We establish a “low rank property” for Sobolev mappings that pointwise solve a first-order nonlinear system of PDEs, whose smooth solutions have the so-called “contact property”. As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group (Formula presented.), and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of (Formula presented.). This provides a complete solution to a question raised in a paper by Balogh et al. (Ergodic Theory Dynam Syst 26(3):621–651, 2006). Our approach differs from the previous ones. Its technical aspect consists in performing an “exterior differentiation by blow-up”, where the standard distributional exterior differentiation is not possible. This method extends to higher-dimensional Sobolev mappings, taking values in higher-dimensional Heisenberg groups.

A Low Rank Property and Nonexistence of Higher-Dimensional Horizontal Sobolev Sets

MONGODI, SAMUELE
2015

Abstract

We establish a “low rank property” for Sobolev mappings that pointwise solve a first-order nonlinear system of PDEs, whose smooth solutions have the so-called “contact property”. As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group (Formula presented.), and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of (Formula presented.). This provides a complete solution to a question raised in a paper by Balogh et al. (Ergodic Theory Dynam Syst 26(3):621–651, 2006). Our approach differs from the previous ones. Its technical aspect consists in performing an “exterior differentiation by blow-up”, where the standard distributional exterior differentiation is not possible. This method extends to higher-dimensional Sobolev mappings, taking values in higher-dimensional Heisenberg groups.
Hausdorff measure; Heisenberg group; Nonisotropic submanifolds; Geometry and Topology
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1010748
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