TITLE

The Ma-Trudinger-Wang curvature for natural mechanical actions

AUTHOR(S)
Lee, Paul; McCann, Robert
PUB. DATE
May 2011
SOURCE
Calculus of Variations & Partial Differential Equations;May2011, Vol. 41 Issue 1/2, p285
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
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 ( A3w) 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 ( A3s) of Ma, Trudinger and Wang, or in some cases has positive cross-curvature. In particular, the quartic potential of the anharmonic oscillator satisfies ( A3s) in the perturbative regime.
ACCESSION #
59439118

 

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