Some results on maps that factor through a tree

We give a necessary and sufficient condition for a map defined on a simply-connected quasiconvex metric space to factor through a tree. In case the target is the Euclidean plane and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number function. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.Comment: 23 page

Matchings in metric spaces, the dual problem and calibrations modulo 2

We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for 1-chains with coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric spaces and present a notion of "matching dimension" which arises naturally.Comment: We corrected some typos and clarified some of the notations and formulations. The new version uses the New York Journal of Mathematics templat

Partial regularity of almost minimizing rectifiable G chains in Hilbert space

We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $\ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that $\{\|g\| : g \in G \}$ is discrete and closed.Comment: 96 page

Integration of Hölder forms and currents in snowflake spaces

For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral $${\int_M f \, dg_1 \wedge \cdots \wedge dg_n}$$ in case the functions $${f, g_1, \ldots, g_n}$$ 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 $${\alpha \in (\tfrac{n}{n+1},1]}$$ 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 $${n \geq 1}$$ and $${\alpha \leq \tfrac{n}{n+1}}$$ the vector space of n-dimensional currents in (X, d α) is zer

Functions of bounded fractional variation and fractal currents

Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $\alpha$ if there is a finite constant $C \geq 0$ such that the estimate $$\biggl|\int u(x) \det D(f,g_1,\dots,g_{n-1})_x \, dx\biggr| \leq C\operatorname{Lip}^\alpha(f) \operatorname{Lip}(g_1) \cdots \operatorname{Lip}(g_{n-1})$$ holds for all Lipschitz functions $f,g_1,\dots,g_{n-1}$ on $\mathbb R^n$. Among such functions are characteristic functions of domains with fractal boundaries and H\"older continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney's flat chains and as multilinear functionals in the setting of Ambrosio-Kirchheim currents. Consequently we discuss extensions to H\"older differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to H\"older maps defined on domains with fractal boundaries.Comment: 53 pages, 1 figur

Distortion of spheres and surfaces in space

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $\pi/2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.Comment: 9 pages, 1 figur