Abstract. Suppose G is a Gromov hyperbolic group, and ∂∞G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H 3. 1.

Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.

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. 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-Lipschitz if | f (a) − f (b) | ≤ L|a − b|

Abstract. Given a Carnot-Carathéodory metric space (Rn, dcc) generated by vector fields {Xi} m i=1 satisfying Hörmander’s condition, we prove in theorem A that any absolute minimizer u ∈ W 1,∞ cc (Ω) to F(v, Ω) = supx∈Ω f(x, Xv(x)) is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation (1.7). If the Carnot-Carathédory space is a Carnot group G and f is independent of x-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13) under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic ∞-Laplacian equation is established in G. Variational problems in L ∞ are very important because of both its analytic difficulties and their frequent appearance in applications, see the survey article [B] by Barron. The study began with Aronsson’s papers [A1, 2]. The simplest model is to consider minimal Lipschitz extensions (or MLE): for a bounded, Lipschitz domain Ω ⊂ R n and g ∈ Lip(Ω),

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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. 1.

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We study a new bi-Lipschitz invariant λ(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor controlled by λ(M). We prove that λ(M) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, some groups of exponential growth and certain classes of Riemannian manifolds of bounded geometry. On the other hand we construct an example of a Riemann surface M of bounded geometry for which λ(M) = ∞.

Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The non-smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 p-parabolicity and p-hyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and p-parabolicity . . . . . . . . . . . . . . . . .

Abstract. We establish pointwise characterizations of functions in the Hardy-Sobolev spaces H 1,p within the range p ∈ (n/(n + 1), 1]. In particular, a locally integrable function u belongs to H 1,p (R n) if and only if u ∈ L p (R n) and it satisfies the Hajlasz type condition |u(x) − u(y) | ≤ |x − y|(h(x) + h(y)), x, y ∈ R n \ E, where E is a set of measure zero and h ∈ L p (R n). We also investigate Hardy-Sobolev spaces on subdomains and extend Hardy inequalities to the case p ≤ 1. 1.