Infinitesimal Hilbertianity of weighted Riemannian manifolds

TL;DR

This paper proves that weighted Riemannian and reversible Finsler manifolds have Sobolev spaces that are Hilbert spaces, extending the understanding of their geometric and functional analytic structure.

Contribution

It establishes that weighted Riemannian manifolds are infinitesimally Hilbertian and embeds tangent modules of weighted reversible Finsler manifolds into measurable tangent sections.

Findings

Weighted Riemannian manifolds are infinitesimally Hilbertian.

Tangent modules of weighted reversible Finsler manifolds can be isometrically embedded.

Sobolev spaces on these manifolds are Hilbert spaces.

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 (\`a 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 $\mu$.