Pansu pullback and rigidity of mappings between Carnot groups

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 poi…

Sobolev mappings between nonrigid Carnot groups

We consider mappings between Carnot groups. In this paper, which is a continuation of "Pansu pullback and rigidity of mappings between Carnot groups" (arXiv:2004.09271), we focus on Carnot groups which are nonrigid in the sense of Ottazzi-Warhurst. We show that quasisymmetric homeomorphisms are reducible in the sense that they preserve a special type of coset foliation, unless the group is isomorphic to R^n or a real or complex Heisenberg group (where the assertion fails). We use this to prove the quasisymmetric rigidity conjecture for such groups. The starting point of the proof is the pullba…

Carnot metrics, Dynamics and Local Rigidity

This paper develops new techniques for studying smooth dynamical systems in the presence of a \CC metric. Principally, we employ the theory of Margulis-Mostow, Métivier, Mitchell and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth \CC metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter applicatio…

Rigidity of flag manifolds

Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally Sobolev mappings with nondegenerate Pansu differential, are rigid when $n \geq 4$; this settles the Regularity Conjecture for such groups. This result is deduced from a rigidity theorem for the manifold of complete flags in $R^n$. Similar results also hold in the complex and quaternion cases.

在5维Cartan群上通过div-curl型估计比较三种可能的亚椭圆Laplacian

Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates

本文在具有2个生成元的3步自由Carnot群（5维Cartan群）上提出三种亚椭圆Hodge-Laplacian定义，并比较它们在获得精确div-curl型不等式方面的适用性，以克服Rumin外微分非齐次性带来的困难。