Davide Vittone
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Submanifolds with boundary in sub-Riemannian Heisenberg Groups Open
We discuss the notion of submanifolds with boundary with intrinsic $C^1$ regularity in sub-Riemannian Heisenberg groups and we provide some examples. Eventually, we present a Stokes' Theorem for such submanifolds involving the integration …
A note on the diameter of small sub-Riemannian balls Open
We observe that the diameter of small (in a locally uniform sense) balls in $C^{1,1}$ sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to $C^0$, the diameter is a…
Submanifolds with boundary and Stokes’ Theorem in Heisenberg groups Open
We introduce and study the notion of -regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes’ Theorem for -regular submanifolds with boundary that takes into account Rumin’s c…
Stepanov Differentiability Theorem for intrinsic graphs in Heisenberg groups Open
We prove a Stepanov differentiability type theorem for intrinsic graphs in sub-Riemannian Heisenberg groups.
Besicovitch's 1/2 problem and linear programming Open
We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywh…
Submanifolds with boundary and Stokes' Theorem in Heisenberg groups Open
We introduce and study the notion of $C^1_\mathbb{H}$-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for $C^1_\mathbb{H}$-regular submanifolds with boundary t…
The Sard problem in step 2 and in filiform Carnot groups Open
We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimensio…
Lipschitz Functions on Submanifolds of Heisenberg Groups Open
We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation …
The Sard problem in step 2 and in filiform Carnot groups, Open
We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimensio…
Lipschitz graphs and currents in Heisenberg groups Open
The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For the purpose of proving such a result, we settle several…
A rectifiability result for finite-perimeter sets in Carnot groups Open
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franc…
Nowhere differentiable intrinsic Lipschitz graphs Open
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher t…
Area of intrinsic graphs and coarea formula in Carnot Groups Open
We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $C^1$ regularity ($C^1_H$). Our first main result is an area formula for $C^1_H$ intrinsic graphs; as an application, we deduce density properties for Hausdorff measur…
A dynamical approach to the Sard problem in Carnot groups Open
We introduce a dynamical-systems approach for the study of the Sard problem in sub-Riemannian Carnot groups. We show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems. As an application, we po…
A rectifiability result for finite-perimeter sets in Carnot groups Open
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franc…
An elementary proof of the rank-one theorem for BV functions Open
We provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation.
A compactness result for BV functions in metric spaces Open
We prove a compactness result for bounded sequences (u_j) of functions with bounded variation in metric spaces (X,d_j) where the space X is fixed but the metric may vary with j. We also provide an application to Carnot–Carathéodory spaces.
Fine properties of functions with bounded variation in Carnot-Carathéodory spaces Open
We study properties of functions with bounded variation in Carnot-Ca\-ra\-théo\-do\-ry spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their…
A compactness result for BV functions in metric spaces Open
We prove a compactness result for bounded sequences $(u_j)_j$ of functions with bounded variation in metric spaces $(X,d_j)$ where the space $X$ is fixed but the metric may vary with $j$. We also provide an application to Carnot-Carathéodo…
On the codimension of the abnormal set in step two Carnot groups Open
In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisa…
Extremal polynomials in stratified groups Open
We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpo…
On Tangent Cones to Length Minimizers in Carnot--Carathéodory Spaces Open
We give a detailed proof of some facts about the blow-up of horizontal curves in Carnot-Carathéodory spaces.
On tangent cones to length minimizers in Carnot-Caratheodory spaces Open
We give a detailed proof of some facts about the blow-up of horizontal curves in Carnot--Caratheodory spaces.
Rank-one theorem and subgraphs of BV functions in Carnot groups Open
We prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups $\mathbb H^n$ for $n\geq 2$. The main tools are properties relating th…
Variation formulas for H-rectifiable sets Open
We compute a first- and second-variation formula for the area of H-rectifiable sets in the Heisenberg group along a contact flow. In particular, the formula holds for sets with locally finite H-perimeter, with no further regularity.