V
C
Last name: VILLANI
First name: Cédric
Given a smooth compact Riemannian surface, we prove that if a suitable convexity assumption on the tangent focal cut loci is satisfied, then all injectivity domains are semiconvex.
We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 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.
In this paper we investigate the regularity of optimal transport maps for the squared distance cost on Riemannian manifolds. First of all, we provide some general necessary and sufficient conditions for a Riemannian manifold to satisfy the so-called Transport Continuity Property. Then, we show that on surfaces these conditions coincide....
We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 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.
In this paper we investigate the regularity of optimal transport maps for the squared distance cost on Riemannian manifolds. First of all, we provide some general necessary and sufficient conditions for a Riemannian manifold to satisfy the so-called Transport Continuity Property. Then, we show that on surfaces these conditions coincide....
Given a smooth compact Riemannian surface, we prove that if a suitable convexity assumption on the tangent focal cut loci is satisfied, then all injectivity domains are semiconvex.
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 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the...
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 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the...