Lower Bounds on the Kobayashi Metric Near a Point of Infinite Type

Abstract

Abstract2015 Mathematica Josephina, Inc. Under a potential-theoretical hypothesis named f-property which holds for all pseudoconvex domains of finite type and many examples of infinite type, we give a new method for constructing a family of bumping functions and hence plurisubharmonic peak functions with good estimates. The rate of lower bounds on the Kobayashi metric follows by the estimates of peak functions. The application to the continuous extendibility of proper holomorphic maps is given.

Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type

“…In Section 2, we construct a weak Hölder, uniformly and strictly plurisubharmonic defining function via the work of the second author about the existence of the bumping functions and the plurisubharmonic peak functions on a domain which enjoys the f -Property of [Kha13]. This particular defining function is the crucial point in the establishing the existence of the solution to the complex MongeAmpère equation.…”

Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in $\mathbb C^n$

“…Here, the first inequality follows by φ p (w ν ) → x, the second follows by (5), the fourth follows by (16), and the last one follows by z ∈ E z 0 (x, R).…”

“…Recently, the first author [16] obtained lower bounds on the Kobayashi metric for a general class of pseudoconvex domains in C n , that contains all domains of finite type and many domains of infinite type.…”

“…The f -property is defined in [17,16] as Definition 1. We say that domain Ω has the f -property if there exists a family of functions {ψ η } such that (i) |ψ η | ≤ 1, C 2 , and plurisubharmonic on Ω; (ii) i∂∂ψ η ≥ c 1 f (η −1 ) 2 Id and |Dψ δ | ≤ c 2 η −1 on {z ∈ Ω : −η < δ Ω (z) < 0} for some constants c 1 , c 2 > 0, where δ Ω (z) is the euclidean distance from z to the boundary ∂Ω.…”

“…If domain is reduced to be convex of finite type m, then the t 1/m -property holds [18]. Furthermore, there is a large class of infinite type pseudoconvex domains that satisfy an f -properties [17,16]. For example (see [17]), the log 1/α -property holds for both the complex ellipsoid of infinite type…”