# Shrinkwrapping and the taming of hyperbolic 3-manifolds

## Abstract

We introduce a new technique for finding CAT surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.

## 0. Introduction

During the period 1960–1980, Ahlfors, Bers, Kra, Marden, Maskit, Sullivan, Thurston and many others developed the theory of *geometrically finite ends* of hyperbolic It remained to understand those ends which are not geometrically finite; such ends are called -manifolds.*geometrically infinite*.

Around 1978 William Thurston gave a conjectural description of geometrically infinite ends of complete hyperbolic An example of a geometrically infinite end is given by an infinite cyclic covering space of a closed hyperbolic 3-manifold which fibers over the circle. Such an end has cross sections of uniformly bounded area. By contrast, the area of sections of geometrically finite ends grows exponentially in the distance from the convex core. -manifolds.

For the sake of clarity we will assume throughout this introduction that where is parabolic free. Precise statements of the parabolic case will be given in §7.

Thurston’s idea was formalized by Bonahon Reference Bo and Canary Reference Ca with the following.

Here by we mean as usual a geodesic metric space for which geodesic triangles are “thinner” than comparison triangles in hyperbolic space. If the metrics pulled back by the , are smooth, this is equivalent to the condition that the Riemannian curvature is bounded above by See .Reference BH for a reference. Note that by Gauss–Bonnet, the area of a surface can be estimated from its Euler characteristic; it follows that a simply degenerate end has cross sections of uniformly bounded area, just like the end of a cyclic cover of a manifold fibering over the circle.

Francis Bonahon Reference Bo observed that geometrically infinite ends are exactly those ends possessing an exiting sequence of closed geodesics. This will be our working definition of such ends throughout this paper.

The following is our main result.

Consequently we have:

In 1974 Marden Reference Ma showed that a geometrically finite hyperbolic 3-manifold is *topologically tame*, i.e., is the interior of a compact He asked whether all complete hyperbolic 3-manifolds with finitely generated fundamental group are topologically tame. This question is now known as the -manifold.*Tame Ends Conjecture* or *Marden Conjecture*.

Ian Agol Reference Ag has independently proven Theorem 0.4.

There have been many important steps towards Theorem 0.2. The seminal result was obtained by Thurston (Reference T, Theorem 9.2) who proved Theorems 0.3 and 0.4 for certain algebraic limits of quasi-Fuchsian groups. Bonahon Reference Bo established Theorems 0.2 and 0.4 when is freely indecomposable, and Canary Reference Ca proved that topological tameness implies geometrical tameness. Results in the direction of 0.4 were also obtained by Canary–Minsky Reference CaM, Kleineidam–Souto Reference KS, Evans Reference Ev, Brock–Bromberg–Evans–Souto Reference BBES, Ohshika Reference Oh, Brock–Souto Reference BS and Souto Reference So.

Thurston first discovered how to obtain analytic conclusions from the existence of exiting sequences of surfaces. Thurston’s work as generalized by Bonahon Reference Bo and Canary Reference Ca combined with Theorem 0.2 yields a positive proof of the Ahlfors’ Measure Conjecture Reference A2.

Theorem 0.5 is one of the many analytical consequences of our main result. Indeed, Theorem 0.2 implies that a complete hyperbolic 3-manifold with finitely generated fundamental group is *analytically tame* as defined by Canary Reference Ca. It follows from Canary that the various results of Reference Ca, §9 hold for .

Our main result is the last step needed to prove the following monumental result, the other parts being established by Alhfors, Bers, Kra, Marden, Maskit, Mostow, Prasad, Sullivan, Thurston, Minsky, Masur–Minsky, Brock–Canary–Minsky, Ohshika, Kleineidam–Souto, Lecuire, Kim–Lecuire–Ohshika, Hossein–Souto and Rees. See Reference Mi and Reference BCM.

The following result was conjectured by Bers, Sullivan and Thurston. Theorem 0.4 is one of many results, many of them recent, needed to build a proof. Major contributions were made by Alhfors, Bers, Kra, Marden, Maskit, Mostow, Prasad, Sullivan, Thurston, Minsky, Masur–Minsky, Brock–Canary–Minsky, Ohshika, Kleineidam–Souto, Lecuire, Kim–Lecuire–Ohshika, Hossein–Souto, Rees, Bromberg and Brock–Bromberg.

The main technical innovation of this paper is a new technique called *shrinkwrapping* for producing surfaces in hyperbolic 3-manifolds. Historically, such surfaces have been immensely important in the study of hyperbolic 3-manifolds; e.g., see Reference T, Reference Bo, Reference Ca and Reference CaM.

Given a locally finite set of pairwise disjoint simple closed curves in the 3-manifold we say that the embedded surface , is -*incompressible* rel. if every compressing disc for meets at least twice. Here is a sample theorem.

In fact, we prove the stronger result that is * -minimal* (to be defined in §1), which implies in particular that it is intrinsically .

Here is the main technical result of this paper.

Theorem 0.4 can now be deduced from Theorem 0.9 and Souto Reference So; however, we prove that Theorem 0.9 implies Theorem 0.4 using only 3-manifold topology and elementary hyperbolic geometry.

The proof of Theorem 0.9 blends elementary aspects of minimal surface theory, hyperbolic geometry, and 3-manifold topology. The method will be demonstrated in §4 where we give a proof of Canary’s theorem. The first-time reader is urged to begin with that section.

This paper is organized as follows. In §1 and §2 we establish the shrinkwrapping technique for finding surfaces in hyperbolic 3-manifolds. In §3 we prove the existence of simple geodesics exiting the end of parabolic free manifolds. In §4 we prove Canary’s theorem. This proof will model the proof of the general case. The general strategy will be outlined at the end of that section. In §5 we develop the topological theory of end reductions in 3-manifolds. In §6 we give the proofs of our main results. In §7 we give the necessary embellishments of our methods to state and prove our results in the case of manifolds with parabolic cusps. -separated

## 1. Shrinkwrapping

In this section, we introduce a new technical tool for finding surfaces in hyperbolic called -manifolds,*shrinkwrapping*. Roughly speaking, given a collection of simple closed geodesics in a hyperbolic -manifold and an embedded surface a surface , is obtained from by *shrinkwrapping rel. * if it is homotopic to can be approximated by an isotopy from , supported in and is the least area subject to these constraints. ,

Given mild topological conditions on (namely to be defined below) the shrinkwrapped surface exists, and is -incompressibility, with respect to the path metric induced by the Riemannian metric on .

We use some basic analytical tools throughout this section, including the Gauss–Bonnet formula, the coarea formula, and the Arzela–Ascoli theorem. At a number of points we must invoke results from the literature to establish existence of minimal surfaces (Reference MSY), existence of limits with area and curvature control (Reference CiSc), and regularity of the shrinkwrapped surfaces along (Reference Ri, Reference Fre). General references are Reference CM, Reference Js, Reference Fed and Reference B.

### 1.1. Geometry of surfaces

For convenience, we state some elementary but fundamental lemmas concerning curvature of (smooth) surfaces in Riemannian -manifolds.

We use the following standard terms to refer to different kinds of minimal surfaces:

Note that we do not require that our minimal or locally least area surfaces are complete.

Any subsurface of a globally least area surface is locally least area, and a locally least area surface is minimal. A smooth surface is minimal iff its mean curvature vector field vanishes identically. For more details, consult Reference CM, especially chapter 5.

The intrinsic curvature of a minimal surface is controlled by the geometry of the ambient manifold. The following lemma is formula 5.6 on page 100 of Reference CM.

In particular, if the Riemannian curvature on is bounded from above by some constant then the curvature of a minimal surface , in is also bounded above by .

The following lemma is just the usual Gauss–Bonnet formula:

Many simple proofs exist in the literature. For example, see Reference Js.

If is merely piecewise with finitely many corners , and external angles the Gauss–Bonnet formula must be modified as follows: ,

Observe for a geodesic triangle with external angles that Lemma 1.4 implies

Notice that the geodesic curvature vanishes precisely when is a geodesic, that is, a critical point for the length functional. More generally, let be the normal bundle of in oriented so that the inward unit normal is a positive section. The exponential map restricted to , defines a map

for small where , and , for small is the boundary in of the tubular neighborhood of Then .

Note that if

for small

### 1.2. Comparison geometry

For basic elements of the theory of comparison geometry, see Reference BH.

There is a slight issue of terminology to be aware of here. In a *surface*, a triangle is a polygonal disk with *path metric space*, a triangle is just a union of

By Lemma 1.4 applied to geodesic triangles, one can show that a

More generally, suppose

Notice by Lemma 1.2 that a smooth surface

### 1.3. Statement of shrinkwrapping theorem

The remainder of this section will be taken up with the proof of Theorem 1.10.

### 1.4. Deforming metrics along geodesics

A surface

The basic idea in the proof of Theorem 1.10 is to search for a least area representative of the isotopy class of the surface

In fact, for technical reasons which will become apparent in §1.8, the deformations described below are not quite adequate for our purposes, and we must consider metrics which are deformed *twice* — firstly, a mild deformation which satisfies curvature pinching

We are really only interested in the behaviour of the metrics

The deformed metrics

### 1.5. Constructing the homotopy

As a first approximation, we wish to construct surfaces in *mean convexity*. A co-oriented surface in a Riemannian *mean convex* if the mean curvature vector of the surface always points to the negative side of the surface, where it does not vanish. Totally geodesic surfaces and other minimal surfaces are examples of mean convex surfaces, with respect to any co-orientation. Such surfaces act as *barriers* for minimal surfaces, in the following sense: suppose that *local*. See Reference MSY for a more thorough discussion of barrier surfaces.