϕ, v ε − C ε } for all ε > 0, where the v ε come from the nef condition (see below), and the C ε are large constants such that ϕ ε = ϕ near the boundary -this is because we don't actually need the estimates in part (a) to hold everywhere, only near the boundary. In this manner we recover, and actually improve upon, [28,Theorem 1.3].…”

“…The new boundary estimate is established in Proposition 2.3 -it is an a priori bound for the tangentnormal derivatives along the Berman path [2]. We prove Corollaries 1.2 and 1.3 in Section 3, and briefly discuss the case of geodesic rays -we mainly observe that the results in [28] still apply in this generality. Finally, we include an appendix containing some estimates for the Dirichlet problem for the ω-Laplacian when the boundary data is degenerating, which will be needed in the proof of Theorem 1.1.…”

“…Proof of Theorem 1.1. Our strategy will be the same as in [28], which is a combination of the techniques in [2] and the estimates in [12]. We begin by approximating V by the following envelopes:…”

“…By [2, Proposition 2.4], [28,Proposition 4.5], the u ε,β converge uniformly to V ε as β → ∞ (the convergence is not uniform in ε, but this will not be a problem as we will send β to infinity before sending ε to zero). Note that as u ε,β ∈ PSH(M, α + εω) with u ε,β | ∂M = ϕ ε | ∂M , by (2.6), we have:…”

“…Our main result follows the approach in [2,8,23,6,28] with several technical improvements -we state our main theorem now, delaying some notation and definitions until the end of the section. Theorem 1.1.…”