Abstract. For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral M f dg 1 ∧ · · · ∧ dgn in case the functions f, g 1 , . . . , gn are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for α ∈ ( n n+1, 1] the n-dimensional locally normal currents in a locally compact metric space (X, d) represent a subspace of the n-dimensional currents in (X, d α ). On the other hand, for n ≥ 1 and α ≤ n n+1 the vector space of n-dimensional currents in (X, d α ) is zero.

Abstract:We give a necessary and su cient condition for a map de ned on a simply-connected quasi-convex metric space to factor through a tree. In case the target is the Euclidean plane and the map is Hölder continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over winding number functions. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carathéodory metric and the Hölder exponent of the map is bigger than 2/3, the map factors through a tree.

It is a folk conjecture that for α > 1/2 there is no α-Hölder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R 2 into the Heisenberg group that is Hölder continuous of order strictly greater than 1/2. The Heisenberg group here is equipped with its Carnot-Carathéodory distance. We show that, in the case that such a surface exists, it cannot be of essential bounded variation and it intersects some vertical line in at least a topological Cantor set.

Extending the notion of bounded variation, a function u ∈ L 1 c (R n ) is of bounded fractional variation with respect to some exponent α if there is a finite constant C ≥ 0 such that the estimate 2010 Mathematics Subject Classification. 49Q15, 28A75, 26A16.

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$, as introduced by Chen. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.