Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

Quasiconformal Planes with Bi-lipschitz Pieces and Extensions of Almost Affine Maps

A quasiplane f(V ) is the image of an n-dimensional Euclidean subspace V of R (1 ≤ n ≤ N − 1) under a quasiconformal map f : R → R . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a biLipschitz n-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of R. One main novelty of these results is that we analyz...

Almost Bi-lipschitz Embeddings and Almost Homogeneous Sets

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (biLipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but ‘almost homogeneous’. We therefore study the problem of embedding an almost homogeneous subset ...

Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem and Brennan's Conjecture

Flag-transitive Extensions of Dual Affine Planes

(with 1 , q , ̀ ) . An infinite family of simply connected examples for this diagram with q 5 2 is obtained by truncating Coxeter complexes of type D n . Two dif ferent finite families with three members and q 5 4 arise from the Steiner systems for the Mathieu groups M 2 2 , M 2 3 and M 2 4 . The geometries of these two families are simply connected . Three more simply connected examples of rank...

Evolution Dynamics of Conformal Maps with Quasiconformal Extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M = Diff S/Rot S embedded into T . As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. Some applications to Hele-Shaw flows of viscous fluids are given.