Two-step Carnot Groups Research Articles

On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets

In the setting of two-step Carnot groups we show a “cone property” forhorizontally convex sets. Namely, we prove that, given a horizontally convex set C,a pair of points P ¬ C and Q ¬ int(C), both belonging to a horizontal line , thenan open truncated subRiemannian cone around and with vertex at P is containedin C.We apply our result to the problem of classification of horizontally monotone setsin Carnot groups. We are able to show that monotone sets in the direct product H×Rof the Heisenberg group with the real line have hyperplanes as boundaries.

Coarea Formula for Functions on 2-Step Carnot Groups with Sub-Lorentzian Structure

We consider C1-functions defined on two-step Carnot groups with a sub-Lorentzian structure defined by one horizontal direction with a negative squared length along it, and prove a nonholonomic coarea formula. A result of interest in itself concerns the correctness of the problem statement, namely, the level sets have to be spacelike.

Sufficient maximality conditions for surfaces on two-step sub-lorentzian structures

We deduce sufficient maximality conditions for graph surfaces on two-step Carnot groups with sub-Lorentzian structure.

On minimal surfaces on two-step Carnot groups

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.

Minimal graph-surfaces on arbitrary two-step Carnot groups

We establish basic properties of minimal graph-surfaces constructed from classes of mappings defined on two-step Carnot groups. The research methods include solving the question, specific for the case under study, on correctness of the problem statement. The main result on necessary minimality conditions is formulated in terms of a sub-Riemannian analog of mean curvature.

On Minimal Surfaces on Two-Step Carnot Groups

On Minimal Surfaces on Two-Step Carnot Groups

Sufficient Maximality Conditions for Surfaces on Two-Step Sub-Lorentzian Structures

Sufficient maximality conditions for graph surfaces on two-step Carnot groups with a sub-Lorentzian structure are deduced.

Level Sets of Classes of Mappings of Two-Step Carnot Groups in a Nonholonomic Interpretation

We obtain the metric properties of level sets of certain sufficiently smooth mappings of two-step Carnot groups which are Holder in the sub-Riemannian sense. In particular, we calculate the Hausdorff dimension of these sets and prove a coarea formula of a new type.

Минимальные поверхности-графики на произвольных двуступенчатых группах Карно

Минимальные поверхности-графики на произвольных двуступенчатых группах Карно

Metric Properties of Level Surfaces of Holder Mappings Defined on Two-Step Carnot Groups

Metric Properties of Level Surfaces of Holder Mappings Defined on Two-Step Carnot Groups

Graphs of Lipschitz Mappings on Two-Step Sub-Lorentzian Structures with Multidimensional Time

A sub-Lorentzian area formula is proved for graph mappings constructed on the basis of intrinsically Lipschitz mappings defined on two-step Carnot groups.

Metric Properties of Level Surfaces of Hölder Mappings Defined on Two-Step Carnot Groups

For a model case of Holder mappings of two-step nilpotent graded groups, the metric properties of level surfaces are established. A version of the coarea formula for noncontact mappings is deduced as an application.

Level sets of classes of mappings of two-step Carnot groups in a nonholonomic interpretation

Level sets of classes of mappings of two-step Carnot groups in a nonholonomic interpretation

Regularity of the distance function to smooth hypersurfaces in some two-step Carnot groups

Regularity of the distance function to smooth hypersurfaces in some two-step Carnot groups

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(∇f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincare inequality and coercive estimates as well as the generalized Poincare inequality on the general Carnot groups.

Geometric analysis on quaternion anisotropic Carnot groups

This paper considers examples of two-step Carnot groups related to the quaternions and studies their geometric properties. We employ the Hamiltonian formalism to obtain parametric equations of geodesics and estimate the cardinality of the set of geodesic curves joining two arbitrary points of the group. We also calculate the length of geodesics and determine the geodesics coinciding with the shortest curves in the Carnot‐ Caratheodory metric. Fundamental solutions of the heat and Laplace equations are also given.