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

Abstract

In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded. Communicated by F. C. Marques.

On the Gauss map of equivariant immersions in hyperbolic space

“…The 1-form 𝜔 indeed satisfies the two properties of a connection: the invariance under the ℝaction follows immediately from the invariance of g 𝑇 1 ℍ 𝑛 (Lemma 2.3) and of 𝜒 (Equation ( 14)); the second property follows from Equation (10), namely, 𝜔(𝜒 (𝑥,𝑣) ) = g 𝑇 1 ℍ 𝑛 (𝜒 (𝑥,𝑣) , 𝜒 (𝑥,𝑣) ) = 1.…”

“…By (10) and ( 14), the norm of vectors proportional to 𝜒 (𝑥,𝑣) is preserved. Together with (12) and (13), vectors of the form 𝑑𝜑 𝑡 (𝑤  ) and 𝑑𝜑 𝑡 (𝑤  ) are orthogonal to 𝑑𝜑 𝑡 (𝜒 (𝑥,𝑣) ) = 𝜒 𝜑 𝑡 (𝑥,𝑣) .…”

On n-superharmonic functions and some geometric applications

“…It is known that m = 1 2π C K e 2u dz and m ∈ [0, 1] due to [24,43], where m = 0 implies u is a constant. The proof of the above result in [11] relies on two important ingredients that are deep results in geometric analysis and partial differential equation. One is the noncollapsing result of Croke-Karcher [26, Theorem A] in 1988 for complete surfaces with nonnegative Gauss curvature; the other is asymptotic estimates for nonnegative solutions to Gauss curvature type equations of Taliaferro in [63, Theorem 2.1] (see also his previous work [61,62]).…”

“…Taliaferro's estimates in [61][62][63] are the major work in the theory of local behavior of a class of subharmonic functions near an isolated singular point. And [11,Lemma 4.2] turns out to be essential to the proof of [11,Main Theorem] in 2 dimensions that a complete, nonnegatively curved, immersed surface in hyperbolic 3-space is necessarily properly embedded, except coverings of equidistant surfaces, which was conjectured by Epstein and Alexander-Currier in [3,4,[28][29][30] around 1990.…”

“…2.5.2 for a very brief introduction of complete and globally strictly convex hypersurfaces in hyperbolic space (cf. [3,4,11,12]). For convenience of readers, we first recall Busemann coordinates in hyperbolic space.…”