Calculus of Variations and Geometric Measure Theory
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V. Magnani - J. Tyson - D. Vittone

On transversal submanifolds and their measure

created by vittone on 28 Nov 2012
modified by magnani on 24 Jul 2015

[BibTeX]

Published Paper

Inserted: 28 nov 2012
Last Updated: 24 jul 2015

Journal: J. Anal. Math.
Volume: 125
Pages: 319–351
Year: 2015

Abstract:

We study the class of transversal submanifolds in Carnot groups. We characterize their blow-ups at transversal points and prove a negligibility theorem for their ``generalized characteristic set'', with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that $C^1$ submanifolds in Carnot groups are generically transversal, the previous results prove that the ``intrinsic measure'' of $C^1$ submanifolds is generically equivalent to their Carnot-Carathéodory Hausdorff measure. As a result, the restriction of this Hausdorff measure to the submanifold can be replaced by a more manageable integral formula, that should be seen as a ``sub-Riemannian mass''. Another consequence of these results is an explicit formula, only depending on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of $C^1$ transversal submanifolds.

Tags: GeMeThNES
Keywords: stratified groups, Submanifolds, Hausdorff measure


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