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Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

Part of: Global differential geometry Linear function spaces and their duals Infinite-dimensional manifolds

Published online by Cambridge University Press:  27 September 2019

Danka Lučić  and
Enrico Pasqualetto
Danka Lučić
Affiliation:
SISSA, via Bonomea 265, 34136 Trieste, Italy Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyvaskyla, Finland Email: [email protected]@jyu.fi
Enrico Pasqualetto
Affiliation:
SISSA, via Bonomea 265, 34136 Trieste, Italy Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyvaskyla, Finland Email: [email protected]@jyu.fi
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Abstract

The main result of this paper is the following: any weighted Riemannian manifold $(M,g,\unicode[STIX]{x1D707})$, i.e., a Riemannian manifold $(M,g)$ endowed with a generic non-negative Radon measure $\unicode[STIX]{x1D707}$, is infinitesimally Hilbertian, which means that its associated Sobolev space $W^{1,2}(M,g,\unicode[STIX]{x1D707})$ is a Hilbert space.

We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold $(M,F,\unicode[STIX]{x1D707})$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\unicode[STIX]{x1D707}$.

By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.

Keywords

infinitesimal HilbertianitySobolev spaceFinsler manifoldsmooth approximation of Lipschitz functions

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 58B20: Riemannian, Finsler and other geometric structures
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 118 - 140
Copyright
© Canadian Mathematical Society 2019

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