Robert J. McCann

The geometry of optimal transportation

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal measures . . . . . . . . . . . . . . . . . . . 126 Part I. Strictly convex costs . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. Existence and uniqueness of optimal maps . . . . . . . . . . . . . 133 4. Characterization of the optimal map . . . . . . . . . . . . . . . . . 137 Part II. Costs which are strictly concave as a function of d i s t a n c e . . . 141 5. The role of optimal maps . . . . . . . . . . . . . . . . . . . . . . . . 141 6. Uniqueness of optimal solutions . . . . . . . . . . . . . . . . . . . . 144 Part III. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A. Legendre transforms and conjugate costs . . . . . . . . . . . . . . 148 B. Examples of c-concave potentials . . . . . . . . . . . . . . . . . . . 152 C. Regularity of c-concave potentials . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Polar factorization of maps on Riemannian manifolds

Abstract. Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If

Existence and uniqueness of monotone measure-preserving maps

s : M \to M

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates

is merely Borel and never maps positive volume into zero volume, we show

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

s = t \circ u

Continuity, curvature, and the general covariance of optimal transportation

factors uniquely a.e. into the composition of a map

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

t(x) = {\rm exp}_x[-\nabla\psi(x)]

Exact solutions to the transportation problem on the line

and a volume-preserving map

Monge’s transport problem on a Riemannian manifold

u : M \to M

RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION

, where