On the Gauss map of equivariant immersions in hyperbolic space

Abstract

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values in the space of oriented geodesics of H n + 1 $\mathbb {H}^{n+1}$ , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion G $G$ of the universal cover of an n $n$ -manifold M $M$ , equivariant for some group representation of π 1 ( M ) $\pi _1(M)$ in Isom ( H n + 1 ) $\mathrm{Isom}(\mathbb {H}^{n+1})$ , is the Gauss map of an equivariant immersion in H n + 1 $\mathbb {H}^{n+1}$ . We fully answer this question for immersions with principal curvatures in ( − 1 , 1 ) $(-1,1)$ : while the only local obstructions are the conditions that G $G$ 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 $M$ compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.