Compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds

Slutskiy, Dmitriy

Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 2, pp. 309-353

We show the existence of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on its boundary coincides with a prescribed surface metric of curvature K≥-1 in the sense of A. D. Alexandrov.

This result extends the existence part of the classical result by Alexandrov and Pogorelov on the realization of a convex domain with a prescribed boundary metric in ℍ3 in the case where ℍ3 is replaced by a quasi-Fuchsian manifold and therefore the topology of a convex domain is not trivial.

Nous montrons l’existence d’un tel domaine compact convexe dans une variété quasi-fuchsienne que la métrique induite sur son bord coïncide avec une métrique prescrite de courbure K≥-1 au sens de A. D. Alexandrov.

Ce résultat étend la partie d’existence d’un résultat classique par Alexandrov et Pogorelov sur la réalisation d’un domaine convexe avec une metrique de bord prescrite dans ℍ3 dans le cas où ℍ3 est remplacé par une variété quasi-fuchsienne et donc la topologie d’un domaine convexe n’est pas triviale.

Classification : 53C45, 20H10, 53C42, 30F40, 57M10, 57M40
Keywords: quasi-Fuchsian manifold, convex compact domain, Alexandrov space, induced metric

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