Geodesic planes in hyperbolic 3-manifolds

Abstract

This paper initiates the study of rigidity for immersed, totally geodesic planes in hyperbolic 3-manifolds M of infinite volume. In the case of an acylindrical 3-manifold whose convex core has totally geodesic boundary, we show that the closure of any immersed geodesic plane is a properly immersed submanifold of M. On the other hand, we show that rigidity fails for quasifuchsian manifolds.

Appendices

Appendix A: Quasifuchsian groups

In this section we show that for a quasifuchsian group $\Gamma$, the closure of a geodesic plane in $M = \Gamma \backslash {\mathbb H}^3$ passing through the convex core can be rather wild. The case of a Fuchsian group was treated in the Introduction, and we will show the same phenomena persists under a small bending deformation.

Bending.  Let $\Sigma _0 = \Gamma _0 \backslash {\mathbb H}^2$ be a closed hyperbolic surface of genus $g \ge 2$. We can identify $\Sigma _0$ with the convex core of the 3-manifold $M_0= \Gamma _0 \backslash {\mathbb H}^3$, which can be written as

$$\begin{aligned} M_0 \cong \Sigma _0 \times {\mathbb R}\end{aligned}$$

in cylindrical coordinates. Here the first coordinate is given by the nearest-point projection $\pi : M_0 \rightarrow \Sigma _0$, and the second is given by the signed distance from $\Sigma _0$.

Let $\beta \subset \Sigma _0$ be a simple closed curve. Then for all angles $\theta$ sufficiently small, there is a quasifuchsian manifold $M_\theta = \Gamma _\theta \backslash {\mathbb H}^3$ and a path isometry

$$\begin{aligned} j_\theta : \Sigma _0 \rightarrow \Sigma _\theta \subset \partial {\text {core}}(M_\theta ), \end{aligned}$$

such that $\Sigma _\theta$ is bent with a dihedral angle of $\theta$ along the image of $\beta$, and otherwise totally geodesic. The manifold $M_\theta$ is unique up to isometry. See e.g. [27, Ch. 8], [4, 8, 15] for this construction.

Choose r sufficiently small that $\beta$ has an embedded annular collar neighborhood $B_r$ of width 2r. Removing this collar, we obtain a surface with boundary

$$\begin{aligned} \Sigma _0(r) = \Sigma _0 - B_r \subset \Sigma _0, \end{aligned}$$

and a closed submanifold

$$\begin{aligned} M_0(r) = \Sigma _0(r) \times {\mathbb R}\subset M_0. \end{aligned}$$

There is a unique extension of $j_\theta | \Sigma _0(r)$ to an orientation-preserving isometric immersion

$$\begin{aligned} J_\theta : M_0(r) \rightarrow M_\theta , \end{aligned}$$

which sends geodesics normal to $\Sigma _0(r)$ to geodesics normal to $\Sigma _\theta (r) = j_\theta (\Sigma _0(r))$.

Proposition A.1

Assume $M_\theta$ is a quasifuchsian manifold, and $\sin (|\theta |/2) < \tanh (r)$. Then the map $J_\theta : M_0(r) \rightarrow M_\theta$ is a proper, isometric embedding.

Sketch of the proof

Qualitatively, the statement above is nearly immediate from the divergence of geodesics normal to $\Sigma _0$ (see Fig. 4). The extreme case $\sin (\theta /2) = \tanh (r)$ corresponds to a hyperbolic right triangle with base of length r and one ideal vertex; its internal angles are $(0,\pi ,\pi -\theta /2)$. $\square$

Fig. 4

Geodesics normal to $\Sigma _0$ and at least distance r from the bending locus remain disjoint under small bendings. (Diagram is in the universal cover.)

Planes in quasifuchsian manifolds.  Now let $\gamma \subset \Sigma _0$ be an immersed geodesic such that ${\overline{\gamma }}$ is disjoint from $\beta$. Then we can also choose r sufficiently small that ${\overline{\gamma }}$ lies in $\Sigma _0(r)$. As discussed in Sect. 1, the locus $P = \gamma \times {\mathbb R}\subset M_0$ is an immersed geodesic plane with

$$\begin{aligned} {\overline{P}}= {\overline{\gamma }}\times {\mathbb R}\subset M_0. \end{aligned}$$

But in fact we have ${\overline{P}}\subset M_0(r)$. Thus $P_\theta = J_\theta (P)$ is an immersed geodesic plane in $M_\theta$. By Proposition A.1., for all $\theta$ sufficiently small, $J_\theta$ maps a neighborhood of ${\overline{P}}$ isometrically to a neighborhood of ${\overline{P}}_\theta$. This shows:

Corollary A.2

For all $\theta$ sufficiently small, the quasifuchsian manifold $M_\theta$ contains an immersed geodesic plane $P_\theta$, passing through its convex core, such

$$\begin{aligned} {\overline{P}}_\theta \cong {\overline{P}}= {\overline{\gamma }}\times {\mathbb R}\subset M_0. \end{aligned}$$

As noted in Sect. 1, the closure of a geodesic can be rather wild, and the same is true for geodesics with ${\overline{\gamma }}$ disjoint from $\beta$. In particular, the closure of $P_\theta$ need not be a submanifold of $M_\theta$, so we have no analogue of Corollary 1.2.

Appendix B: Bounded geodesic planes

In this section we record a rigidity theorem for bounded geodesic planes in an arbitrary hyperbolic 3-manifold. Shah’s theorem on geodesic planes in a compact hyperbolic 3-manifold [25] follows as a special case.

Theorem B.1

Let $P \subset M$ be a bounded geodesic plane in an arbitrary hyperbolic 3-manifold $M = \Gamma \backslash {\mathbb H}^3$. Then either:

• P is a compact, immersed hyperbolic surface; or

• M is compact, and $\overline{P} = M$. More precisely, frames tangent to P are dense in ${{\text {F}}}M$.

Proof

The proof will follow the same lines as the proof in Sect. 9 above.

Let $\Lambda \subset S^2$ be the limit set of $\Gamma$, and let $C \subset S^2$ be a circle corresponding to a lift of P to the universal cover of M. To prove the theorem, we must show that $\Gamma C \subset {{\mathcal {C}}}$ is either closed or dense. Note we have $C \subset \Lambda$, because P is bounded in M. Thus every circle D in $\overline{\Gamma C}$ is also contained in $\Lambda$.

Assume $\Gamma C$ is not closed. We will show that $\overline{\Gamma C} = {{\mathcal {C}}}_\Lambda$.

Since $\Gamma C$ is not closed, $\Gamma$ is Zariski dense in G, and there is limit set on both sides of C. (Otherwise C would be the boundary of a component of $\Omega = S^2-\Lambda$, and hence $\Gamma C$ would be discrete.)

If $\Gamma D$ is closed for some $D \in \overline{\Gamma C}$, then $\overline{\Gamma C } = {{\mathcal {C}}}_\Lambda$ by Theorems 5.1 and 6.2. So we may assume that C is an exotic circle, i.e. $\overline{\Gamma D} \ne \Gamma D$ for all $D \in \overline{\Gamma C}$. Then the corresponding compact, H-invariant set $X \subset {{\text {F}}}M$ contains no closed H-orbit.

Since P is bounded, and C is contained in $\Lambda$, X is a compact subset of the renormalized frame bundle ${{\text {RF}}}M$.

The argument of Sect. 9 now goes through verbatim, to show that $\overline{\Gamma C} = {{\mathcal {C}}}_\Lambda$. Indeed, the argument is somewhat simpler: since $X \subset {{\text {RF}}}M$, we have $Y^* = Y$, and hence there is no need to consider thick sets (i.e. we can take $T_n = U$ when applying Theorems 8.1 and 8.2).

To complete the proof, note that every circle in $\overline{\Gamma C}$ is contained in the limit set. Thus $\overline{\Gamma C} = {{\mathcal {C}}}_\Lambda$ implies $\Lambda = S^2$ and $\overline{\Gamma C} = {{\mathcal {C}}}$. Consequently frames tangent to P are dense in ${{\text {F}}}M$; and since P is bounded, M is compact. $\square$

Using Theorem 11.1, we obtain the following analogue of Corollary 11.2:

Corollary B.2

Let $K \subset M$ be a compact subset of a hyperbolic 3-manifold. Then either:

• The set of compact geodesic surfaces in M contained in K is finite; or

• Compact geodesic surfaces are dense in M, and M is compact.

Remarks

One could also deduce these results from Ratner’s works [21, 22], using [2, Thm. 1] or modifying [22, Thm 2.1] to handle the case where $\Gamma$ is not a lattice.