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.     

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     

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.     

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.     

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.     

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.     

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H ₂-convex function in vanishing on ∂Ω one has

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.     

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.     

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.     

Compensated compactness for differential forms in Carnot groups and applications

In this paper we prove a compensated compactness theorem for differential forms of the intrinsic complex of a Carnot group. The proof relies on an Ls-Hodge decomposition for these forms. Because of the lack of homogeneity of the intrinsic exterior differential, Hodge decomposition is proved using the parametrix of a suitable 0-order Laplacian on forms.     

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).     

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.