A geometric approach to regularity for optimal transport maps

The first regularity results for maps optimizing an open class of cost were proved by Ma-Trudinger-Wang (2005) using a key inequality which they introduced (based on a classical strategy of Pogorelov). Away from the boundary, this inequality controls the size of the derivative of any smooth optimal map. After reviewing some history concerning minimal surfaces in signed geometries, we describe a new derivation of this estimate with Brendle, Leger and Rankin (2024) which relies in part on Kim, McCann and Warren's (2010) observation that the graph of an optimal map becomes a volume maximizing non-timelike submanifold when the product of the source and target domains is endowed with a suitable geometry that combines both the densities being transported and the cost being optimized. This unexpectedly links optimal transport to the Plateau problem in geometry with split signature (and more philosophically, with observer independence in Einstein's theory of gravity). The key difficulty is to uniformize ellipticity by showing the maximizing non-timelike submanifold is in fact (uniformly) spacelike.

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