By John M. Mackay

ISBN-10: 0821852299

ISBN-13: 9780821852293

Conformal measurement measures the level to which the Hausdorff measurement of a metric house might be diminished through quasisymmetric deformations. brought by means of Pansu in 1989, this idea has proved tremendous fruitful in a various variety of parts, together with geometric functionality concept, conformal dynamics, and geometric staff idea. This survey leads the reader from the definitions and simple thought via to energetic study purposes in geometric functionality conception, Gromov hyperbolic geometry, and the dynamics of rational maps, among different parts. It reports the speculation of size in metric areas and of deformations of metric areas. It summarizes the elemental instruments for estimating conformal measurement and illustrates their software to concrete difficulties of self reliant curiosity. quite a few examples and proofs are supplied. operating from uncomplicated definitions via to present learn components, this publication can be utilized as a advisor for graduate scholars attracted to this box, or as a invaluable survey for specialists. heritage wanted for a possible reader of the booklet contains a operating wisdom of actual and complicated research at the point of first- and second-year graduate classes

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**Extra resources for Conformal Dimension: Theory and Application**

**Sample text**

We restrict our discussion to the real, complex and quaternionic categories only since the Cayley hyperbolic plane requires a diﬀerent treatment. ) Let K ∈ {R, C, H} be one of the classical division algebras. Recall that the quaternions are the unique four-dimensional algebra over R spanned by 1 and the symbols i, j, k, subject to the rules i2 = j 2 = k 2 = ijk = −1. 1) x, y = xj yj j=1 where x = (x1 , . . , xn ) and x denotes the conjugate of x ∈ K. Denote by ||x|| = x, x 1/2 the corresponding norm.

Let M be a compact Riemannian manifold given as the quotient of HKn , K = R, by a group of isometries Γ acting discretely and properly discontinuously. If nk + k − 2 1≤a< , k = dimR K, nk − 1 then M admits no metric with (−a2 , −1)-pinched sectional curvature. In order to explain the proof, we recall the notion of volume entropy, which captures the asymptotic growth rate of the volumes of balls in an open manifold. 8. Let (M, g) be an open Riemannian manifold, and let dg be the associated distance function on M .

15 are folklore, implicit in early work of Gehring [57], [58] and V¨ais¨al¨a [174]. The local quasisymmetry of quasiconformal maps is sometimes known as the egg yolk principle; Heinonen [76, Chapter 11] attributes the term to McMullen. The precise deﬁnition of quasisymmetric maps in the metric space context was ﬁrst given by Tukia and V¨ais¨al¨ a [166]. However, the notion was implicit in many earlier works. In particular, quasisymmetric maps on the real line (as the boundary maps of quasiconformal maps of the upper half plane) have been well studied since at least the work of Ahlfors and Beurling [14].

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