Towards Differential Calculus in stratified groups

Abstract

In this paper, we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We can ideally divide our work into five main parts. The first one develops the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the major tools in several proofs. The second part is the characterization of graded group-valued continuously Pansu differentiable mappings through a system of nonlinear first order PDEs, namely, the "contact equations". This first result requires a preliminary characterization of absolutely continuous curves in graded groups where the contact equations play a central role. Through this approach we also establish a quantitative estimate for the Pansu difference quotient, that is crucial to obtain the mean value inequality. It is important to emphasize the potential of contact equations to tackle Lipschitz extension problems and embedding problems of surfaces modeled on a fixed group. The third part corresponds to the mean value inequality and its consequences. The first of these ones is the inverse mapping theorem, that is used in turn to achieve the rank theorem for graded group-valued continuously Pansu differentiable mappings. Another important application of the mean value inequality is in the proof of the implicit function theorem for graded group-valued continuously Pansu differentiable mappings. This is the central part of this work, since it represents our initial motivation. On the other hand, both the implicit function theorem and the rank theorem also require some algebraic conditions on the Pansu differential. These ones lead us to the fourth part of the paper, where we study factorizations of stratified groups into inner semidirect products. In fact, not all homogeneous group homomorphisms induce a natural splitting of the group. On the other hand, this splitting is a necessary condition to represent sets defined by regular mappings as intrinsic graphs. This is connected to the fifth part of the paper, where we introduce intrinsically regular sets modeled on groups, as a natural consequence of both the rank theorem and the implicit function theorem. Notice that in the case of commutative stratified groups, we obtain the classical notion of manifold. We show that these sets admit an intrinsic tangent cone at every point. On the other hand, the metric structure of regular images may differ from the one of regular level sets, since factorization induced by the intrinsic tangent cone is not "commutative". Finally, we provide some examples, where our tools can be used. We show that all Legendrian submanifolds of graded groups can be characterized as regular images modeled on Euclidean spaces. An example of new geometry where our results can be applied is the real 6-dimensional complexified Heisenberg group. Here we classify all possible intrinsically regular sets, seen either as images or as level sets