Quasiregular Mappings on Sub-Riemannian Manifolds

We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot–Carathéodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives).     

Polar coordinates 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. Received: 26 June 2001; in final form: 14 January 2002/Published online: 5 September 2002     

A sufficient condition for nonrigidity of Carnot groups

In this article we consider contact mappings on Carnot groups. Namely, we are interested in those mappings whose differential preserves the horizontal space, defined by the first stratum of the natural stratification of the Lie algebra of a Carnot group. We give a sufficient condition for a Carnot group G to admit an infinite dimensional space of contact mappings, that is, for G to be nonrigid. A generalization of Kirillov’s Lemma is also given. Moreover, we construct a new example of nonrigid Carnot group. This research was partly supported by the Swiss National Science Foundation. The author would like to thank H. M. Reimann for the helpful advices and the constant support.     

On the structure of finite perimeter sets in step 2 Carnot groups

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure. First author was supported by University of Bologna, Italy, funds for selected research topics; second and third authors were supported by MURST, Italy, and University of Trento, Italy.     

Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type

We prove the hypoellipticity for systems of H?rmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set. Received October 23, 2000 / final version received February 5, 2002?Published online May 15, 2002 The work of the first author was partially supported by NSF Grant No. DMS-9800794 The work of the second author was partially supported by NSF Grants No. DMS-9706892 and DMS-0070492     

Analytic continuation for Beltrami systems, Siegel's Theorem for UQR maps, and the Hilbert-Smith Conjecture

A uniformly quasiregular mapping, is a mapping of the m-sphere with the property that it and all its iterates have uniformly bounded distortion. Such maps are rational with respect to some bounded measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We begin by investigating the analogue of Siegel's theorem on the local conjuga cy of rotational dynamics. We are led to consider the analytic continuation properties of solutions to the highly nonlinear first order Beltrami systems. We reduce these problems to a central and well known conjecture in the theory of transformation groups; namely the Hilbert-Smith conjecture, which roughly asserts that effective transformation groups of manifolds are Lie groups. Our affirmative solution to this problem then implies unique analytic continuation and Siegel's theorem. Received: 14 September 2000 / Revised version: 23 November 2001 / Published online: 5 September 2002 RID="*" ID="*" Research supported in part by grants from the Marsden Fund and Royal Society (NZ).     

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].     

The existence of harmonic maps from an annulus toS 2 with symmetric boundary value

In this paper, we use the deformation method andG-equivariant theory to prove the existence and multiplicity of harmonic maps from an annulus to the unit sphere inℝ ³ with symmetric boundary value. In particular, we can get infinitely many homotopically different harmonic maps if the boundary value isS ¹-equivariant and nonconstant. This research partially supported by the NNSF, P.R. China.     

Bi-Lipschitz maps in <Emphasis Type="Italic">Q</Emphasis>-regular Loewner spaces

By applying the theory of quasiconformal maps in measure metric spaces that was introduced by Heinonen-Koskela, we characterize bi-Lipschitz maps by modulus inequalities of rings and maximal, minimal derivatives in Q-regular Loewner spaces. Meanwhile the sufficient and necessary conditions for quasiconformal maps to become bi-Lipschitz maps are also obtained. These results generalize Rohde’s theorem in ℝ ⁿ and improve Balogh’s corresponding results in Carnot groups. This research is supported by China NSF (Grant No. 10271077)     

Smooth Quasiregular Maps with Branching in R n

According to a theorem of Martio, Rickman and Väisälä, all nonconstant C^n/(n-2)-smooth quasiregular maps in Rⁿ, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R³. We prove that the order of smoothness is sharp in R⁴. For each n≥5 we construct a C^1+ε(n)-smooth quasiregular map in Rⁿ with nonempty branch set.     

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.     

First-order regularity of convex functions on Carnot Groups

We prove that h-convex functions on Carnot groups of step two are locally Lipschitz continuous with respect to any intrinsic metric. We show that an additional measurability condition implies the local Lipschitz continuity of h-convex functions on arbitrary Carnot groups. To the Memory of Q. G.     

Boundary measures of Markov chains

It is known that each Markov chain has associated with it a polytope and a family of Markov measures indexed by the interior points of the polytope. Measure-preserving factor maps between Markov chains must preserve the associated families. In the present paper, we augment this structure by identifying measures corresponding to points on the boundary of the polytope. These measures are also preserved by factor maps. We examine the data they provide and give examples to illustrate the use of this data in ruling out the existence of factor maps between Markov chains. E. Cawley was partially supported by the Modern Analysis joint NSF grant in Berkeley. S. Tuncel was partially supported by NSF Grant DMS-9303240.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

On equilibrium solutions of diffusion equations with nonlinear boundary conditions

The existence or the nonexistence of nonconstant stable equilibrium solutions for a diffusion equation with nonlinear Neumann boundary conditions is studied. We prove the nonexistence of nonconstant stable equilibria when the nonlinearity has a small Lipschitz constant or a second derivative of constant sign or either when the domain is a ball. We construct an example of existence for a connected domain with several disconnected boundary components.This work is partially supported by the project PB91-0497-C02-02 of the DGICYT (Spain).     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

Incidence-polytopes with toroidal cells

A type of partially ordered structures called incidence-polytopes generalizes the notion of polyhedra in a combinatorial sense. The concept includes all regular polytopes as well as many well-known configurations. We use hyperbolic geometry to derive certain types of incidence-polytopes whose cells are isomorphic to maps of type {4, 4}, {6, 3}, or {3, 6} on a torus. For these structures we give a criterion on the finiteness in terms of groups of 2 × 2 matrices, leading among other things to the explicit recognition of the groups in some interesting special cases.Dedicated to H. S. M. Coxeter on the occasion of his 80th birthday.Research supported by NSERC Canada Grant A8857.     

Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x∈G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they show in Math. Ann. 321, 479–531, 2001 and J. Geom. Anal. 13, 421–466, 2003 that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace. The second author was partially supported by NSF grant DMS-0701515.     

Porosity,Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.