Every metric space which is compact in the associated length metric is the continuous image of [0,1] by a map which is metrically differentiable almost everywhere, and is the continuous image of [0,1]^n for each n>1 by a map which is metrically differentiable almost everywhere and lies in a suitably defined Sobolev space W^{1,n}([0,1]^n,X).

We study global conformal Assouad dimension in the Heisenberg group H^n. For each α ∈ {0} ∪ [1,2n+2], there is a bounded set in H^n with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of H^n. On the other hand, for any set S in H^n with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets F(S), taken over all quasiconformal maps F of H^n, equals zero. We also consider dilatation-dependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in self-similar fractal geometry and tilings in H^n and regularity of the Carnot-Caratheodory distance from smooth hypersurfaces.

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Caratheodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Caratheodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Caratheodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.

We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli-Garofalo-Nhieu or Lu-Manfredi-Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.

A hyperspace of a metric (or topological) space X is a collection of subsets of X, equipped with a natural metric (e.g., the Hausdorff metric) or topology. We study Gromov hyperbolicity, quasisymmetric equivalence and bi-Lipschitz embeddings of hyperspaces of closed subsets of metric spaces.

We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in l^2. If X is a countable compact metric space containing at most n non-isolated points, there is a Lipschitz embedding of K(X) in R^(n+1); in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K(I) of the unit interval I contains a bi-Lipschitz copy of a certain self-similar doubling series-parallel graph studied by Laakso, Lang--Plaut, and Lee--Mendel--Naor, and consequently admits no bi-Lipschitz embedding into any uniformly convex Banach space. Schori and West proved that K(I) is homeomorphic with the Hilbert cube, while Hohti showed that K(I) is not bi-Lipschitz equivalent with a variety of metric Hilbert cubes.

A metric space is called an L^p-metric space, 1 ≤ p ≤ ∞, if it satisfies the L^p-triangle inequality, and is said to be a snowflake space if it is bi-Lipschitz equivalent with an L^p-metric space for some p>1. Suppose that X is compact and doubling. Then X is a snowflake if and only if X admits a bi-Lipschitz embedding in a uniformly convex Banach space and no weak tangent to X contains a rectifiable curve. We give several equivalent conditions for the snowflake property, and examples distinguishing these conditions under weaker hypotheses. As a corollary we prove that the polygasket PG(N) is a snowflake for N=5 or N ≥ 7.

According to a theorem of Rickman, all nonconstant C^(n/(n-2))-smooth quasiregular maps in R^n, n≥2, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in dimension three. We prove that the order of smoothness is sharp in dimension four. For each n≥5 we construct a C^(1+&eps;(n))-smooth quasiregular map in R^n with nonempty branch set.

We survey recent developments in the theory of quasiconformal maps in metric spaces and present open questions and conjectures in the area.

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano.

We consider horizontal iterated function systems in the Heisenberg group H, i.e., collections of Lipschitz contractions of H with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals. We study questions related to connectivity of horizontal fractals, and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV surfaces in H that is in contrast with the nonexistence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim.

The Sierpinski gasket and other self-similar fractal subsets of R^n, n≥2, can be mapped by quasiconformal self-maps of R^n onto sets of Hausdorff dimension arbitrarily close to one. In R^2 we construct explicit mappings. In R^n, n≥3, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we identify geometric conditions whose validity ensures that (i) isomorphic systems have quasisymmetrically equivalent invariant sets, and (ii) one-parameter isotopies of systems have invariant sets which are equivalent under global quasiconformal maps.

Smale's Mean Value Conjecture asserts that for every polynomial P of degree d satisfying P(0)=0, there exists a critical point θ of P for which |P(θ)/θ|≤K|P'(0)|, where K=(d-1)/d. A stronger conjecture due to Tischler asserts that |1/2-P(θ)/(θ⋅P'(0))|≤ K_1 for some critical point θ, where K_1=1/2-1/d. We give examples of polynomials in each degree d≥5 for which the conjecture fails. We also prove estimates for certain weighted L^p-averages of the quantities |1/2-P(θ)/(θ⋅P'(0)|.

Using Cheeger's differentiability theorem for Lipschitz functions on metric measure spaces, we construct a conformal analogue of the Martin boundary for relatively compact domains in locally compact metric measure spaces which are locally Q-regular for some Q>1 and support a (1,p)-Poincaré inequality for some p≠Q. For relatively compact uniform domains which have uniformly Q-fat complement we show that the conformal Martin boundary maps surjectively onto the topological boundary. We also investigate the behavior of the conformal Martin boundary under conformal and quasiconformal maps.

This article is a survey of recent work (partially joint with I. Holopainen and J. J. Manfredi) on nonlinear potential theory in Carnot groups.

We describe a procedure for constructing "polar coordinates" in a certain class of Carnot groups. We show that our construction can be carried out in groups of Heisenberg type and we give explicit formulas for the polar coordinate decomposition in that setting. The construction makes use of nonlinear potential theory, specifically, fundamental solutions for the p-sub-Laplace operators. As applications of this result we obtain exact capacity estimates, representation formulas, and an explicit sharp constant for the Moser-Trudinger inequality. We also obtain topological and measure-theoretic consequences for quasiregular mappings.

We obtain sharp weighted Moser--Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for x-symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form ||K*_W L||_{r,W} ≤ ||K||_{p,W} ||L||_{q,W}, 1+1/r=1/p+1/q, for first-layer radial weights W on a general Carnot group G and functions K,L:G → R with L first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.

For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q-Laplacian whose exponential is a homogeneous norm on G. This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser-Trudinger inequality.

In any Carnot (nilpotent stratified Lie) group G of homogeneous dimension Q, Green's function u for the nonlinear operator A(ξ)=|ξ|^(Q-2)ξ exists and is unique. We prove that there exists a constant γ=γ(G)>0 so that N=e^(-γu) is a homogeneous norm in G. Then the extremal lengths of spherical ring domains (measured with respect to N) can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the Carnot-Caratheodory metric. Applications include regularity properties of quasiconformal mappings and a geometric characterization of bi-Lipschitz mappings.

We prove that quasiconformal maps onto domains which satisfy a quasihyperbolic boundary condition are globally Hölder continuous in the internal metric. The primary improvement here over existing results along these lines is that no assumptions are made on the source domain. We reduce the problem to the verification of a capacity estimate in domains satisfing a quasihyperbolic boundary condition, which we establish using a combination of a chaining argument involving the Poincaré inequality on Whitney cubes together with Frostman's theorem. We also discuss related results where the quasihyperbolic boundary condition is slightly weakened; in this case the Hölder continuity of quasiconformal maps is replaced by uniform continuity with a modulus of continuity which we calculate explicitly.

We prove that a domain in R^n, n≥2, whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient β≤1 is a (q,p)-Poincaré domain for all p and q satisfying p ∈[1,∞)∩(n-nβ,n) and q∈[p,β p^*), where p^*=np/(n-p) denotes the Sobolev conjugate exponent. An elementary example shows that the given ranges for p and q are sharp. The proof makes use of estimates for a variational capacity and Frostman's theorem. When p=2 we give an application to the solvability of the Neumann problem on domains with irregular boundaries. We also discuss the relationship between this growth condition on the quasihyperbolic metric and the s-John condition.

Recent developments in geometry have highlighted the need for abstract formulations of the theory of quasiconformal mappings in Euclidean spaces. We modify Pansu's generalized notion of the modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension Q>1, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three different definitions for quasiconformality: an infinitesimal metric condition, a semi-global metric condition, and a potential-theoretic characterization.

We illustrate our results with several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper Q-regular Loewner space, Q>1, is not quasiconformally equivalent to any subdomain. (In the Euclidean case this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

We answer a question of Jost on the validity of Poincaré inequalities for metric space-valued functions in a Dirichlet domain. We also investigate the relationship between Dirichlet functions and elements of the Sobolev-type spaces of functions introduced by Korevaar and Schoen.

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.

We show that for each α ∈ [1,d) and K<∞ there is a subset X of R^d such that dim(f(X))≥α=dim(X) for every K-quasiconformal map, but such that dim(g(X)) can be made as small as we wish for some quasiconformal g, i.e., the conformal dimension of X is zero. These sets are used to construct new examples of minimal sets for conformal dimension and sets where the conformal dimension is not attained.

We show that the self-similar planar set X known as the ``antenna set'' has the property that inf_f dim(f(X))=1, where the infimum is over all quasiconformal mappings of the plane and dim denotes Hausdorff dimension, but that this infimum is not attained by any quasiconformal map, indeed, is not attained for any quasisymmetric map into any metric space.

We establish basic analytic properties of locally quasisymmetric homeomorphisms f:G→Y, where G is a domain in the Euclidean space R^n, n≥2, and Y is a metric space with locally finite Hausdorff n-measure. We show that such maps are ACL and satisfy a reverse Hadamard inequality. Furthermore, they are ``Sobolev functions'' (in a sense recently introduced by Reshetnyak) and satisfy the following form of Lusin's condition (N): if E⊂G has Lebesgue measure zero, then f(E) has Hausdorff n-measure zero. This extends work of Väisälä, who showed that these facts hold when the image embeds in some (high-dimensional) Euclidean space.

We study the relationship between the Assouad dimension and quasisymmetric mappings, showing that spaces of dimension strictly less than one can be quasisymmetrically deformed onto spaces of arbitrarily small dimension. We conjecture that this fact holds also for the Hausdorff dimension, and our results yield several corollaries which provide partial support for this conjecture. The proofs make use of connections between Assouad dimension, porosity, and ultrametrics.

For any 1≤α≤n, there is a compact set E⊂R^n of (Hausdorff) dimension α whose dimension cannot be lowered by any quasiconformal self-map of R^n. We conjecture that no such set exists when &alpha<1. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.

A homeomorphism f:X→Y between metric spaces is called quasisymmetric if it satisfies the three-point condition of Tukia and Väisälä. It has been known since the 1960's that when X=Y=R^n, n>1, the class of quasisymmetric maps coincides with the class of quasiconformal maps, i.e., the homeomorphisms f:R^n→R^n which quasipreserve the conformal moduli of all families of curves. We prove that quasisymmetry implies quasiconformality when X and Y are locally compact and connected and have Hausdorff dimension Q>1 quantitatively. The main conceptual tool in the proof is a discrete version of the conformal modulus due to Pansu.