Properly immersed surfaces in hyperbolic $3$-manifolds

Abstract

We study complete finite topology immersed surfaces Σ in complete Riemannian 3-manifolds N with sectional curvature K N ≤ −a 2 ≤ 0, such that the absolute mean curvature function of Σ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface Σ must be proper in N and its total curvature must be equal to 2πχ(Σ). If N is a hyperbolic 3-manifold of finite volume and Σ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of Σ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in N .Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42. Key words and phrases: Calabi-Yau problem, hyperbolic 3-manifolds, asymptotic injectivity radius, bounded mean curvature, isoperimetric inequality. Definition 1.1. Let e be an end of a complete Riemannian surface Σ whose injectivity radius function we denote by I Σ : Σ → (0, ∞). Let E(e) be the collection of proper subdomains E ⊂ Σ, with compact boundary, that represents e. We define the (lower) asymptotic injectivity radius of e byNote that if Σ has an end e which admits a one-ended representative E, then I ∞ Σ (e) = lim inf E I Σ | E . If Σ has finite topology, then every end of Σ has a one-ended representative which is an annulus (i.e., a surface with the topology of S 1 × [0, ∞)), hence this simpler definition can be used. Moreover, Lemma 5.1 in the Appendix shows that if Σ has nonpositive Gaussian curvature and an end e has a representative E which is an annulus, then for every divergent sequence of points {p n } n∈N on E,We define the mean curvature function of an immersed two-sided surface Σ with a given unit normal field in a Riemannian 3-manifold to be the pointwise average of its principal curvatures; note that if Σ does not have a unit normal field, then the absolute value |H Σ | of the mean curvature function of Σ still makes sense because a unit normal field locally exists on Σ and under a change of this local choice, the principal curvatures change sign.Theorem 1.2. Let N be a complete 3-manifold with sectional curvature K N ≤ −a 2 ≤ 0, for some a ≥ 0. Let ϕ : Σ → N be an isometric immersion of a complete surface Σ with finite topology, whose mean curvature function satisfies |H ϕ | ≤ a. Then Σ has nonpositive Gaussian curvature and the following hold:A. If N is simply connected, then I ∞ Σ (e) = ∞ for every end e of Σ.B. If N has positive injectivity radius Inj(N ) = δ > 0, then every end e of Σ satisfies I ∞ Σ (e) ≥ δ. In particular, Σ has positive injectivity radius.C. If I Σ is bounded, then Σ has finite total curvature Σ K Σ = 2πχ(Σ), where χ(Σ) is the Euler characteristic of Σ. Furthermore, for each annular end representative E of Σ, the induced map ϕ * : π 1 (E) → π 1 (N ) on fundamental groups is injective.

Totally umbilic surfaces in hyperbolic 3-manifolds of finite volume

“…As already observed, a surface Σ appears as a totally umbilic surface with mean curvature H Σ ≥ 1 in a hyperbolic manifold N of finite volume if and only if Σ is a geodesic sphere with H Σ > 1 or it is a flat torus or a flat Klein bottle in a cusp end of N when H Σ = 1. On the other hand, if Σ is a totally umbilic surface with H Σ ∈ [0, 1), which must be proper by item 1, then Corollary 4.7 of [7] implies that the Euler characteristic of Σ is negative, completing the proof of items 2 and 3.…”

“…Suppose that Σ is a totally umbilic surface in a hyperbolic 3-manifold N of finite volume, with finite negative Euler characteristic and H Σ ∈ [0, 1). Then, item 4a follows immediately from [7,Corollary 4.7].…”

“…Note that the main statement of item 4 follows from the above discussion and the fact that any properly immersed, infinite topology surface with constant mean curvature H ∈ [0, 1) in a hyperbolic 3-manifold of finite volume has unbounded norm of its second fundamental form; see item 4 of [7,Theorem 1.3] for this unboundedness property.…”

A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

“…Theorem 1.1 contrasts with [6,Proposition 4.8], where it is shown that, for any H ≥ 1 and any noncompact hyperbolic 3-manifold N of finite volume, there exists a complete, properly immersed annulus with constant mean curvature H. Therefore, the hypothesis of embeddedness in Theorem 1.1 is necessary. Moreover, in [1], together with Adams, we proved that for any H ∈ [0, 1) and any surface S of finite negative Euler characteristic, there exists a hyperbolic 3-manifold of finite volume with a proper embedding of S with constant mean curvature H. Furthermore, there are examples of closed surfaces in hyperbolic 3-manifolds of finite volume for any H ≥ 1; namely geodesic spheres and tori and Klein bottles in its cusp ends.…”

“…We continue the study of properly immersed surfaces of constant mean curvature H in hyperbolic 3-manifolds of finite volume that began with the works of Collin, Hauswirth, Mazet and Rosenberg [2,4] in the minimal case, and was extended to the H ∈ (0, 1) case by the authors [6].…”