Recent advances in Geometric Analysis

Prescribing the Gauss curvature of hyperbolic convex bodies

speaker: Philippe Castillon (Université de Montpellier )

abstract: The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure $\mu$, the Alexandrov problem consists in proving the existence and uniqueness of a convex body whose curvature measure is $\mu$.

In the Euclidean space, this problem was first solved by Alexandrov, and it was observed later that it is equivalent to an optimal transport problem on the sphere. In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain how the optimal transport approach leads to a non-linear Kantorovich problem on the sphere and how to solve it.

Joint work with Jérôme Bertrand.

timetable:
Thu 7 Jun, 9:00 - 10:00, Aula Dini
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