On Ahlfors’ weak finiteness 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...

Generation of the Ahlfors Five Islands Theorem

On the Weak-type Constant of the Beurling-ahlfors Transform

(1.2) ‖B‖p = p∗ − 1, where 1 < p <∞ and p∗ = max{p, p p−1}, is partly motivated by its relation to the Gehring-Reich conjecture proved by Astala in [1]. (The lower bound of (p∗ − 1) was obtained by Lehto [16] in 1965.) Recent work has revealed B as an exemplary junction between Fourier analysis and probability, and martingale methods established by Burkholder have led to the present best known ...

A Theorem of Ahlfors for Hyperbolic Spaces

L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball B3 has finitely many sides, then the normalized Lebesgue measure of L(G) is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.

generation of the ahlfors five islands theorem

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