On the Ma–trudinger–wang Curvature on Surfaces
نویسندگان
چکیده
We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger–Wang condition is stable under C perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.
منابع مشابه
New Examples Satisfying Ma-Trudinger-Wang Conditions
In this paper, we study the Ma–Trudinger–Wang (MTW) conditions for cost functions c which are of the form c = l ◦ d, where d is a Riemannian distance function with constant sectional curvature. In this case, the MTW conditions are equivalent to some computable conditions on the function l. As a corollary, we give some new costs on Riemannian manifolds of constant negative curvature for which th...
متن کاملThe Ma-trudinger-wang Curvature for Natural Mechanical Actions
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 crosscurvature, and i...
متن کاملCounterexamples to Continuity of Optimal Transportation on Positively Curved Riemannian Manifolds
Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.
متن کاملNearly round spheres look convex
We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C topology, has uniformly convex injectivity domains — so M appears uniformly convex in any exponential chart. The proof is based on the Ma–Trudinger–Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
متن کاملContinuity, Curvature, and the General Covariance of Optimal Transportation
Let M and M̄ be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ρ̄. For subdomains M and M̄ of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4(M × M̄) to guarantee smoothness of the optimal map pushing ρ forward to ρ̄; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions ...
متن کامل