On the Gauss map of equivariant immersions in hyperbolic space
Résumé
Given an oriented immersed hypersurface in hyperbolic space H^{n+1}, its Gauss map is defined with values in the space of oriented geodesics of H^{n+1}, which is endowed with a natural para-Kähler structure. In this paper we address the question of whether an immersion G of the universal cover of an n-manifold M , equivariant for some group representation of π_1(M) in Isom(H^{n+1}), is the Gauss map of an equivariant immersion in H^{n+1}. We fully answer this question for immersions with principal curvatures in (−1, 1): while the only local obstructions are the conditions that G is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations , the first in terms of the Maslov class, and the second (for M compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.
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