# A.D. Alexandrov imbedding theorem for metrics with symmetry

A well known theorem due to A.D. Alexandrov says that any metric on the 2-sphere $$S^2$$ with curvature at least -1 (in the sense of Alexandrov) can be isometrically realized either as convex surface in the hyperbolic 3-space or as a doubled 2-dimensional convex set there (I omit the precise description of the latter notion).

REMARK: I do not know if the uniquness of the isometric imbedding holds up to isometries of the hyperbolic space. For isometric imbeddings into Euclidean space such a uniqueness was proven by Pogorelov.

Assume that the metric as above on $$S^2$$ is invariant under the antipodal involution $$x\mapsto -x$$. Is it true that there exists an isometric imbedding into the hyperbolic space which intertwines the antipodal involution and the reflection of the hyperbolic space with respect to some fixed point? In particular the convex surface in the image would have center of symmetry.

Approximate the metric by a sequence of symmetric hyperbolic cone-metrics (i. e. locally hyperbolic with cone points of angles $$< 2\pi$$). Each of these metrics has a unique, and therefore centrally symmetric, realization as the boundary of a convex polytope in the hyperbolic space. From this sequence of polytopes one can choose (by Blaschke's selection principle) a converging subsequence. The limit convex body is centrally symmetric, and its boundary is isometric to our given metric.