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arXiv · 2311.10208

A geometric approach to apriori estimates for optimal transport maps

Simon Brendle·Flavien Léger·Robert J. McCann·Cale Rankin
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Abstract

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.

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Simon Brendle, Flavien Léger, Robert J. McCann, Cale Rankin. 2023-11-16. A geometric approach to apriori estimates for optimal transport maps. https://arxiv.org/abs/2311.10208

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