2009/08/31 by Paul W. Y. Lee, Robert J. McCann, Lee, Paul W. Y. +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.0908.4460
The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the least action corresponding to a harmonic oscillator has zero cross-curvature, and in particular satisfies the necessary and sufficient condition \Athreew for the continuity of optimal maps. We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition \Athrees of Ma, Trudinger and Wang, or in some cases has positive cross-curvature. In particular, the quartic potential of the anharmonic oscillator satisfies \Athrees in the perturbative regime.