On the Gauss map of equivariant immersions in hyperbolic space

Abstract

Given an oriented immersed hypersurface in hyperbolic space double-struckHn+1$\mathbb {H}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of double-struckHn+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 π1(M)$\pi _1(M)$ in Isomfalse(Hn+1false)$\mathrm{Isom}(\mathbb {H}^{n+1})$, is the Gauss map of an equivariant immersion in double-struckHn+1$\mathbb {H}^{n+1}$. We fully answer this question for immersions with principal curvatures in false(−1,1false)$(-1,1)$: 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.

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

“…Points i) and ii) are well known. For point i), see [25] or [24,Proposition 4.15,Remark 4.22]. Let S be the lift of S = ι(S) to the universal cover H 3 .…”

“…Let S be the lift of S = ι(S) to the universal cover H 3 . To show point ii), the fundamental property is that S stays in the concave side of any tangent horosphere (see [24,Lemma 4.11]), hence a fortiori on the concave side of any tangent metric ball centered at a point P outside S. This implies that the geodesics orthogonal to S are pairwise disjoint and form a global foliation in lines of M . Moreover, the distance from S is realized along the orthogonal geodesic through P .…”

“…Observe that if S = ι(S) has small principal curvatures, then all equidistant surfaces ζ(S × {r}) also have small principal curvatures ( [25,Chapter 3] or[24, Corollary 4.4]). Hence one can repeat the above argument replacing S with ζ(Σ × {r}), and conclude (4.25) for all r1, r2.To prove points iii) and iv), observe that, with our convention on the mean curvature (see Section 4.1.3), the principal curvatures λ1(r), λ2(r) of the embedding ιr := ζ(•, r) : S → M at the point p satisfy the formula:λi(p, r) = tanh(µi(p) + r) ,(4.26)which is monotone increasing in r, where λi(p, 0) = tanh µi(p) ∈ (−1, 1).…”