Open AccessPosted Content
Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
TLDR
In this article, the authors investigated Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary and proved several measure rigidity results for some important functional and geometric inequalities.Abstract:
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In particular, using $L^1$-optimal transportation theory, we prove that ${\rm CD}(K, \infty)$ condition implies geodesical convexity.read more
Citations
More filters
Posted Content
Boundary regularity and stability for spaces with Ricci bounded below
TL;DR: In this paper, the authors studied the structure and stability of boundaries in noncollapsed RCD spaces, that is, metric-measure spaces with lower Ricci curvature bounded below.
Journal ArticleDOI
On the topology and the boundary of N–dimensional RCD(K,N) spaces
Vitali Kapovitch,Andrea Mondino +1 more
TL;DR: In this paper, the authors established topological regularity and stability of N-dimensional RCD(K,N) spaces up to a small singular set and introduced the notion of a boundary of such spaces and studied its properties, including its behavior under Gromov-Hausdorff convergence.
Journal ArticleDOI
New differential operator and non-collapsed $RCD$ spaces
TL;DR: In this article, the authors give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in the heat kernel of RCD spaces.
Posted Content
On the structure of RCD spaces with upper curvature bounds
TL;DR: In this article, a structure theory for RCD spaces with curvature bounded above in Alexandrov sense is developed, and it is shown that any such space is a topological manifold with boundary whose interior is equal to the set of regular points.
Journal ArticleDOI
Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
TL;DR: In this paper, the authors investigated Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary and proved measure rigidity results related to optimal transport.
References
More filters
Journal ArticleDOI
On the geometry of metric measure spaces. II
TL;DR: In this article, a curvature-dimension condition CD(K, N) for metric measure spaces is introduced, which is more restrictive than the curvature bound for Riemannian manifolds.
Journal ArticleDOI
Ricci curvature for metric-measure spaces via optimal transport
John Lott,Cédric Villani +1 more
TL;DR: In this paper, a notion of a length space X having nonnegative N-Ricci curvature, for N 2 [1;1], or having 1-RICci curvatures bounded below by K, for K2 R, was given.
Journal ArticleDOI
Differentiability of Lipschitz Functions on Metric Measure Spaces
TL;DR: In this paper, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, without Abstracts. (Without Abstract) (without Abstract)
Book ChapterDOI
(σ, ρ)-calculus
TL;DR: In this article, a discrete-time system with time indexed by t = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28
Book ChapterDOI
Semi-Riemannian Geometry
TL;DR: In this paper, the basics of differentiable manifolds and semi-Riemannian geometry for the applications in general relativity are developed. But the applicability of these manifolds to general relativity is not discussed.