Section: New Results
Conjugate and cut loci computations and applications
Participants : Bernard Bonnard, Olivier Cots, Jean-Baptiste Caillau, Alessio Figalli [Univ. of Texas at Austin, USA] , Thomas Gallouët [MEPHYSTO project-team] , Ludovic Rifford.
Many optimal control problems from mechanics or quantum systems (see  and the last paragraph of section 6.1 ) lead to studying some king of singular metrics, sometimes known as almost-Riemannian. This led us to consider, in  , metrics on the two-sphere of revolution of the following find: they are Riemannian on each open hemisphere whereas one term of the corresponding tensor becomes infinite on the equator. Length minimizing curves can be computed and structure results on the cut and conjugate loci given, extending those in  . These results rely on monotonicity and convexity properties of the quasi-period of the geodesics; such properties are studied on an example with elliptic transcendency. A suitable deformation of the round sphere allows to reinterpretate the equatorial singularity in terms of concentration of curvature and collapsing of the sphere.
It is known that convexity of the injectivity domain (the boundary of which is sent by the exponential map to the first cut locus) and the “Ma–Trudinger–Wang condition” (an positivity condition on the Ma–Trudinger–Wang tensor) both play a very important role in the continuity of solutions of optimal transport problems. This led to study these properties on their own, and it is still an open question to decide under which conditions the latter implies the former. In  , it is proved that the MTW condition implies the convexity of injectivity domains on a smooth nonfocal compact Riemannian manifold. This improves a previous result by Loeper and Villani.