Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

2018/09/29 by Danka Lučić, Enrico Pasqualetto · 13 citations
Mathematics · #Geometric Analysis and Curvature Flows #Hilbert space #Infinitesimal #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Riemannian geometry #Sobolev space #Tangent #Tangent bundle #Tangent space #Unit tangent bundle #math.DG #msc:46E35 #msc:53C23 #msc:58B20

paper · pdf · doi:10.4153/s0008439519000328

published in Canadian Mathematical Bulletin 63(1), 118-140 (Cambridge University Press)

openalex created_date 2018/09/27 · arxiv created 2018/09/29 · openalex publication_date 2019/06/24 · arxiv updated 2020/02/19 · openalex updated_date 2026/08/08

Abstract

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.

Citations

• Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Lecture Notes On Differential Calculus on \sf RCD Spaces