We construct an intrinsic regular surface in the first Heisenberg group ${ℍ}^1\equiv {ℝ}^3$ equipped wiht its Carnot-Carathéodory metric which has euclidean Hausdorff dimension $2.5$. Moreover we prove that each intrinsic regular surface in this setting is a $2$-dimensional topological manifold admitting a $\frac{1}{2}$-Hölder continuous parameterization.

Let N be a simply connected nilpotent Lie group and let $S=N⋊{\left(ℝ⁺\right)}^d$ be a semidirect product, ${\left(ℝ⁺\right)}^d$ acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that $li{m}_{t\to \infty }{t}^{\chi ₀}\nu x:\left|x\right|>t=C>0$. In particular, this applies to classical Poisson kernels on symmetric spaces,...

We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.

If N is a simply connected real nilpotent Lie group with a Γ-rational complex structure, where Γ is a lattice in N, then [...] for each s, t.We study relations between invariant complex structures and Hodge numbers of compact nilmanifolds from a viewpoint of Lie algberas.

The purpose of this paper is to investigate Ricci-flatness of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. We first show that if $〈,〉$ is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group $N$, then the restriction of $〈,〉$ to the center of the Lie algebra of $N$ is degenerate. We then characterize the 2-step nilpotent Lie groups which can be endowed with a Ricci-flat left-invariant Lorentzian metric, and we deduce from this that a Heisenberg Lie group $H_{2n+1}$ can be...