The Kähler geometry of certain optimal transport problems

Abstract

Let $X$ and $Y$ be domains of ${ℝ}^n$ equipped with probability measures $\mu$ and $\nu$, respectively. We consider the problem of optimal transport from $\mu$ to $\nu$ with respect to a cost function $c:X\times Y\to ℝ$. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly relatively $c$-convex with respect to each other and the cost function has nonnegative MTW tensor. For cost functions of the form $c\left(x,y\right)=\mathrm{\Psi }\left(x-y\right)$ for some convex function $\mathrm{\Psi }:\mathcal{ℳ}\to ℝ$, we find an associated Kähler manifold on $T\mathcal{ℳ}$ whose orthogonal antibisectional curvature is proportional to the MTW tensor. We also show that relative $c$-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection on $\mathcal{ℳ}$. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann (J. Eur. Math. Soc. 12:4 (2010), 1009–1040).

We provide several applications of this work. In particular, we find a complete Kähler surface with nonnegative orthogonal antibisectional curvature that is not a Hermitian symmetric space or biholomorphic to ${ℂ}^2$. We also address a question in mathematical finance raised by Pal and Wong (2018, arXiv:1807.05649) on the regularity of pseudoarbitrages, or investment strategies which outperform the market.