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


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2015-07

AUTHORS

Valentino Magnani, Jan Malý, Samuele Mongodi

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 H1, and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of H1. 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. More... »

PAGES

1444-1458

References to SciGraph publications

  • 2008-01. Non-horizontal submanifolds and coarea formula in JOURNAL D'ANALYSE MATHÉMATIQUE
  • 2006-03. Blow-up of regular submanifolds in Heisenberg groups and applications in CENTRAL EUROPEAN JOURNAL OF MATHEMATICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s12220-014-9478-1

    DOI

    http://dx.doi.org/10.1007/s12220-014-9478-1

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    https://app.dimensions.ai/details/publication/pub.1012951198


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