Analytic properties of quasiconformal mappings on Carnot groups

The research was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00378) and the International Science Foundation (Grant RAT000).     

The class of mappings with bounded specific oscillation,and integrability of mappings with bounded distortion on Carnot groups

This paper is the first of the author’s three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class W _p,loc ¹ , where p → ∞ as the distortion coefficient tends to 1.     

Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings

We study the properties of the mappings on a Carnot group which induce, via the change-of-variables rule, the isomorphisms of Sobolev spaces with the summability exponent different from the Hausdorff dimension of the group.     

On Liouville’s theorem for locally quasiregular mappings inR n

Letf:R ⁿ→Rⁿ be locally quasiregular in the sense that the restriction off to any ball |x|<r has finite inner dilatationK ₁(r). Suppose that the growth condition ∫^∞r^-1K₁(r)^1/(1-n) holds. Then Liouville’s theorem is valid:If f is bounded, f is a constant. An example shows that this growth condition is relatively sharp.     

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.     

Parametrizations of level surfaces of real-valued mappings of Carnot groups

We study the regularity of the parametrizations of level surfaces of continuously horizontally differentiable real-valued mappings of Carnot groups. Equations that describe a regular hypersurface in terms of parametrizations are obtained.     

On Lichtenstein’s theorem about globally conformal mappings

We present a new variational proof of the well-known fact that every Riemannian metric on a two-dimensional, simply connected domain with boundary can be represented by globally conformal parameters. From this the corresponding result for a metric on S² is derived.Received: 1 September 2004, Accepted: 13 September 2004, Published online: 10 December 2004Mathematics Subject Classification: 49Q05, 53A10, 53C42     

Discreteness and openness of mappings with distortion in Orlicz spaces on ℍ-type Carnot groups

We prove discreteness and openness of the mappings with finite distortion defined on ℍ-type Carnot groups whose distortion functions belong to certain Orlicz spaces. Similar results in Euclidean spaces were established bymany authors under different assumptions about the mappings and their distortion functions. The text was submitted by the authors in English.     

Rank-one theorem and subgraphs of BV functions in Carnot groups

We prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups ${\mathbb{H}}^n$ for $n\ge 2$. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.     

Liouville theorem for harmonic maps with potential

Let M, N be complete manifolds, u:M→N be a harmonic map with potential H, namely, a critical point of the functional , where e(u) is the energy density of u. We will give a Liouville theorem for u with a class of potentials H's. Received: Received: 10 July 1997     

Liouville theorem for harmonic maps with potential

LetM, N be complete manifolds,u:M →N be a harmonic map with potentialH, namely, a critical point of the functionalE _H (u)=∫ _M [e(u) − H(u)], wheree(u) is the energy density ofu. We will give a Liouville theorem foru with a class of potentialsH’s. Research supported in part by NNSFC, SFECC and NSFCCNU.     

Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups

In any Carnot (nilpotent stratified Lie) group G of homogeneous dimension Q, Green's function u for the Q-Laplace equation exists and is unique. We prove that there exists a constant so that 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-Carathéodory metric. Applications include regularity properties of quasiconformal mappings and a geometric characterization of bi-Lipschitz mappings. Received: 18 September 2000/ revised version: 19 November 2001 / Published online: 17 June 2002