A Note on the Hyperconvexity of Pseudoconvex Domains Beyond Lipschitz Regularity

Abstract

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 194(4) 1987) beyond Lipschitz regularity.

Linear invariants of complex manifolds and their plurisubharmonic variations

“…It is well known that two pseudoconvex domains (or even Stein manifolds) Ω 1 and Ω 2 are biholomorphic if and only if O(Ω 1 ) and O(Ω 2 ), the spaces of holomorphic functions, are isomorphic as C-algebras with unit. This implies that the holomorphic structure of a pseudoconvex domain is uniquely determined by algebraic structure of the space of holomorphic functions on it.…”

“…Let Ω 1 ⊂ C n and Ω 2 ⊂ C m be bounded hyperconvex domains. Suppose that there is a p > 0, p = 2, 4, 6 · · · , such that (1) there is a linear isometry T : A p (Ω 1 ) → A p (Ω 2 ), and (2) the p-Bergman kernels of Ω 1 and Ω 2 are exhaustive, then m = n and there exists a unique biholomorphic map F : Ω 1 → Ω 2 such that |T φ • F ||J F | 2/p = |φ|, ∀φ ∈ A p (Ω 1 ), where J F is the holomorphic Jacobian of F . If n = 1, the assumption of hyperconvexity can be dropped.…”

“…Let Ω 1 ⊂ C n and Ω 2 ⊂ C m be bounded hyperconvex domains. Suppose that there is a linear isometry T : A p (Ω 1 ) → A p (Ω 2 ) for some p ∈ (0, 2) , then m = n and there exists a unique biholomorphic map F : Ω 1 → Ω 2 such that |T φ • F ||J F | 2/p = |φ|, ∀φ ∈ A p (Ω 1 ).…”

“…It is known that bounded pseudoconvex domains with Lipschitz boundary are hyperconvex [12]. Even more, bounded pseudoconvex domains with α-Hörder boundary for all α < 1 are hyperconvex [1]. Recall that a bounded domain Ω ⊂ C n is called homogeneous regular, a concept proposed by Liu-Sun-Yau [27], if there is a constant c ∈ (0, 1) such that for any z ∈ Ω, there is a holomorphic injective map f : Ω → B n with f (z) = 0 and B n (c) ⊂ f (Ω), where B n (c) = {z ∈ C n ; z < c}.…”

The Geometry of m-Hyperconvex Domains

“…Kerzman and Rosay also studied which pseudoconvex domains are hyperconvex. We shall not address this question here (see e.g., the introduction of [5] for an up-to-date account of this question). Carlehed et al [17] showed in 1999 the equivalence between (1) and (4).…”