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On the Gauss map of equivariant immersions in hyperbolic space

Christian El Emam, Andrea Seppi

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 $π_1(M)$ 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.

On the Gauss map of equivariant immersions in hyperbolic space

Abstract

Given an oriented immersed hypersurface in hyperbolic space , its Gauss map is defined with values in the space of oriented geodesics of , which is endowed with a natural para-Kähler structure. In this paper we address the question of whether an immersion of the universal cover of an -manifold , equivariant for some group representation of in , is the Gauss map of an equivariant immersion in . We fully answer this question for immersions with principal curvatures in : while the only local obstructions are the conditions that 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 compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.

Paper Structure

This paper contains 38 sections, 44 theorems, 167 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Context in literature
  3. Integrability of immersions in $\mathcal{G}(\mathbb{H}^{n})$
  4. Local integrability
  5. Global integrability
  6. $\rho$-integrability
  7. Nearly-Fuchsian representations
  8. The geometry of $\mathcal{G}(\mathbb{H}^{n})$ and $T^1\mathbb{H}^n$
  9. Overview of the proofs
  10. Relation with geometric flows
  11. The space of geodesics of hyperbolic space
  12. Hyperboloid model
  13. Para-Sasaki metric on the unit tangent bundle
  14. A para-Kähler metric on the space of geodesics
  15. The Gauss map of hypersurfaces in $\mathbb{H}^{n+1}$
  16. ...and 23 more sections

Key Result

Lemma 2.1

Given $(x,v)\in T^1\mathbb{H}^n$, the horizontal lift $v^\mathcal{H}$ coincides with the infinitesimal generator $\chi_{(x,v)}$ of the geodesic flow, and has the expression:

Figures (12)

  • Figure 1: The normal bundle $N^1Q$ of a $k$-dimensional totally geodesic submanifold $Q$ in $\mathbb{H}^{n+1}$ (here $k=1$ and $n=2$). On the right: after composing with the geodesic flow $\varphi_t$ for $t\neq 0$, one obtains an equidistant cylinder.
  • Figure 2: A schematic picture of the argument in the proof of Lemma \ref{['lemma curve small acc']}. On the left, for $t\in(-\epsilon,\epsilon)$ the image of the curve $\gamma(t)$ lies in the concave side of the horosphere tangent to $\gamma$ at $t=0$. On the right, the same holds in fact for every $t$, for otherwise one would obtain a contradiction with the first part of the proof at the minimum point $t_{\min}$.
  • Figure 3: Schematically, an immersion $\sigma$ tangent at one point to a metric sphere (whose principal curvatures are larger than $1$), a horosphere (equal to $1$) and a $r$-cap (smaller than $1$). The image of $\sigma$ is contained in the concave side of the three of them.
  • Figure 4: A sketch of the proof of the first part of Proposition \ref{['prop injectivity']}, namely the injectivity of $\sigma$. If $\sigma(p)=\sigma(q)=y_0$ for $p\neq q$, then the image $\sigma\circ\gamma$ of a $\mathrm{I}$-geodesic connecting $p$ and $q$ would be tangent to a metric ball centered at $y_0$, which contradicts the assumption that $\sigma$ has small principal curvatures.
  • Figure 5: The setting of the proof that $\sigma$ is proper in Proposition \ref{['prop injectivity']}: in the upper half-plane model, the image of $\sigma$ is contained below the horosphere $\{x_{n+1}=1\}$ and in the outer side of the horosphere $x_1^2+\ldots+x_{n}^2+(x_{n+1}-\frac{1}{2})^2= \frac{1}{4}$. On the right, the neighbourhood $U$ of $y_0$, where the image of $\sigma$ is proved to be the graph of a function $h:B(0,\epsilon)\to{\mathbb R}$.
  • ...and 7 more figures

Theorems & Definitions (126)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 116 more