Compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds. (Domaines convexes compacts avec des métriques de bord prescrites dans les variétés quasi-fuchsiennes.) (English. French summary) Zbl 1415.53053
The author states in the abstract of the paper: “We show the existene of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on boundary coincides with a prescribed surface metric of curvature \(K\ge-1\) 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 metric in \(\mathbb{H}^3\) in the case, where \(\mathbb{H}^3\) is replaced by a quasi-Fuchsian manifold and therefore the topology of a convex domain is not trivial.”
This result extends the existence part of the classical result by Alexandrov and Pogorelov on the realization of a convex domain with a prescribed metric in \(\mathbb{H}^3\) in the case, where \(\mathbb{H}^3\) is replaced by a quasi-Fuchsian manifold and therefore the topology of a convex domain is not trivial.”
Reviewer: Themistocles M. Rassias (Athens)
MSC:
| 53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 57M10 | Covering spaces and low-dimensional topology |
| 57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |
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