The splitting theorem in non-smooth context

2013/02/22 by Nicola Gigli, Gigli, Nicola · 11 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1302.5555

Abstract

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of R and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space W^1,2(X,d,m), which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• On the differential structure of metric measure spaces and applications

Cited by

• A study of the metric measure space of probability measures via a purely atomic superposition principle
• Inradius collapsed manifolds with a lower Ricci curvature bound
• Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
• Lipschitz continuity of harmonic maps between \rm RCD(K,N) spaces and \rm CAT(κ) spaces
• Harmonic functions on metric measure spaces
• Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• A sufficient condition to a regular set of positive measure on RCD spaces
• Noncompact RCD(0,N) spaces with linear volume growth
• Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian
• Infinitesimal Hilbertianity of locally CAT(κ)-spaces

Related

• An overview on the proof of the splitting theorem in non-smooth context
• Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian
• Examples of CD(0,N) spaces with non-constant dimension
• The splitting lemmas for nonsmooth functionals on Hilbert spaces
• The Abresch-Gromoll inequality in a non-smooth setting