On the Gauss map of equivariant immersions in hyperbolic space

TL;DR

This paper investigates when a Gauss map of an equivariant hypersurface in hyperbolic space corresponds to an actual immersed hypersurface, providing global criteria involving Maslov class and Hamiltonian symplectomorphisms.

Contribution

It offers a complete characterization of when an equivariant Lagrangian and Riemannian immersion in the space of geodesics arises as a Gauss map of an immersed hypersurface in hyperbolic space, with new global obstructions.

Findings

Local conditions: G must be Lagrangian and Riemannian.

Global obstructions characterized by Maslov class.

For compact M, obstruction related to Hamiltonian symplectomorphisms.

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-K\"ahler 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.