Transport maps, non-branching sets of geodesics and measure rigidity

Kell, Martin

doi:10.1016/j.aim.2017.09.003

Cited by 20 publications

(21 citation statements)

References 29 publications

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“…Firstly we show the absolute continuity of the reference measure and the regularity of its density. The following theorem improves the results proved by Cavalletti-Mondino [10] and Kell [23] for essentially non-branching MCP(K, N) spaces. To the knowledge of the author, this is the first measure rigidity result without dimension bound.…”

Section: Introductionsupporting

confidence: 83%

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

Han

2019

Preprint

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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 related to optimal transport on Riemannian manifolds, which completely characterize CD(K, ∞) condition and non-collapsed CD(K, N ) condition on Riemannian manifolds with boundary. In particular, we reveal the measure rigidity of Riemannian interpolation inequality proved by Cordero-Erausquin, McCann and Schmuckenschläger. We prove that logconcave measures are the only reference measures so that displacement convexity holds on Riemannian manifolds. This is the first measure rigidity result concerning synthetic dimension-free CD condition, which is new even on R n . Using L 1 -optimal transportation theory, we prove that CD condition yields geodesical convexity, which is also an unsolved problem even on Euclidean plan.

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Section: Introductionsupporting

confidence: 83%

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

Han

2019

Preprint

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“…Since f (t) → 1 as t → 0 we see that the densities of µ t for sufficiently small t can be uniformly bounded. Now the claim follows directly from the Self-Intersection Lemma in [15,Lemma 6.4].…”

Section: It Follows Thatmentioning

confidence: 95%

“…Using the exponential map of L one sees that there exists C > 0 such that c L (x i+1 , y i ) ≤ − C √ n . Then one has for every point [10], see also [14,12,15]. Note, however, there is no unique equivalent to the assumption of achronality resp.…”

Section: 2mentioning

confidence: 99%

On the existence of dual solutions for Lorentzian cost functions

Kell

Suhr

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided. resultsLet M be a smooth manifold. Throughout the article one fixes a complete Riemannian metric h on M , though local changes to the metric will be allowed. Consider a continuous function L :denotes the zero section in T M ) and positive homogenous of degree 2 such that the second fiber derivative is non-degenerate with index dim M − 1. Let C be a causal structure of (M, L), see [21], and define the Lagrangian L on T M by settingThe pair (M, L) is referred to as a Lorentz-Finsler manifold.One calls an absolutely continuous curve γ : I → M (C-)causal ifγ ∈ C for almost all t ∈ I. Note that this condition already implies that the tangent vector is contained in C whenever it exists.Denote with J + (x) the set of points y ∈ M such that there exists a causal curve γ : [a, b] → M with γ(a) = x and γ(b) = y. Two points x and y will be called causally related if y ∈ J + (x). Note that this relation is in general asymmetric. Define the setand J − (y) : {x ∈ M | y ∈ J + (x)}. A Lorentz-Finsler manifold is said to be causal if it does not admit a closed causal curve, i.e. J + ∩ △ = ∅ for △ := {(x, x)| x ∈ M }. Definition 2.1. A causal Lorentz-Finsler manifold (M, L) is globally hyperbolic if the sets J + (x) ∩ J − (y) are compact for all x, y ∈ M .

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“…With the new notion of essential non-branching introduced by Rajala and Sturm, Cavalletti and Mondino further proved the existence of optimal transport maps for essentially non-branching spaces with the measure contraction property [9]. Continuing from the work of Cavalletti and Huesmann [8] and the work of Cavalletti and Mondino [9], Kell proved that in a metric space endowed with qualitatively non-degenerate measure, and therefore especially in spaces satisfying the measure contraction property, the condition of being essentially non-branching is equivalent with having the existence and uniqueness of the optimal transport maps [14].…”

Section: Introductionmentioning

confidence: 90%