Juha Heinonen

Quasiconformal maps in metric spaces with controlled geometry

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both relevant and viable. The first main issue is the problem of definition, which we next describe. Quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount. This requirement makes sense in every metric space. Given a homeomorphism f from a metric space X to a metric space Y , then for x∈X and r>0 set

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality.

Sobolev mappings with integrable dilatations

We show that each quasi-light mapping f in the Sobolev space W1n(Ω, Rn) satisfying ¦Df(x)¦n ≦K(x, f)J(x, f) for almost every x and for some KεLr(Ω), r>n-1, is open and discrete. The assumption that f be quasilight can be dropped if, in addition, it is required that fε W1p(ω, Rn) for some p > = n + 1/ (n-2). More generally, we consider mappings in the John Ball classes Axxxp,q(Ω), and give conditions that guarantee their discreteness and openness.

Definitions of quasiconformality

SummaryWe establish that the infinitesimal “H-definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even inRn where we obtain that the “limsup” condition in theH-definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.

Quasiregular maps on Carnot groups

In this paper we initiate the study of quasiregular maps in a sub-Riemannian geometry of general Carnot groups. We suggest an analytic definition for quasiregularity and then show that nonconstant quasiregular maps are open and discrete maps on Carnot groups which are two-step nilpotent and of Heisenberg type; we further establish, under the same assumption, that the branch set of a nonconstant quasiregular map has Haar measure zero and, consequently, that quasiregular maps are almost everywhere differentiable in the sense of Pansu. Our method is that of nonlinear potential theory. We have aimed at an exposition accessible to readers of varied background.

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.

Thirty-three yes or no questions about mappings, measures, and metrics

Most problems in the ensuing list are of fairly recent origin. None of them seem easy and some are likely to be very difficult. The formulation of each problem is such that it can be answered by one word only: either yes or no. (Strictly speaking, it is conceivable that within the same question, the answer sometimes depends on the dimension.) We offer no conjectures or guesses. In many cases, the particular question is just a chosen concise representative from a whole group of related open problems. Whenever known, we shall point out the original source of a question. Otherwise, the question is either a folk question, a modification of a folk question, or suggested by one or both of the authors. We apologize in advance for all omissions and misquotes. Practically all the problems require some background definitions; many concepts that are being used have only recently been introduced, and are perhaps not so widely known. Typically, the question is stated first, and the relevant definitions and references are given right afterwords. To keep this essay brief, we give little or no motivation here. For this purpose, we kindly invite the reader to consult the literature as referred to in the text. This list of questions was born at the Institut des Hautes Études Scientifiques in August 1996. The choices we made were necessarily partial but still somewhat arbitrary. There certainly are many more problems around these topics that we deem equally worthy.

On the locally branched Euclidean metric gauge

A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to produce the branched coordinates for the gauge.

The Gehring–Hayman inequality for quasihyperbolic geodesics

The quasihyperbolic metric in a proper subdomain D of R n is defined by where the infimum is taken over all rectifiable arcs γ in D joining x and y . There always exists an arc, called a quasihyperbolic geodesic in D , for which the infimum above is attained. We refer to [ 3 ], [ 4 ], [ 16 ], and [ 17 ] for the motivation and basic properties of the quasihyperbolic metric.

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.