Convex hull property and exclosure theorems for H-minimal hypersurfaces in carnot groups

In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition

Geometric inequalities in Carnot groups

Let \GG be a sub-Riemannian $k$-step Carnot group of homogeneous dimension $Q$. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in \GG, endowed with the \HH-perimeter measure.Comment: 26 page

Hypersurfaces and variational formulas in sub-Riemannian Carnot groups

AbstractIn this paper we study smooth immersed non-characteristic submanifolds (with or without boundary) of k-step sub-Riemannian Carnot groups, from a differential-geometric point of view. The methods of exterior differential forms and moving frames are extensively used. Particular emphasis is given to the case of hypersurfaces. We state divergence-type theorems and integration by parts formulas with respect to the intrinsic measure σHn−1 on hypersurfaces. General formulas for the first and the second variation of the measure σHn−1 are proved

Stable H-Minimal Hypersurfaces

We prove some stability results for smooth non-characteristic H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the first and second variation of the H-perimeter measure together with some non-trivial geometric identities

The distributional divergence of horizontal vector fields vanishing at infinity on Carnot groups

We define a BV -type space in the setting of Carnot groups (i.e., simply connected Lie groups with stratified nilpotent Lie algebra) that allows one to characterize all distributions F for which there exists a continuous horizontal vector field {\Phi}, vanishing at infinity, that solves the equation divH{\Phi} = F. This generalize to the setting of Carnot groups some results by De Pauw and Pfeffer, [12], and by De Pauw and Torres, [13], for the Euclidean setting.Comment: 24 page

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

We study geometric properties of the Carnot-Carath\ue9odory signed distance \u3b4s to a smooth hypersurface S in some 2-step Carnot groups. In particular, a sub-Riemannian version of Gauss' Lemma is proved