The Ahlfors estimate

Estimate of the Cauchy Integral over Ahlfors Regular Curves

We obtain the complete characterization of those domains G ⊂ C which admit the so called estimate of the Cauchy integral, that is to say, ̨̨̨R ∂G f(z) dz ̨̨̨ ≤ C(G) ‖f‖∞ γ(E) for all E ⊂ G and f ∈ H∞(G \ E), where γ(E) is the analytic capacity of E. The corresponding result for continuous functions f and the continuous analytic capacity α(E) is also proved.

Generalized Ahlfors Functions

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γz}z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Let p ∈ Σ and let F be the family of all bounded holomorphic functions f on Σ such that f(p) ≥ 0 and f(z) ∈ γ̂z for almost every z ∈ ∂Σ. Then there exists a smooth up to the boundary holomorphic function f0 ∈ F w...

Ahlfors Theorems for Differential Forms

and Applied Analysis 3 Let y1, . . . , yk be an orthonormal system of coordinates in R, 1 ≤ k ≤ n. Let A be a domain in R, and let B be an n − k -dimensional Riemannian manifold. We consider the manifold N A × B. 2. Boundary Sets Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We als...

Generation of the Ahlfors Five Islands Theorem

On the Ahlfors Finiteness Theorem

The goal of this note is to give a proof of the Ahlfors Finiteness theorem which requires just the bare minimum of the complex analysis: (a) the existence theorem for the Beltrami equation and (b) the Rado-Cartan uniqueness theorem for holomorphic functions. However our proof does require some (by now standard) 3-dimensional topology and Greenberg's algebraic trick to deal with the triply-punct...