535 research outputs found
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 . 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
Some properties of H\"older surfaces in the Heisenberg group
It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder
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 Hoelder continuous of order strictly greater than 1/2. The Heisenberg group
here is equipped with its Carnot-Caratheodory 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.Comment: 18 pages, 1 figur
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 chain in is dense in its support,
whenever the group of coefficients is so that is
discrete and closed.Comment: 96 page
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 . In this note we show that distortion minimisers exist among convex
embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove
that is a sharp lower bound on the distortion of embedded closed
surfaces of positive genus.Comment: 9 pages, 1 figur
Integration of Hölder forms and currents in snowflake spaces
For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral in case the functions 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 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 and 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 is of bounded fractional variation with respect to some exponent
if there is a finite constant such that the estimate holds for all Lipschitz functions
on . 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
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