Shrinkwrapping and the taming of hyperbolic 3-manifolds

By Danny Calegari and David Gabai

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 -manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.

Around 1978 William Thurston gave a conjectural description of geometrically infinite ends of complete hyperbolic -manifolds. 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.

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.

Definition 0.1.

An end of a hyperbolic -manifold is simply degenerate if it has a closed neighborhood of the form where is a closed surface, and there exists a sequence of surfaces exiting which are homotopic to in . This means that there exists a sequence of maps such that the induced path metrics induce structures on the ’s, and is homotopic to a homeomorphism onto via a homotopy supported in .

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.

Theorem 0.2.

An end of a complete hyperbolic -manifold with finitely generated fundamental group is simply degenerate if there exists a sequence of closed geodesics exiting .

Consequently we have:

Theorem 0.3.

Let be a complete hyperbolic -manifold with finitely generated fundamental group. Then every end of is geometrically tame; i.e., it is either geometrically finite or simply degenerate.

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

Theorem 0.4.

If is a complete hyperbolic -manifold with finitely generated fundamental group, then is topologically tame.

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.

If is a finitely generated Kleinian group, then the limit set is either or has Lebesgue measure zero. If , then acts ergodically on .

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.

Theorem 0.6 (Classification Theorem).

If is a complete hyperbolic -manifold with finitely generated fundamental group, then is determined up to isometry by its topological type, the conformal boundary of its geometrically finite ends and the ending laminations of its geometrically infinite ends.

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.

Theorem 0.7 (Density Theorem).

If is a complete hyperbolic -manifold with finitely generated fundamental group, then is the algebraic limit of geometrically finite Kleinian groups.

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.

Theorem 0.8 (Existence of shrinkwrapped surface).

Let be a complete, orientable, parabolic free hyperbolic -manifold, and let be a finite collection of pairwise disjoint simple closed geodesics in . Furthermore, let be a closed embedded -incompressible surface rel. which is either nonseparating in or separates some component of from another. Then is homotopic to a surface via a homotopy

such that

(1)

,

(2)

is an embedding disjoint from for ,

(3)

,

(4)

If is any other surface with these properties, then .

We say that is obtained from by shrinkwrapping rel. , or if is understood, is obtained from by shrinkwrapping.

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 0.9.

Let be an end of the complete orientable hyperbolic -manifold with finitely generated fundamental group. Let be a -dimensional compact core of , the component of facing and . If there exists a sequence of closed geodesics exiting , then there exists a sequence of surfaces of genus exiting such that each is homologically separating in . That is, each homologically separates from .

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 -separated 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.

Notation 0.10.

If , then denotes a regular neighborhood of in and denotes the interior of . If is a topological space, then denotes the number of components of . If are topological subspaces of a third space, then denotes the intersection of with the complement of .

1. Shrinkwrapping

In this section, we introduce a new technical tool for finding surfaces in hyperbolic -manifolds, called 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 -incompressibility, to be defined below) the shrinkwrapped surface exists, and is 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:

Definition 1.1.

A smooth surface in a Riemannian -manifold is minimal if it is a critical point for area with respect to all smooth compactly supported variations. It is locally least area (also called stable) if it is a local minimum for area with respect to all smooth, compactly supported variations. A closed, embedded surface is globally least area if it is an absolute minimum for area amongst all smooth surfaces in its isotopy class.

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.

Lemma 1.2 (Monotonicity of curvature).

Let be a minimal surface in a Riemannian manifold . Let denote the curvature of , and the sectional curvature of . Then restricted to the tangent space ,

where denotes the second fundamental form of .

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:

Lemma 1.3 (Gauss–Bonnet formula).

Let be a Riemannian surface with (possibly empty) boundary . Let denote the Gauss curvature of , and the geodesic curvature along . Then

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:

Lemma 1.4 (Gauss–Bonnet with corners).

Let be a Riemannian surface with boundary which is piecewise and has external angles at finitely many points . Let and be as above. Then

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 is a surface with sectional curvature bounded above by , then by integrating this formula we see that the ball of radius in about a point satisfies

for small .

1.2. Comparison geometry

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

Definition 1.5 (Comparison triangle).

Let be a geodesic triangle in a geodesic metric space . Let be given. A -comparison triangle is a geodesic triangle in the complete simply-connected Riemannian -manifold of constant sectional curvature , where the edges and satisfy

Given a point on one of the edges of , there is a corresponding point on one of the edges of the comparison triangle, satisfying

and

Remark 1.6.

Note that if , the comparison triangle might not exist if the edge lengths are too big, but if the comparison triangle always exists and is unique up to isometry.

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

Definition 1.7 (CAT()).

Let be a closed surface with a path metric . Let denote the universal cover of , with path metric induced by the pullback of the path metric . Let be given. is said to be if for every geodesic triangle in , and every point on the edge , the distance in from to is no more than the distance from to in a -comparison triangle.

By Lemma 1.4 applied to geodesic triangles, one can show that a surface with sectional curvature satisfying everywhere is with respect to the Riemannian path metric. This fact is essentially due to Alexandrov; see Reference B for a proof.

More generally, suppose is a surface which is outside a closed, nowhere dense subset . Furthermore, suppose that holds in , and suppose that the formula from Lemma 1.4 holds for every geodesic triangle with vertices in (which is a dense set of geodesic triangles). Then the same argument shows that is . See, e.g., Reference Re, §8, pp. 135–140 for more details and a general discussion of metric surfaces with (integral) curvature bounds.

Definition 1.8 (-minimal surfaces).

Let be given. Let be a complete Riemannian -manifold with sectional curvature bounded above by , and let be an embedded collection of simple closed geodesics in . An immersion

is -minimal if it is smooth with mean curvature in and is metrically with respect to the path metric induced by from the Riemannian metric on .

Notice by Lemma 1.2 that a smooth surface with mean curvature in is , so a minimal surface (in the usual sense) is an example of a -minimal surface.

1.3. Statement of shrinkwrapping theorem

Definition 1.9 (-incompressibility).

An embedded surface in a -manifold disjoint from a collection of simple closed curves is said to be -incompressible rel. if any essential compressing disk for must intersect in at least two points. If is understood, we say is -incompressible.

Theorem 1.10 (Existence of shrinkwrapped surface).

Let be a complete, orientable, parabolic free hyperbolic -manifold, and let be a finite collection of pairwise disjoint simple closed geodesics in . Furthermore, let be a closed embedded -incompressible surface rel. which is either nonseparating in or separates some component of from another. Then is homotopic to a -minimal surface via a homotopy

such that

(1)

,

(2)

is an embedding disjoint from for ,

(3)

,

(4)

if is any other surface with these properties, then .

We say that is obtained from by shrinkwrapping rel. , or if is understood, is obtained from by shrinkwrapping.

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

Remark 1.11.

In fact, for our applications, the property we want to use of our surface is that we can estimate its diameter (rel. the thin part of ) from its Euler characteristic. This follows from a Gauss–Bonnet estimate and the bounded diameter lemma (Lemma 1.15, to be proved below). In fact, our argument will show directly that the surface satisfies Gauss–Bonnet; the fact that it is is logically superfluous for the purposes of this paper.

1.4. Deforming metrics along geodesics

Definition 1.12 (-separation).

Let be a collection of disjoint simple geodesics in a Riemannian manifold . The collection is -separated if any path with endpoints on and satisfying

is homotopic rel. endpoints into . The supremum of such is called the separation constant of . The collection is weakly -separated if

whenever are distinct components of . The supremum of such is called the weak separation constant of .

Definition 1.13 (Neighborhood and tube neighborhood).

Let be given. For a point , we let denote the closed ball of radius about , and let denote, respectively, the interior and the boundary of . For a closed geodesic in , we let denote the closed tube of radius about , and let denote, respectively, the interior and the boundary of . If denotes a union of geodesics , then we use the shorthand notation

Remark 1.14.

Topologically, is a sphere and is a torus, for sufficiently small . Similarly, is a closed ball, and is a closed solid torus. If is -separated, then is a union of solid tori.

Lemma 1.15 (Bounded Diameter Lemma).

Let be a complete hyperbolic -manifold. Let be a disjoint collection of -separated embedded geodesics. Let be a Margulis constant for dimension , and let denote the subset of where the injectivity radius is at most . If is a -incompressible -minimal surface, then there is a constant and such that for each component of , we have

Furthermore, can only intersect at most components of .

Proof.

Since is -incompressible, any point either lies in or is the center of an embedded -disk in , where

Since is , Gauss–Bonnet implies that the area of an embedded -disk in has area at least .

This implies that if , then

The proof now follows by a standard covering argument.

A surface satisfying the conclusion of the Bounded Diameter Lemma is sometimes said to have diameter bounded by modulo .

Remark 1.16.

Note that if is a Margulis constant, then consists of Margulis tubes and cusps. Note that the same argument shows that, away from the thin part of and an -neighborhood of , the diameter of can be bounded by a constant depending only on and .

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 , subject to the constraint that the track of this isotopy does not cross . Unfortunately, is not complete, so the prospects for doing minimal surface theory in this manifold are remote. To remedy this, we deform the metric on in a neighborhood of in such a way that we can guarantee the existence of a least area surface representative with respect to the deformed metric and then take a limit of such surfaces under a sequence of smaller and smaller such metric deformations. We describe the deformations of interest below.

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 , and which is totally Euclidean in a neighborhood of , and secondly a deformation analogous to the kind described below in Definition 1.17, which is supported in this totally Euclidean neighborhood. Since the reason for this “double perturbation” will not be apparent until §1.8, we postpone discussion of such deformations until that time.

Definition 1.17 (Deforming metrics).

Let be such that is -separated. Choose some small with . For we define a family of Riemannian metrics on in the following manner. The metrics agree with the hyperbolic metric away from some fixed tubular neighborhood .

Let

be the function whose value at a point is the hyperbolic distance from to . We define a metric on which agrees with the hyperbolic metric outside , and on is conformally equivalent to the hyperbolic metric, as follows. Let be a bump function, which is equal to on the interval , which is equal to on the intervals and , and which is strictly increasing on and strictly decreasing on . Then define the ratio

We are really only interested in the behaviour of the metrics as . As such, the choice of is irrelevant. However, for convenience, we will fix some small throughout the remainder of §1.

The deformed metrics have the following properties:

Lemma 1.18 (Metric properties).

The metric satisfies the following properties:

(1)

For each there is an satisfying such that the union of the tori is totally geodesic for the metric.

(2)

For each component and each , the metric restricted to admits a family of isometries which preserve and acts transitively on the unit normal bundle (in ) to .

(3)

The area of a disk cross section on is .

(4)

The metric dominates the hyperbolic metric on -planes. That is, for all -vectors , the area of is at least as large as the hyperbolic area of .

Proof.

Statement (2) follows from the fact that the definition of has the desired symmetries. Statements (3) and (4) follow from the fact that the ratio of the metric to the hyperbolic metric is pinched between and . Now, a radially symmetric circle linking of radius has length in the hyperbolic metric, and therefore has length

in the metric. For sufficiently small (but fixed) , this function of has a local minimum on the interval . It follows that the family of radially symmetric tori linking a component of has a local minimum for area in the interval . By property (2), such a torus must be totally geodesic for the metric.

Notation 1.19.

We denote length of an arc with respect to the metric as , and area of a surface with respect to the metric as .

1.5. Constructing the homotopy

As a first approximation, we wish to construct surfaces in which are globally least area with respect to the metric. There are various tools for constructing least area surfaces in Riemannian -manifolds under various conditions, and subject to various constraints. Typically one works in closed -manifolds, but if one wants to work in -manifolds with boundary, the “correct” boundary condition to impose is mean convexity. A co-oriented surface in a Riemannian -manifold is said to be 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 is a mean convex surface and is a minimal surface. Suppose furthermore that is on the negative side of . Then if and are tangent, they are equal. One should stress that this barrier property is local. See Reference MSY for a more thorough discussion of barrier surfaces.

Lemma 1.20 (Minimal surface exists).

Let be as in the statement of Theorem 1.10. Let be as in Lemma 1.18, so that is totally geodesic with respect to the metric. Then for each , there exists an embedded surface isotopic in to , and which is globally -least area among all such surfaces.

Proof.

Note that with respect to the metrics, the surfaces described in Lemma 1.18 are totally geodesic and therefore act as barrier surfaces. We remove the tubular neighborhoods of bounded by these totally geodesic surfaces and denote the result by throughout the remainder of this proof. We assume, after a small isotopy if necessary, that does not intersect for any , and therefore we can (and do) think of as a surface in . Notice that is a complete Riemannian manifold with totally geodesic boundary. We will construct the surface in , in the same isotopy class as (also in ).

If there exists a lower bound on the injectivity radius in with respect to the metric, then the main theorem of Reference MSY implies that either such a globally least area surface can be found, or is the boundary of a twisted -bundle over a closed surface in , or else can be homotoped off every compact set in .

First we show that these last two possibilities cannot occur. If is nonseparating in , then it intersects some essential loop with algebraic intersection number . It follows that cannot be homotoped off and does not bound an -bundle. Similarly, if are distinct geodesics of separated from each other by , then the ’s can be joined by an arc which has algebraic intersection number with the surface . The same is true of any homotopic to ; it follows that cannot be homotoped off the arc , nor does it bound an -bundle disjoint from , and therefore does not bound an -bundle in .

Now suppose that the injectivity radius on is not bounded below. We use the following trick. Let be obtained from the metric by perturbing it on the complement of some enormous compact region so that it has a flaring end there, and such that there is a barrier -minimal surface close to , separating the complement of in from . Then by Reference MSY there is a globally