Some Results on Maps That Factor through a Tree

Abstract

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.

Constructing Hölder maps to Carnot groups

“…It follows from Züst's result [12] mentioned above that if α > 2 3 then there is no locally α-Hölder homeomorphism from R 3 to H. On the other hand, a smooth or C 2 homeomorphism from R 3 to H is locally 1 2 -Hölder. The Hölder equivalence problem asks for the maximum α such that there is an α-Hölder homeomorphism from R 3 to H. Such a map must be topologically nontrivial; for instance, it must be proper and degree 1. In this paper, we construct topologically nontrivial maps from subsets of R 2 and R 3 to H with Hölder exponent less than, but arbitrarily close to, 2 3 .…”

“…These conditions arise from the fact that any surface embedded in H has topological dimension 2 but Hausdorff dimension at least 3 by [4, 2.1], so if α > 2 3 , then the image of a surface under an α-Hölder map cannot be a surface. Indeed, Züst [12] showed that if M is a simply-connected Riemannian manifold and f : M → H is α-Hölder with α > 2 3 , then f factors through a metric tree. Moreover, Le Donne and Züst [7] proved that if α > 1 2 then any α-Hölder surface in H (if it exists) must intersect many vertical lines in a topological Cantor set.…”

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In this paper, we construct Hölder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group H. Pansu and Gromov [4] observed that any surface embedded in H has Hausdorff dimension at least 3, so there is no α-Hölder embedding of a surface into H when α > 2 3 . Züst [12] improved this result to show that when α > 2 3 , any α-Hölder map from a simply-connected Riemannian manifold to H factors through a metric tree. In the present paper, we show that Züst's result is sharp by constructing ( 23 −ϵ)-Hölder maps from D 2 and D 3 to H that do not factor through a tree.

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On exterior differential systems involving differentials of Hölder functions

“…It is not clear whether all maps g satisfying (2.10) have necessarily to be in the form g = f • h as in the above lemma. A weaker result of this ilk however holds due to theorems 1.2 and 1.1 from [13]. Namely, if m = k = 2, i.e.…”

“…We notice that in the case α = β, this turns out to be equivalent to the problem of non-trivial horizontal surfaces in the Heisenberg group (using e.g. the results in [13]), which has been solved in [18].…”