Seppo Rickman

Geometric branched covers between generalized manifolds

We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension. 0. Introduction It is a difficult problem to determine when a given metric space is locally bi-Lipschitz equivalent to an open subset of Euclidean space. Recall that a map f : X → Y is L-Lipschitzif | f (a)− f (b)| ≤ L|a− b| for each pair of pointsa,b ∈ X, and for someL ≥ 1 independent of the points. A homeomorphismf is L-bi-Lipschitzif both f and f −1 areL-Lipschitz. (Generally, in this paper, we use the distance notation |x − y| in every metric space.) In 1979 L. Siebenmann and D. Sullivan [ SS] noted the curious fact that there are, for eachn ≥ 5, compact puren-dimensional polyhedra that are topological manifolds but do not admit local bi-Lipschitz parametrizations. The double suspension of a homology sphere with nontrivial π1 serves as an example. After some interesting positive results due to T. Toro [ T1], [T2], Semmes exhibited a family of geometrically nice metrics onS3 for which no local bi-Lipschitz parametrizations exist (see [S4], [S3]). At this point, it is not clear if a simple geometric characterization can be found for the metric spaces that admit local bi-Lipschitz parametrizations. (Dimensionn = 2 could be special here (cf. [ S1], [DS2], [HK2]; see, however, Rem. 0.6.) DUKE MATHEMATICAL JOURNAL Vol. 113, No. 3, c ©2002 Received 17 February 2000. Revision received 29 May 2001. 2000Mathematics Subject Classification . Primary 57M12; Secondary 30C65. Heinonen’s work supported by National Science Foundation grant number DMS 9970427.

The analogue of Picard's theorem for quasiregular mappings in dimension three

The theory of quasiregular mappings has turned out to be the fight extension of the geometric parts of the theory of analytic functions in the plane to real n-dimensional space. The study of these mappings was initiated by Re~etnjak around 1966 and his main contributions to the theory is presented in the recent book [8]. For the basic theory of quasiregular mappings we refer to [2], [3], [13]. The definition is given in Section 2.1. In 1967 Zofi~ [14] raised the question of the validity o f a Picards theorem on omitted values for quasiregular mappings. Such a theorem appeared in 1980 in the following form.

Quasiregular maps S3 → S3 with wild branch sets

Abstract Two examples of quasiregular maps S3 → S3 that branch on a wild Cantor set are constructed. As an application it is shown that certain interesting 3-dimensional metric spaces recently constructed by Semmes admit Lipschitz branched covers onto S3. Moreover, it is shown that a uniformly quasiconformal group of Freedman and Skora acting on S3 and not topologically conjugate to a Mobius group is quasiregularly semiconjugate to a Mobius group.

Classification of Riemannian manifolds in nonlinear potential theory

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50s by Ahlfors and Tôki. Our main objective is to show that such inclusions are proper also in the generalized case.

Existence of quasiregular mappings

Quasiregular mappings are roughly quasiconformal mappings without the homeomorphism requirement. In the Euclidean n-space R n , n ≥ 2, the definition is given as follows. A continuous map f: G → R n of a domain G in R n is called quasiregular (qr) if (1) f is in the local Sobolev space W n,loc 1 (G), i-.e. f has weak order partial derivatives which are locally in L n, and (2) there exists K, 1 ≤ K < ∞, such that

Failure of the Denjoy theorem for quasiregular maps in dimension ≥3

{\left| {f\prime (x)} \right|^{n}}K{J_{f}}(x)a.e.

Mappings into the n -Sphere with Punctures

(1.1) .

Applications of Modulus Inequalities

Quasiregular mappings were introduced and studied by Yu. G. Reshetnyak in a series of articles that began to appear in 1966. Reshetnyak used the term “mappings with bounded distortion” and defined them by means of the so called analytic definition of quasiconformal mappings leaving out the homeomorphism requirement. One of Reshetnyak’s main result in the theory is that, if not constant, a quasiregular mapping is discrete and open. This means that such mappings are branched covers in a general sense.