# Pseudo-Riemannian Calibrated Geometry and Optimal Transportation

# Pseudo-Riemannian Calibrated Geometry and Optimal Transportation

Given a manifold $M$, there is a naturally occurring pseudo-Riemannian metric and Kähler form on the product $M\times M$. The graph of the solution to the optimal transportation problem for given smooth densities on $M$ is then a calibrated maximal Lagrangian submanifold in $M\times M$, with respect to a conformal metric on $M\times M$. Thus the graph of the optimal map is special Lagrangian in the sense of Hitchin. This variational characterization of optimal transportation is different from the traditional one. The calibrations which detect these special Lagrangians are pseudo-Riemannian analogues of the special Lagrangian calibrations for Calabi-Yau manifolds. Like in the Calabi-Yau case, the moduli space of such submanifolds is itself a manifold of dimension $b_1(M)$. The calibrated submanifolds that are not the graph of the optimal map are graphs of Lie solutions (see recent work of Delanoe) which are maps which locally satisfy the elliptic equation but do not globally minimize the cost integral. This is joint work with Young-Heon Kim and Robert McCann.