2008 DG-MP-PDE Seminar - 05

Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold $M$, we look for a smooth optimal transportation map $G$, pushing one measure to the other at a least total squared distance cost. The recent local $C^2$ estimate of Ma--Trudinger--Wang enabled G. Loeper to treat the standard sphere case. In this talk, we discuss this topic on manifolds with curvature sufficiently close to 1 in $C^2$ norm. This is a joint work with P. Delanoe.