Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces

Abstract

In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.

arXiv commentDedicated to Misha Gromov's 75th birthday

References in corpus (18)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• The splitting theorem in non-smooth context
• Alexandrov meets Lott--Villani--Sturm
• Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Independence on $p$ of weak upper gradients on RCD spaces
• New differential operator and non-collapsed $RCD$ spaces
• Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below
• A weakly second order differential structure on rectifiable metric measure spaces
• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces
• A sufficient condition to a regular set of positive measure on RCD spaces
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds
• Regularity of Lagrangian flows over $RC{D}^{\ast }(K,N)$ spaces
• Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and $RCD(K,\infty )$ spaces

Cited by in corpus (2)

• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Ricci flow smoothing for locally collapsing manifolds