Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below

By Nicola Gigli, T. Rajala, Karl‐Theodor Sturm. 2013

86 citations recorded by the source corpus.

We prove existence and uniqueness of optimal maps on $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) bounds.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/350762912fe10f060182545baafccad35dad44d1

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “Step 2: proof of (3.34) We assume m (Ann R (¯ x )) > 0 or there is nothing to prove, then we put , µ 1 := δ ¯ x and let π ∈ P ( C ([0 , 1] , X)) be the only optimal geodesic plan from µ 0 to µ 1 (see [38]).”
On quotients of spaces with Ricci curvature bounded below (2017, background): “Since RCD ∗(K,N) is fulfilled on the whole (O, d,m), by [36], we know there is a unique dynamical 2-optimal plan ν ∈ GeoOpt(μ0, μ1) and it is induced by a map F : |O| → Geo(|O|) which is μ0-a.”
On a mixture of Brenier and Strassen Theorems (2018, background): “This result has then been extended to various cost functions [24] and state spaces [53, 26, 3, 22, 21, 19, 7, 27] and has had numerous applications in PDE, geometric analysis or probability theory, see [2, 4, 69, 70] and the references therein.”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “Since RCD(K, N) implies MCP(K, N), see [18] and [11], and the L(2) optimal plans are unique, the estimate (6.”
Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, unknown): “Picking μ0 ≤ Cm for some C > 0 and with bounded support, by the results in [13] we know that the only geodesic (μt) from μ0 to μ1 is such that μt ≤ C ′m for any t ∈ [0, 12 ].”
The globalization theorem for the Curvature-Dimension condition (2016, background): “The following pointwise density inequality is a known equivalent definition of CD(K,N) on essentially non-branching spaces (the equivalence follows by combining the results of [29] and [41], see the proof of Proposition 9.”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, methodology): “3 in [32] grants that φ is locally Lipschitz on Ω and that ∂cφ(x) 6= ∅ for every x ∈ Ω.”
From volume cone to metric cone in the nonsmooth setting (2015, background): “But being it not induced by a map, we found a contradiction with the fact that optimal plans on RCD∗(0, N) spaces must be induced by maps (see [19]).”