On the Gauss map of equivariant immersions in hyperbolic space

Abstract

Given an oriented immersed hypersurface in hyperbolic space ${\mathbb{H}}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of ${\mathbb{H}}^{n+1}$, which is endowed with a natural para-Kähler structure. In this paper we address the question of whether an immersion $G$ of the universal cover of an $n$-manifold $M$, equivariant for some group representation of ${\pi }_1(M)$ in $\mathrm{Isom}({\mathbb{H}}^{n+1})$, is the Gauss map of an equivariant immersion in ${\mathbb{H}}^{n+1}$. 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.

arXiv comment55 pages, 12 figures. Minor improvements with respect to the previous version