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Theses

Regularity of optimal transport map on compact Riemannian manifolds

Abstract : In this thesis, we are concerned with the regularity of optimal transport maps on compact Riemannian manifolds. In the first chapter, we give some definitions and recall some facts in Riemannian geometry. In the second chapter, we examine the variation of the curvature on the geodesics. In the third chapter, we study the MTW tensor on compact Riemannian manifold. We show that an improved MTW condition is satisfied on nearly spherical manifold. The proof goes by a careful analysis combined with the perturbative arguments on the spheres. In the fourth chapter, we study the inverse of the Hessian matrix of the squared distance. In the fifth chapter, we prove the smoothness of the optimal transport maps on two classes of compact Riemannian manifold-nearly spherical manifolds and Riemannian products of nearly spherical manifolds. In the last chapter, we provide some perspectives about the optimal transportation in the literature.
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Jian Ye. Regularity of optimal transport map on compact Riemannian manifolds. Analysis of PDEs [math.AP]. Université Paul Sabatier - Toulouse III, 2018. English. ⟨NNT : 2018TOU30354⟩. ⟨tel-02927159⟩

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