Prescription de la courbure de Gauss pour les corps convexes dans les espaces hyperboliques

Date/heure
15 février 2021
14:00 - 15:00

Lieu
Salle de géométrie virtuelle

Oratrice ou orateur
Jérôme Bertrand

Catégorie d'évènement
Séminaire de géométrie différentielle

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov’s problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution.

In this paper, we address Alexandrov’s problem for convex bodies in the hyperbolic space ${\mathbf{H}}^{m+1}$ . After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov’s problem in this setting. Contrary to the Euclidean case, we also prove the uniqueness of such a convex body. The methods for proving existence and uniqueness of the solution to this problem are both new.