On the Gauss Map of Equivariant Immersions in Hyperbolic Space Christian El Emam, Andrea Seppi

On the Gauss map of equivariant immersions in hyperbolic space Christian El Emam, Andrea Seppi

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Christian El Emam, Andrea Seppi. On the Gauss map of equivariant immersions in hyperbolic space. 2020. hal-03000958

HAL Id: hal-03000958 https://hal.archives-ouvertes.fr/hal-03000958 Preprint submitted on 12 Nov 2020

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n n n n “endpoints”in the visual boundary ∂H : this gives an identification (H ) ∼= ∂H ∂H ∆. G n × \ Under this identification the Gauss map Gσ of an immersion σ : M H corresponds to a → pair of hyperbolic Gauss maps G± : M ∂Hn. σ → A parallel research direction, originated by the works of Uhlenbeck [Uhl83] and Epstein [Eps86a, Eps86b, Eps87], concerned the study of immersed hypersurfaces in Hn, mostly in dimension n = 3. These works highlighted the relevance of hypersurfaces satisfying the geometric condition for which principal curvatures are everywhere different from 1, sometimes ± called horospherically convexity: this is the condition that ensures that the hyperbolic Gauss ± maps Gσ are locally invertible. On the one hand, Epstein developed this point of view to give a description “from infinity” of horospherically convex hypersurfaces as envelopes of horo- spheres. This approach has been pursued by many authors by means of analytic techniques, see for instance [Sch02, IdCR06, KS08], and permitted to obtain remarkable classification results often under the assumption that the principal curvatures are larger than 1 in ab- solute value ([Cur89, AC90, AC93, EGM09, BEQ10, BEQ15, BQZ17, BMQ18]). On the other hand, Uhlenbeck considered the class of so-called almost-Fuchsian manifolds, which has been largely studied in [KS07, GHW10, HL12, HW13, HW15, Sep16, San17] afterwards. These are complete hyperbolic manifolds diffeomorphic to S R, for S a closed orientable × surface of genus g 2, containing a minimal surface with principal curvatures different from ≥ 1. These surfaces lift on the universal cover to immersions σ : S H3 which are equivari- ± → ant for a quasi-Fuchsian representation ρ : π (S) Isom(H3) and, by the Gauss-Bonnet 1 → formula, have principal curvatures in ( 1, 1), a condition to whiche we will refer as having − small principal curvatures.

1.2. Integrability of immersions in (Hn). One of the main goals of this paper is to G discuss when an immersion G: M n (Hn+1) is integrable, namely when it is the Gauss → G map of an immersion M Hn+1, in terms of the geometry of (Hn+1). We will distinguish → G three types of integrability conditions, which we list from the weakest to the strongest: An immersion G: M (Hn+1) is locally integrable if for all p M there exists a • → G ∈ n+1 neighbourhood U of p such that G U is the Gauss map of an immersion U H ; | → An immersion G: M (Hn+1) is globally integrable if it is the Gauss map of an • → G immersion M Hn+1; → Given a representation ρ: π (M) Isom(Hn+1), a ρ-equivariant immersion G: M •