Carnot Groups Research Articles

Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups

Abstract 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.

Yamabe-Type Equations on Carnot Groups

This article is concerned with a class of elliptic equations on Carnot groups depending of one real positive parameter and involving a critical nonlinearity. As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic critical equation defined on a smooth and bounded domain $D$ of the {Heisenberg group} $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. Our approach is based on pure variational methods and locally sequentially weakly lower semicontinuous arguments.

Nonlinear elliptic equations on Carnot groups

This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repov\v{s}, Yamabe-type equations on Carnot groups, Potential Anal. 46:2 (2017), 369-383; arXiv:1705.10100 [math.AP]). As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic equation defined on a smooth and bounded domain $D$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. The main approach is based on variational methods.

The parabolic infinite-Laplace equation in Carnot groups

By employing a Carnot parabolic maximum principle, we show existence-uniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.

Modules of systems of measures on polarizable Carnot groups

The paper presents a study of Fuglede’s \(p\)-module of systems of measures in condensers in polarizable Carnot groups. In particular, we calculate the \(p\)-module of measures in spherical ring domains, find the extremal measures, and finally, extend a theorem by Rodin to these groups.

On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker “horizontal” sense.

Rigidity of quasiconformal maps on Carnot groups

AbstractWe show that quasiconformal maps on many Carnot groups must be biLipschitz. In particular, this is the case for 2-step Carnot groups with reducible first layer. These results have implications for the rigidity of quasiisometries between negatively curved solvable Lie groups.

Measure contraction properties of Carnot groups

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number $k+3$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least $N$ such that the $\mathrm{MCP}(0,N)$ is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.

Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if \({f\colon X\to Y}\) is a quasisymmetric mapping between Ahlfors regular spaces, then \({dim_H f(E)\leq dim_H E}\) for “almost every” bounded Ahlfors regular set \({E\subseteq X}\). If additionally, \({X}\) and \({Y}\) are Loewner spaces then \({dim_H f(E)=dim_H E}\) for “almost every" Ahlfors regular set \({E\subset X}\). The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that if \({f}\) is a quasiconformal map of \({\mathbb{R}^N}\), \({N\geq 2}\), then for Lebesgue a.e. \({y\in\mathbb{R}^N}\) we have \({dim_H f(y+E) = dim_H E}\). A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if \({E \subset {\mathbb{R}}}\) is Ahlfors \({d}\)-regular, \({d < 1}\), then some component of \({f(E \times {\mathbb{R}})}\) has dimension at most \({2/(d+1)}\), and we construct examples to show this bound is sharp. In addition, we show there is a \({1}\)-dimensional set \({S\subseteq \mathbb R}\) and planar quasiconformal map \({f}\) such that \({f({\mathbb{R}} \times S)}\) contains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).

Three-spheres theorems for subelliptic quasilinear equations in Carnot groups of Heisenberg-type

We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p p -harmonic type in Carnot groups of Heisenberg type for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty . In the presentation we exhibit the special cases of sub-Laplace equations ( p = 2 p=2 ) and the case p p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.

The weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces in Carnot groups

This paper mainly studies hypersurfaces immersed in a sub-Riemannian step k Carnot group G with a density ef, where f is a smooth function on G. We give the first variation formula about the weighted horizontal area of hypersurfaces. Furthermore on the base of some work we derive the weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces.

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and Lt are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.

Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of the $\mathcal{A}$ -harmonic approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Holder continuous coefficients we obtain directly the optimal Holder exponent for the horizontal gradient of weak solutions on its regular set.

Weighted Poincaré inequalities on half spaces in Carnot groups

Weighted Poincaré inequalities on half spaces in Carnot groups

Homeomorphisms of the Heisenberg group preserving BMO

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.

Reverse Poincaré inequalities, isoperimetry, and Riesz transforms in Carnot groups

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

Subelliptic and parametric equations on Carnot groups

This article concerns a class of elliptic equations on Carnot groups depending on one real parameter. Our approach is based on variational methods. More precisely, we establish the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated Pucci-Serrin theorem and of a local minimum result for differentiable functionals due to Ricceri.

Composition operators in weighted Sobolev spaces on the Carnot group

We study the properties of the mappings inducing bounded change-of-variable operators in weighted Sobolev spaces on the Carnot group. We obtain an analytical description of these mappings in terms of integrability of the weighted distortion function. In some cases we prove that the mapping inducing a bounded operator is piecewise absolutely continuous on almost all horizontal lines.

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175–1198, 2013) and Manfredini (Anal Geom Metric Spaces 1:255–275, 2013) concerning stability of doubling properties, Poincare’ inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hormander vector fields.

Global contact and quasiconformal mappings of Carnot groups

We show that globally defined quasiconformal mappings of rigid Carnot groups are affine, but that globally defined contact mappings of rigid Carnot groups need not be quasiconformal, and a fortiori not affine.