Robert J. McCann

Wilfrid Gangbo; Robert J. McCann

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

Robert J. McCann

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

Robert J. McCann

s : M \to M

José A. Carrillo; Robert J. McCann; Cédric Villani

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

Luis A. Caffarelli; Mikhail Feldman; Robert J. McCann

s = t \circ u

Young-Heon Kim; Robert J. McCann

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

Daniel Matthes; Robert J. McCann; Giuseppe Savaré

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

Robert J. McCann

and a volume-preserving map

Mikhail Feldman; Robert J. McCann

u : M \to M

Robert J. McCann; Peter M. Topping

, where