Weakly horospherically convex hypersurfaces in hyperbolic space

Abstract

Abstract. In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces φ : M n → H n+1 and a class of conformal metrics on domains of the round sphere S n . Some of the key aspects of the correspondence and its consequences have dimensional restrictions n ≥ 3 due to the reliance on an analytic proposition from [5] concerning the asymptotic behavior of conformal factors of conformal metrics on domains of S n . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of [2] to all dimensions n ≥ 2 in a unified way. In the case of a single point boundary ∂ ∞ φ(M ) = {x} ⊂ S n , we improve these results in one direction.

On nonnegatively curved hypersurfaces in $$\mathbb {H}^{n+1}$$

“…Soρ(G −1 (ξ )) → +∞ as ξ → q, which, together with the proof of Lemma 3.2 of [4], implies that ∂ ∞ φ(M) = {q}. We remark that one may derive the same conclusion from [5]. From the embedding theorem of Epstein in [16], we know φ is embedding.…”

“…whereρ = ρ • G −1 : → R. When there is no confusion, we will also refer to this conformal metricĝ h as the horospherical metric. The correspondence between horospherically concave hypersurfaces φ : M n → H n+1 in hyperbolic space and the conformal metricĝ h on the image of the Gauss map G have been promoted in [3][4][5]16,18]. The following result follows from the so-called global correspondence from [4,5,18] and will be useful to our work here.…”

“…In [4,5], based on the global correspondence theorem, we established an extension of the embedding theorem of Epstein [16] as follows: Theorem 2.2 (cf. [4,5]) For n ≥ 2, let φ : M n → H n+1 be a complete uniformly horospherically concave hypersurface with injective hyperbolic Gauss map G : M n → S n . Suppose that the asymptotic boundary ∂ ∞ φ(M) at infinity of the hypersurface is a disjoint union of smooth closed embedded submanifolds in S n .…”

Min‐Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

“…PROOF OF THEOREM 3.3. Since jSch g j < C1 and g is a complete metric, following the results in [2] (see also [3]), there exists t > 0 such that the horospherically concave hypersurface associated to † t D t . n ƒ/ H nC1 is properly embedded with boundary and @ 1 † t D ƒ.…”

“…Our approach relies in a geometric method developed by the third author, Gálvez and Mira in [8], and further developments contained in [1][2][3]6,7], where conformal metrics on spherical domains are represented by hypersurfaces in the hyperbolic space. In order to reduce our problem on locally conformally flat manifolds to conformal metrics on subdomains of the sphere, we use results contained in the work of Spiegel [22] and Li and Nguyen [15] based on the deep theory by Schoen and Yau [20] on the developing map of a locally conformally flat manifold.…”