Optimal Transportation and Differential Geometry (12w5118)
Alessio Figalli (The University of Texas at Austin)
Young-Heon Kim (University of British Columbia)
In recent years, we are observing remarkable interplay between optimal transportation and differential geometry. One of the most striking examples is the result of Sturm, Lott and Villani, which allows to define the class of metric-measure spaces with lower bounded Ricci curvature by asking for convexity inequalities of suitable entropy functionals, and then to prove nontrivial analytic and geometric properties for these spaces. These works exemplify how the ideas of optimal transportation can be applied to geometric problems: with its measure theoretic nature, optimal transportation theory works nicely on general metric spaces.
Along this line, McCann and Topping have attempted to find a `weak' notion of Ricci flow for singular spaces, a fundamental problem that is still far from being fully understood. Even for smooth spaces, the ideas in optimal transportation are used to give new proofs and perspectives for Ricci flow as seen in recent works of Lott and Topping. Optimal transportation ideas are also being adopted in discrete mathematics, and for example, Ollivier has been building up Ricci curvature theory on discrete spaces. Extending Ricci curvature and Ricci flow to more general geometric spaces is still a wide open problem, and these mentioned works hint that optimal transportation theory will play crucial roles in future progress.
For other but closely related aspects of optimal transportation, there are studies of Sturm, Gigli, Ohta, Von Renesse and others, on the heat flow on ``singular'' spaces such as Finsler or Alexandrov spaces. Very recently, Gigli, Kuwada and Ohta have obtained a fundamental result showing the equivalence between gradient flow of energy functional and that of entropy functional on Alexandrov spaces as the underlying space. Apart from allowing to extend the time-discrete scheme of Jordan, Kinderlehrer and Otto to Alexandrov spaces for constructing solutions to the heat equations, a simple corollary of this equivalence is that solutions to the heat equations become instantaneously Lipschitz continuous in space (before this result, only H"older continuity was known). This and other related results are expected to be a source of much further progress in linear and nonlinear diffusion, and geometry of singular spaces in the future.
A totally unexpected breakthrough in geometry has been obtained from the regularity theory of solutions of optimal transportation problem. The so called MTW curvature, which was found by Ma, Trudinger and Wang as a condition for the transportation cost to ensure smoothness of optimal transportation map, has nowadays several important aspects.
First, by proving that nonnegativity of MTW curvature is necessary for smoothness of optimal transportation maps (or simply optimal maps), Loeper showed the (unexpected) result that there are discontinuous optimal maps even between smooth distributions whenever the manifold has negative curvature at at least one point.
Then, Kim and McCann found that the MTW curvature is the Riemann curvature of some suitable pseudo-metric on the product space of source and target domains of optimal maps. Recently, in collaboration with Warren, they have extended their result to find a pseudo-metric with respect to which the graphs of optimal maps give volume maximizing space-like Lagrangian submanifolds, thus giving some hope for relating optimal transportation theory to submanifold theory and symplectic geometry.
Moreover, the MTW curvature was used by Figalli, Rifford and Villani to study convexity of injectivity domains, proving for instance that smooth perturbations of the round sphere have all injectivity domains uniformly convex (this result was unknown before).
There are many open problems regarding MTW curvature, especially concerning its relation to the sectional curvature. Another difficult open problem is to develop a general regularity theory of optimal maps on manifold domains, proving for instance that transport maps are smooth diffeomorphism outside a closed set of measure zero (this result in Euclidean spaces has been recently proved by Figalli and Kim).
A further line of reaseach, which takes its origin in McCann's proof of Brunn-Minkowski inequality via optimal transport, is to apply optimal transport to prove geometric and functional inequalities (like isoperimetric or Sobolev inequalities). For example, recently Figalli, Maggi and Pratelli were able to exploit the optimal transport proof of the Wulff inequality (an anisotropic version of the isoperimetric inequality) to prove a sharp stability estimate and solving a long-standing open problem. One of the main open problems in this area is to understand whether one can find optimal transport proofs also on manifolds (for instance, for the isoperimetric inequality on the sphere or the hyperbolic space),
which may then lead also to new results.
As we have discussed so far, optimal transportation theory has progressed very rapidly in recent years, solving some of the major problems in geometry (and geometric inequalities). Thus it is timely to have a new BIRS workshop on optimal transportation focusing on geometric aspects after having a similar one in April 2010, which covered more broad range of the theory. The plan is to include leading experts in optimal transportation, nonlinear partial differential equations and differential geometry, as well as junior researchers, students and minors. It is expected after two years, to see much further development, with a lot of interesting new results and directions to discuss, even with surprises.