Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$CD(K,∞)-spaces

By T. Rajala, Karl‐Theodor Sturm. 2012

78 citations recorded by the source corpus.

We prove that in metric measure spaces where the entropy functional is $$K$$K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/24586ac75103cce822bb4bd69e8303be433d67f4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “ would like to exclude is the presence of branching structures. Note that the ﬁrst result proving absence ofbranchinggeodesicsassuming a curvaturecondition, in that casestrongCD(K,∞), is contained in [21]. The strategy we will follow is: prove that d-cyclically monotone sets do not have branching st”
On quotients of spaces with Ricci curvature bounded below (2017, background): “We now recall the following definition given for the first time in [66].”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, background): “It was proved in [69] that the RCD∗(K,N) condition implies the essentially non-branching property, so this is a fairly natural assumption in the framework of m.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “on is an enforcement of the so called reduced curvature dimension condition, denoted by CD∗(K,N), that has been introduced in [7], and it also implies that the space is essentially non-branching, see [48]. The reduced CD∗(K,N) condition ask for the same inequality (2.3) of CD(K,N) but coeﬃcients τ(t”
New formulas for the Laplacian of distance functions and applications (2018, background): “ion CD(K,N) pioneered by Lott-Sturm-Villani [42, 56, 57] and than the (weaker) reduced curvature dimension condition CD∗(K,N) [7]. The essential non-branching condition, introduced by T. Rajala-Sturm [52], roughly amounts to ask that W2-geodesics are concentrated on non-branching geodesics. Remark 1”
The globalization theorem for the Curvature-Dimension condition (2016, background): “ or more generally, to require that the L2-Optimal-Transport on (X,d,m) be concentrated (i.e. up to a null-set) on a non-branching subset of geodesics, an assumption introduced by Rajala and Sturm in [67] under the name essentially non-branching (see Section 6 for precise deﬁnitions). For instance, ”
Optimal maps in essentially non-branching spaces (2016, background): “(Geo(X)) is concentrated on a set of non-branching geodesics if there exists a nonbranching Borel set Γ ⊂ Geo(X), such that ν(Γ) = 1. Then we recall the following deﬁnition given for the ﬁrst time in [20]. Deﬁnition 2.2. A metric measure space (X,d,m) is essentially non-branching if for every µ0,µ1 ”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, methodology): “ group [Riz16], any ideal Carnot group [Rif13], any generalized H-type Carnot group of rank k and dimension n [BR18]. • Strong CD∗(K,N) spaces, and in particular RCD∗(K,N) (thus also RCD(K,N)) spaces [RS14]. A.2. Disintegration in essentially non-branching MCP(K,N) spaces. In the proof of the main r”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, background): “8 states their main properties, following [45].”
On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) Metric Measure Spaces (2016, background): “The definition adopted here has been later on improved in [1] (asking the convexity inequality along some geodesic, and then recovering convexity along all geodesics out of the EVIK theorem [17]), see also [30] for new recent developments.”