Alessandro Ottazzi

Sard property for the endpoint map on some Carnot groups

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sards theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sards property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

RIGIDITY OF CARNOT GROUPS RELATIVE TO MULTICONTACT STRUCTURES

We prove a rigidity type result for stratified nilpotent Lie algebras which gives a positive answer to a special case of a conjecture formulated by M. Cowling and of another conjecture formulated by A. Koranyi.

A Liouville Type Theorem for Carnot Groups: A Case Study

L. Capogna and M. Cowling showed that if

Contact and 1-quasiconformal maps on Carnot groups

\phi

Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds

is 1-quasiconformal on an open subset of a Carnot group G, then composition with

CONFORMAL MAPS OF CARNOT GROUPS

\phi

A sufficient condition for nonrigidity of Carnot groups

preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that

Multicontact vector fields on Hessenberg manifolds

\phi

Algebraic prolongation and rigidity of Carnot groups

is in fact

ULTRARIGID TANGENTS OF SUB-RIEMANNIAN NILPOTENT GROUPS

C^\infty