Related papers: Area-minimizing horizontal graphs with low-regular…

We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a…

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces…

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated…

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab $\{-1<x<1\}$. We show that our condition is necessary by introducing a family of…

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique…

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and…

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different…

We present a characterization of minimal cones of class $C^2$ and $C^1$ in the first Heisenberg group $\mathbb{H}^1$, with an additional set of examples of minimal cones that are not of class $C^1$.

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the…

We study the stability of minimizers of weighted $p$-area functionals associated with prescribed $p$-mean curvature surfaces in the Heisenberg group. While existence and uniqueness results are well established, quantitative stability with…

In this paper we study some geometric properties of surfaces in the Heisenberg group, $\mathcal{H}_{3}.$ We obtain, using the Gauss map for Lie groups, a partial classification of minimal graphs in $\mathcal{H}_{3}.$ We also proof the non…

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some…

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups.…

Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the…

We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported…

We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a…

We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a…

Using a geometric construction, we solve Plateau's Problem in the Heisenberg group $\mathbb{H}^{1}$ for intrinsic graphs defined on a convex domain $D$, under a smallness condition either on the boundary $\partial D$ or on the Lipschitz…

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body $K_0 \subseteq \mathbb{R}^{2n}$, for $t$-graphs on a bounded domain $\Omega$ in the Heisenberg group $\mathbb{H}^n$. When the prescribed datum…

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz…