The Memoirs of the AMS is devoted to the publication of new research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers of groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the American Mathematical Society. All papers are peer-reviewed.
Chapters;
Prologue by Luigi Ambrosio;
1. Introduction;
2. Multiples of $\mathrm {b}$ are Kantorovich potentials;
3. The gradient flow of $\mathrm {b}$ preserves the measure;
4. The gradient flow of $\mathrm {b}$ preserves the distance;
5. The quotient space isometrically embeds into the original one;
6. ""Pythagoras' theorem"" holds;
7. The quotient space has dimension $N-1$; A. Infinitesimal Hilbertianity and
behavior of gradient flows; B. Infinitesimal Hilbertianity and behavior of
the distance; C. Eulerian and Lagrangian points of view on lower Ricci
curvature bounds
Nicola Gigli, Universite de Nice, France
