Abstract
Given a compact connected n-dimensional Riemannian manifold, we investigate the smoothness of the optimal transport map between the smooth densities with respect to the squared Riemannian distance cost. The optimal map is characterized by exp(gradu), where the potential function u satisfies a Monge–Ampère type equation. Delanoë [7] showed the smoothness of u on the Riemannian surfaces when the scalar curvature is close to 1 in C 2 norm. In this work, we study the regularity issue on Riemannian manifolds with curvature sufficiently close to curvature of round sphere in C 2 norm in all dimensions and prove that the 𝒞-curvature on such Riemannian manifolds satisfies an improved Ma-Trudinger-Wang condition and the Jacobian of the exponential map is positive. As a consequence, we imply the smoothness of the optimal transport map by the continuity method.
Highlights
The aim of this paper is to show the smoothness of the optimal transport maps G, or equivalently, the smoothness of the optimal transport potential u
We are interested in the high order regularity on closed manifolds. Such regularity result holds on flat manifolds [5], on spheres [29], on complexe or quaternionic projective spaces [7, 13, 23], on product of spheres [12, 13, 23], on nearly spherical manifolds with topology [8, 30] and on 2 dimensional connected manifolds or positively curved Riemannian locally symmetric spaces [7]
Once we prove the A3S condition, we could study the regularity of the optimal transport maps
Summary
Let (M, g) be a compact connected Riemannian manifold without boundary of dimension n 2. McCann [32] developed Brenier’s theory on Riemannian manifolds He showed the optimal transport map is unique and takes the form G(m) = expm(∇u(m)) where u is some c-convex function, that is, ∀ x ∈ M , u(x) = supy∈M (−c(x, y) − v(y)) for some function v on M. We are interested in the high order regularity on closed manifolds Such regularity result holds on flat manifolds [5], on spheres [29], on complexe or quaternionic projective spaces [7, 13, 23] (see [27]), on product of spheres [12, 13, 23], on nearly spherical manifolds with topology [8, 30] and on 2 dimensional connected manifolds or positively curved Riemannian locally symmetric spaces [7]
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