Algebraic prolongation and rigidity of Carnot groups

We discuss the known results on rigidity of Carnot groups using Tanaka’s prolongation theory. We also apply Tanaka’s theory to study rigidity of an extended class of H-type groups which we call J-type groups. In particular we obtain a rigidity criterion giving rise to a rigid class of J-type groups which includes the H-type groups, and thus extends the results of H.M. Reimann. We also construct a noncomplex J-type group which is nonrigid and does not satisfy the rank 1 condition over the reals.     

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     

Subharmonic functions on Carnot groups

 Many results of classical Potential Theory are extended to sub-Laplacians ▵_𝔾 on Carnot groups 𝔾. Some characterizations of ▵_𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵_𝔾-subharmonic functions in ℝ ^N . Received: 21 June 2000 / Revised version: 12 March 2002 / Published online: 2 December 2002 RID="★" ID="★" Investigation supported by University of Bologna. Funds for selected research topics. Mathematics Subject Classification (2000): 31B05, 35J70, 35H20     

Algebraic groups

This paper is a continuation of V. P. Platonov's survey in Vol. 11 ofAlgebra, Topology, Geometry. It consists of two parts, the first of which deals with structural problems and rationality questions for algebraic groups, and the second with the arithmetic theory of algebraic groups. Particular attention is paid to properties of semisimple groups and their groups of rational points.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 21, pp. 80–134, 1983.     

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. Dedicated to Seppo Rickman on his sixtieth birthday J.H. was partially supported by NSF, the Academy of Finland, and the A. P. Sloan Foundation. I.H. was partially supported by the EU HCM contract no. CHRX-CT92-0071.     

Viscosity convex functions on Carnot groups

We prove that any upper semicontinuous v-convex function in any Carnot group is h-convex.

     

Polynomial sub-Riemannian differentiability on Carnot groups

The polynomial sub-Riemannian differentiability of classes of mappings of Carnot groups and graphs is proved. Examples of polynomial sub-Riemannian differentials preserving Hausdorff dimension are given.     

Subelliptic harmonic maps from Carnot groups

For subelliptic harmonic maps from a Carnot group into a Riemannian manifold without boundary, we prove that they are smooth near any -regular point (see Definition 1.3) for sufficiently small . As a consequence, any stationary subelliptic harmonic map is smooth away from a closed set with zero H^Q-2 measure. This extends the regularity theory for harmonic maps (cf. [SU], [Hf], [El], [Bf]) to this subelliptic setting.Received: 24 April 2002, Accepted: 30 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000): 35B65, 58J42     

Extremals in some classes of Carnot groups

Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.     

Coarse differentiation and quantitative nonembeddability for Carnot groups

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot–Carathéodory metric (G,_dcc)$(G,d_{cc})$ to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of G   into a metric space (X,_dX)$(X,d_X)$ if X is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of G  . For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding f:B(n)→X$f:B(n)\to X$, where B(n)$B(n)$ is the ball of radius n of a finitely generated nonabelian torsion-free nilpotent group G. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.     

Cohomology and rigidity of fuchsian groups

We prove the localC ³-rigidity of the standard actions of cocompact lattices in PSL(2,ℝ) on a circle, using the Schwarzian and the duality technique for twisted cocycles. Partially supported by NSF Grant #DMS 9403870.     

Sub-Riemannian calculus on hypersurfaces in Carnot groups

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.     

Weighted Hardy and Rellich inequality on Carnot groups

We prove some weighted Hardy and Rellich inequalities on general Carnot groups with weights associated to the norm constructed by Folland’s fundamental solution of the Kohn sub-Laplacian.