Continuity , Curvature , and the General Covariance of Optimal Transportation

  title={Continuity , Curvature , and the General Covariance of Optimal Transportation},
  author={Young-Heon Kim and R. J. McCann},
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 to be largely dictated by the covariance of the question; it expresses them via non-negativity of… CONTINUE READING
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