Calculus of Variations and Geometric Measure Theory
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E. Le Donne - D. Lučić - E. Pasqualetto

Universal infinitesimal Hilbertianity of sub-Riemannian manifolds

created by pasqualetto on 15 Oct 2019
modified by ledonne on 01 Dec 2019

[BibTeX]

preprint

Inserted: 15 oct 2019
Last Updated: 1 dec 2019

Year: 2019

ArXiv: 1910.05962 PDF

Abstract:

We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.

Tags: GeoMeG

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