Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 233-247

THE TEICHMÜLLER PROBLEM FOR MEAN DISTORTION

Gaven J. Martin

Massey University, Institute of Information and Mathematical Sciences
Albany, Auckland, New Zealand; g.j.martin 'at' massey.ac.nz

Abstract. The classical Teichmüller problem asks one to identify the deformation of a disk which holds the boundary fixed, moves the origin to a given point and which minimises the maximal conformal distortion. The minimiser exists and is quasiconformal - Teichmüller identified the extremal. Here we study the same problem, but instead of the maximal conformal distortion we consider the mean conformal distortion. In this setting many of the usual tools of quasiconformal mappings are lost. In surprising contrast to this classical case, we show that there cannot be a minimiser. However we give asymptotically sharp bounds for the minimal mean distortion and conjectured extrema. These exhibit quite different behaviour to that observed for the maximal conformal distortion and lend themselves to possibly modeling other phenomena in material science, for instance tearing. The key tools for the proofs of the main results are based on our recent joint work with Astala, Iwaniec and Onninen.

2000 Mathematics Subject Classification:

Key words: Quasiconformal, mean distortion, Teichmüller.

Reference to this article: G.J. Martin: The Teichmüller problem for mean distortion. Ann. Acad. Sci. Fenn. Math. 34 (2009), 233-247.

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