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

Non-branching Geodesics and Optimal Maps in Strong Cd(k,∞)-spaces

We prove that in metric measure spaces where the entropy functional is Kconvex 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 ...

Geodesics in the space of measure - preserving maps and plans ∗

We study Brenier’s variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality condi...

Scattering Rigidity with Trapped Geodesics

We prove that the flat product metric on D × S is scattering rigid where D is the unit ball in R and n ≥ 2. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map S : U∂M → U−∂M from unit vectors V at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes V to γ′ V (T0) where γV is the unit speed geodesic d...

Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L metric, which as observed by Arnold [Arn66] solve the Euler equations of inviscid incompressible uids. The method relies on the generalized polar decomposition of Brenier [Bre91], numerically implemented through semi-discrete optimal transport. It is robust enough to e...

Semi-θ-Open Sets and New Classes of Maps