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

Infinitesimal Hilbertianity of weighted Riemannian manifolds

created by pasqualetto on 16 Sep 2018
modified on 29 Sep 2018

[BibTeX]

Preprint

Inserted: 16 sep 2018
Last Updated: 29 sep 2018

Year: 2018

Abstract:

The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,\mu)$ - i.e. endowed with a generic non-negative Radon measure $\mu$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,\mu)$ is a Hilbert space.

We actually prove a stronger result: the abstract tangent module (à la Gigli) associated to any weighted reversible Finsler manifold $(M,F,\mu)$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\mu$.

Keywords: Sobolev space, Infinitesimal Hilbertianity, Finsler manifold, smooth approximation of Lipschitz functions


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