American Mathematical Society

Shrinkwrapping and the taming of hyperbolic 3-manifolds

By Danny Calegari and David Gabai

Abstract

We introduce a new technique for finding CAT left-parenthesis negative 1 right-parenthesis 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 3 -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 3 -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 upper N equals double-struck upper H cubed slash normal upper Gamma where normal upper Gamma is parabolic free. Precise statements of the parabolic case will be given in §7.

Thurston’s idea was formalized by Bonahon ReferenceBo and Canary ReferenceCa with the following.

Definition 0.1

An end script upper E of a hyperbolic 3 -manifold upper N is simply degenerate if it has a closed neighborhood of the form upper S times left-bracket 0 comma normal infinity right-parenthesis where upper S is a closed surface, and there exists a sequence StartSet upper S Subscript i Baseline EndSet of CAT left-parenthesis negative 1 right-parenthesis surfaces exiting script upper E which are homotopic to upper S times 0 in script upper E . This means that there exists a sequence of maps f Subscript i Baseline colon upper S right-arrow upper N such that the induced path metrics induce CAT left-parenthesis negative 1 right-parenthesis structures on the upper S Subscript i ’s, f left-parenthesis upper S Subscript i Baseline right-parenthesis subset-of upper S times left-bracket i comma normal infinity right-parenthesis and f is homotopic to a homeomorphism onto upper S times 0 via a homotopy supported in upper S times left-bracket 0 comma normal infinity right-parenthesis .

Here by CAT left-parenthesis negative 1 right-parenthesis , 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 f Subscript i are smooth, this is equivalent to the condition that the Riemannian curvature is bounded above by negative 1 . See ReferenceBH for a reference. Note that by Gauss–Bonnet, the area of a CAT left-parenthesis negative 1 right-parenthesis 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 ReferenceBo 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 script upper E of a complete hyperbolic 3 -manifold upper N with finitely generated fundamental group is simply degenerate if there exists a sequence of closed geodesics exiting script upper E .

Consequently we have:

Theorem 0.3

Let upper N be a complete hyperbolic 3 -manifold with finitely generated fundamental group. Then every end of upper N is geometrically tame; i.e., it is either geometrically finite or simply degenerate.

In 1974 Marden ReferenceMa showed that a geometrically finite hyperbolic 3-manifold is topologically tame, i.e., is the interior of a compact 3 -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 upper N is a complete hyperbolic 3 -manifold with finitely generated fundamental group, then upper N is topologically tame.

Ian Agol ReferenceAg has independently proven Theorem 0.4.

There have been many important steps towards Theorem 0.2. The seminal result was obtained by Thurston (ReferenceT, Theorem 9.2) who proved Theorems 0.3 and 0.4 for certain algebraic limits of quasi-Fuchsian groups. Bonahon ReferenceBo established Theorems 0.2 and 0.4 when pi 1 left-parenthesis upper N right-parenthesis is freely indecomposable, and Canary ReferenceCa proved that topological tameness implies geometrical tameness. Results in the direction of 0.4 were also obtained by Canary–Minsky ReferenceCaM, Kleineidam–Souto ReferenceKS, Evans ReferenceEv, Brock–Bromberg–Evans–Souto ReferenceBBES, Ohshika ReferenceOh, Brock–Souto ReferenceBS and Souto ReferenceSo.

Thurston first discovered how to obtain analytic conclusions from the existence of exiting sequences of CAT left-parenthesis negative 1 right-parenthesis surfaces. Thurston’s work as generalized by Bonahon ReferenceBo and Canary ReferenceCa combined with Theorem 0.2 yields a positive proof of the Ahlfors’ Measure Conjecture ReferenceA2.

Theorem 0.5

If normal upper Gamma is a finitely generated Kleinian group, then the limit set upper L Subscript normal upper Gamma is either upper S Subscript normal infinity Superscript 2 or has Lebesgue measure zero. If upper L Subscript normal upper Gamma Baseline equals upper S Subscript normal infinity Superscript 2 , then normal upper Gamma acts ergodically on upper S Subscript normal infinity Superscript 2 .

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 upper N with finitely generated fundamental group is analytically tame as defined by Canary ReferenceCa. It follows from Canary that the various results of ReferenceCa, §9 hold for upper N .

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 ReferenceMi and ReferenceBCM.

Theorem 0.6 (Classification Theorem).

If upper N is a complete hyperbolic 3 -manifold with finitely generated fundamental group, then upper N 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 upper N equals double-struck upper H cubed slash normal upper Gamma is a complete hyperbolic 3 -manifold with finitely generated fundamental group, then normal upper Gamma is the algebraic limit of geometrically finite Kleinian groups.

The main technical innovation of this paper is a new technique called shrinkwrapping for producing CAT left-parenthesis negative 1 right-parenthesis surfaces in hyperbolic 3-manifolds. Historically, such surfaces have been immensely important in the study of hyperbolic 3-manifolds; e.g., see ReferenceT, ReferenceBo, ReferenceCa and ReferenceCaM.

Given a locally finite set normal upper Delta of pairwise disjoint simple closed curves in the 3-manifold upper N , we say that the embedded surface upper S subset-of upper N is 2 -incompressible rel. normal upper Delta if every compressing disc for upper S meets normal upper Delta at least twice. Here is a sample theorem.

Theorem 0.8 (Existence of shrinkwrapped surface).

Let upper M be a complete, orientable, parabolic free hyperbolic 3 -manifold, and let normal upper Gamma be a finite collection of pairwise disjoint simple closed geodesics in upper M . Furthermore, let upper S subset-of upper M minus normal upper Gamma be a closed embedded 2 -incompressible surface rel. normal upper Gamma which is either nonseparating in upper M or separates some component of normal upper Gamma from another. Then upper S is homotopic to a CAT left-parenthesis negative 1 right-parenthesis surface upper T via a homotopy

upper F colon upper S times left-bracket 0 comma 1 right-bracket right-arrow upper M

such that

(1)

upper F left-parenthesis upper S times 0 right-parenthesis equals upper S ,

(2)

upper F left-parenthesis upper S times t right-parenthesis equals upper S Subscript t is an embedding disjoint from normal upper Gamma for 0 less-than-or-equal-to t less-than 1 ,

(3)

upper F left-parenthesis upper S times 1 right-parenthesis equals upper T ,

(4)

If upper T prime is any other surface with these properties, then area left-parenthesis upper T right-parenthesis less-than-or-equal-to area left-parenthesis upper T prime right-parenthesis .

We say that upper T is obtained from upper S by shrinkwrapping rel. normal upper Gamma , or if normal upper Gamma is understood, upper T is obtained from upper S by shrinkwrapping.

In fact, we prove the stronger result that upper T is normal upper Gamma -minimal (to be defined in §1), which implies in particular that it is intrinsically CAT left-parenthesis negative 1 right-parenthesis .

Here is the main technical result of this paper.

Theorem 0.9

Let script upper E be an end of the complete orientable hyperbolic 3 -manifold upper N with finitely generated fundamental group. Let upper C be a 3 -dimensional compact core of upper N , partial-differential Subscript script upper E Baseline upper C the component of partial-differential upper C facing script upper E and g equals g e n u s left-parenthesis partial-differential Subscript script upper E Baseline upper C right-parenthesis . If there exists a sequence of closed geodesics exiting script upper E , then there exists a sequence StartSet upper S Subscript i Baseline EndSet of CAT left-parenthesis negative 1 right-parenthesis surfaces of genus g exiting script upper E such that each upper S Subscript i is homologically separating in script upper E . That is, each upper S Subscript i homologically separates partial-differential Subscript script upper E Baseline upper C from script upper E .

Theorem 0.4 can now be deduced from Theorem 0.9 and Souto ReferenceSo; 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 CAT left-parenthesis negative 1 right-parenthesis surfaces in hyperbolic 3-manifolds. In §3 we prove the existence of epsilon -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 upper X subset-of upper Y , then upper N left-parenthesis upper X right-parenthesis denotes a regular neighborhood of upper X in upper Y and int left-parenthesis upper X right-parenthesis denotes the interior of upper X . If upper X is a topological space, then StartAbsoluteValue upper X EndAbsoluteValue denotes the number of components of upper X . If upper A comma upper B are topological subspaces of a third space, then upper A minus upper B denotes the intersection of upper A with the complement of upper B .

1. Shrinkwrapping

In this section, we introduce a new technical tool for finding CAT left-parenthesis negative 1 right-parenthesis surfaces in hyperbolic 3 -manifolds, called shrinkwrapping. Roughly speaking, given a collection of simple closed geodesics normal upper Gamma in a hyperbolic 3 -manifold upper M and an embedded surface upper S subset-of upper M minus normal upper Gamma , a surface upper T subset-of upper M is obtained from upper S by shrinkwrapping upper S rel. normal upper Gamma if it is homotopic to upper S , can be approximated by an isotopy from upper S supported in upper M minus normal upper Gamma , and is the least area subject to these constraints.

Given mild topological conditions on upper M comma normal upper Gamma comma upper S (namely 2 -incompressibility, to be defined below) the shrinkwrapped surface exists, and is CAT left-parenthesis negative 1 right-parenthesis with respect to the path metric induced by the Riemannian metric on upper M .

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 (ReferenceMSY), existence of limits with area and curvature control (ReferenceCiSc), and regularity of the shrinkwrapped surfaces along normal upper Gamma (ReferenceRi, ReferenceFre). General references are ReferenceCM, ReferenceJs, ReferenceFed and ReferenceB.

1.1. Geometry of surfaces

For convenience, we state some elementary but fundamental lemmas concerning curvature of (smooth) surfaces in Riemannian 3 -manifolds.

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

Definition 1.1

A smooth surface normal upper Sigma in a Riemannian 3 -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 ReferenceCM, 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 ReferenceCM.

Lemma 1.2 (Monotonicity of curvature).

Let normal upper Sigma be a minimal surface in a Riemannian manifold upper M . Let upper K Subscript normal upper Sigma denote the curvature of normal upper Sigma , and upper K Subscript upper M the sectional curvature of upper M . Then restricted to the tangent space upper T normal upper Sigma ,

upper K Subscript normal upper Sigma Baseline equals upper K Subscript upper M Baseline minus one-half StartAbsoluteValue upper A EndAbsoluteValue squared comma

where upper A denotes the second fundamental form of normal upper Sigma .

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

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

Lemma 1.3 (Gauss–Bonnet formula).

Let normal upper Sigma be a upper C cubed Riemannian surface with (possibly empty) upper C cubed boundary partial-differential normal upper Sigma . Let upper K Subscript normal upper Sigma denote the Gauss curvature of normal upper Sigma , and kappa the geodesic curvature along partial-differential normal upper Sigma . Then

integral Underscript normal upper Sigma Endscripts upper K Subscript normal upper Sigma Baseline equals 2 pi chi left-parenthesis normal upper Sigma right-parenthesis minus integral Underscript partial-differential normal upper Sigma Endscripts kappa d l period

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

If partial-differential normal upper Sigma is merely piecewise upper C cubed , with finitely many corners p Subscript i and external angles alpha Subscript i , the Gauss–Bonnet formula must be modified as follows:

Lemma 1.4 (Gauss–Bonnet with corners).

Let normal upper Sigma be a upper C cubed Riemannian surface with boundary partial-differential normal upper Sigma which is piecewise upper C cubed and has external angles alpha Subscript i at finitely many points p Subscript i . Let upper K Subscript normal upper Sigma and kappa be as above. Then

integral Underscript normal upper Sigma Endscripts upper K Subscript normal upper Sigma Baseline equals 2 pi chi left-parenthesis normal upper Sigma right-parenthesis minus integral Underscript partial-differential normal upper Sigma Endscripts kappa d l minus sigma-summation Underscript i Endscripts alpha Subscript i Baseline period

Observe for a b c a geodesic triangle with external angles alpha 1 comma alpha 2 comma alpha 3 that Lemma 1.4 implies

integral Underscript a b c Endscripts upper K equals 2 pi minus sigma-summation Underscript i Endscripts alpha Subscript i Baseline period

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

phi colon partial-differential normal upper Sigma times left-bracket 0 comma epsilon right-bracket right-arrow normal upper Sigma

for small epsilon , where phi left-parenthesis dot comma 0 right-parenthesis equals Id vertical-bar Subscript partial-differential normal upper Sigma Baseline , and phi left-parenthesis partial-differential normal upper Sigma comma t right-parenthesis for small t is the boundary in normal upper Sigma of the tubular t neighborhood of partial-differential normal upper Sigma . Then

integral Underscript partial-differential normal upper Sigma Endscripts kappa d l equals minus StartFraction d Over d t EndFraction vertical-bar Subscript t equals 0 Baseline l e n g t h left-parenthesis phi Subscript t Baseline left-parenthesis partial-differential normal upper Sigma right-parenthesis right-parenthesis period

Note that if normal upper Sigma is a surface with sectional curvature bounded above by negative 1 , then by integrating this formula we see that the ball upper B Subscript t Baseline left-parenthesis p right-parenthesis of radius t in normal upper Sigma about a point p element-of normal upper Sigma satisfies

a r e a left-parenthesis upper B Subscript t Baseline left-parenthesis p right-parenthesis right-parenthesis greater-than-or-equal-to 2 pi left-parenthesis hyperbolic cosine left-parenthesis t right-parenthesis minus 1 right-parenthesis greater-than pi t squared

for small t greater-than 0 .

1.2. Comparison geometry

For basic elements of the theory of comparison geometry, see ReferenceBH.

Definition 1.5 (Comparison triangle).

Let a 1 a 2 a 3 be a geodesic triangle in a geodesic metric space upper X . Let kappa element-of double-struck upper R be given. A kappa -comparison triangle is a geodesic triangle a 1 overbar a 2 overbar a 3 overbar in the complete simply-connected Riemannian 2 -manifold of constant sectional curvature kappa , where the edges a Subscript i Baseline a Subscript j and a Subscript i Baseline overbar a Subscript j Baseline overbar satisfy

l e n g t h left-parenthesis a Subscript i Baseline a Subscript j Baseline right-parenthesis equals l e n g t h left-parenthesis a Subscript i Baseline overbar a Subscript j Baseline overbar right-parenthesis period

Given a point x element-of a 1 a 2 on one of the edges of a 1 a 2 a 3 , there is a corresponding point x overbar element-of a 1 overbar a 2 overbar on one of the edges of the comparison triangle, satisfying

l e n g t h left-parenthesis a 1 x right-parenthesis equals l e n g t h left-parenthesis a 1 overbar x overbar right-parenthesis

and

l e n g t h left-parenthesis x a 2 right-parenthesis equals l e n g t h left-parenthesis x overbar a 2 overbar right-parenthesis period

Remark 1.6

Note that if kappa greater-than 0 , the comparison triangle might not exist if the edge lengths are too big, but if kappa less-than-or-equal-to 0 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 3 geodesic edges. In a path metric space, a triangle is just a union of 3 geodesic segments with common endpoints.

Definition 1.7 (CAT( kappa )).

Let upper S be a closed surface with a path metric g . Let upper S overTilde denote the universal cover of upper S , with path metric induced by the pullback of the path metric g . Let kappa element-of double-struck upper R be given. upper S is said to be CAT left-parenthesis kappa right-parenthesis if for every geodesic triangle a b c in upper S overTilde , and every point z on the edge b c , the distance in upper S overTilde from a to z is no more than the distance from a overbar to z overbar in a kappa -comparison triangle.

By Lemma 1.4 applied to geodesic triangles, one can show that a upper C cubed surface normal upper Sigma with sectional curvature upper K Subscript normal upper Sigma satisfying upper K Subscript normal upper Sigma Baseline less-than-or-equal-to kappa everywhere is CAT left-parenthesis kappa right-parenthesis with respect to the Riemannian path metric. This fact is essentially due to Alexandrov; see ReferenceB for a proof.

More generally, suppose normal upper Sigma is a surface which is upper C cubed outside a closed, nowhere dense subset upper X subset-of normal upper Sigma . Furthermore, suppose that upper K Subscript normal upper Sigma Baseline less-than-or-equal-to kappa holds in normal upper Sigma minus upper X , and suppose that the formula from Lemma 1.4 holds for every geodesic triangle with vertices in normal upper Sigma minus upper X (which is a dense set of geodesic triangles). Then the same argument shows that normal upper Sigma is CAT left-parenthesis kappa right-parenthesis . See, e.g., ReferenceRe, §8, pp. 135–140 for more details and a general discussion of metric surfaces with (integral) curvature bounds.

Definition 1.8 ( normal upper Gamma -minimal surfaces).

Let kappa element-of double-struck upper R be given. Let upper M be a complete Riemannian 3 -manifold with sectional curvature bounded above by kappa , and let normal upper Gamma be an embedded collection of simple closed geodesics in upper M . An immersion

psi colon upper S right-arrow upper M

is normal upper Gamma -minimal if it is smooth with mean curvature 0 in upper M minus normal upper Gamma and is metrically CAT left-parenthesis kappa right-parenthesis with respect to the path metric induced by psi from the Riemannian metric on upper M .

Notice by Lemma 1.2 that a smooth surface upper S with mean curvature 0 in upper M is CAT left-parenthesis kappa right-parenthesis , so a minimal surface (in the usual sense) is an example of a normal upper Gamma -minimal surface.

1.3. Statement of shrinkwrapping theorem

Definition 1.9 ( 2 -incompressibility).

An embedded surface upper S in a 3 -manifold upper M disjoint from a collection normal upper Gamma of simple closed curves is said to be 2 -incompressible rel. normal upper Gamma if any essential compressing disk for upper S must intersect normal upper Gamma in at least two points. If normal upper Gamma is understood, we say upper S is 2 -incompressible.

Theorem 1.10 (Existence of shrinkwrapped surface).

Let upper M be a complete, orientable, parabolic free hyperbolic 3 -manifold, and let normal upper Gamma be a finite collection of pairwise disjoint simple closed geodesics in upper M . Furthermore, let upper S subset-of upper M minus normal upper Gamma be a closed embedded 2 -incompressible surface rel. normal upper Gamma which is either nonseparating in upper M or separates some component of normal upper Gamma from another. Then upper S is homotopic to a normal upper Gamma -minimal surface upper T via a homotopy

upper F colon upper S times left-bracket 0 comma 1 right-bracket right-arrow upper M

such that

(1)

upper F left-parenthesis upper S times 0 right-parenthesis equals upper S ,

(2)

upper F left-parenthesis upper S times t right-parenthesis equals upper S Subscript t is an embedding disjoint from normal upper Gamma for 0 less-than-or-equal-to t less-than 1 ,

(3)

upper F left-parenthesis upper S times 1 right-parenthesis equals upper T ,

(4)

if upper T prime is any other surface with these properties, then area left-parenthesis upper T right-parenthesis less-than-or-equal-to area left-parenthesis upper T prime right-parenthesis .

We say that upper T is obtained from upper S by shrinkwrapping rel. normal upper Gamma , or if normal upper Gamma is understood, upper T is obtained from upper S 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 upper T is that we can estimate its diameter (rel. the thin part of upper M ) 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 upper T satisfies Gauss–Bonnet; the fact that it is CAT left-parenthesis negative 1 right-parenthesis is logically superfluous for the purposes of this paper.

1.4. Deforming metrics along geodesics

Definition 1.12 ( delta -separation).

Let normal upper Gamma be a collection of disjoint simple geodesics in a Riemannian manifold upper M . The collection normal upper Gamma is delta -separated if any path alpha colon upper I right-arrow upper M with endpoints on normal upper Gamma and satisfying

l e n g t h left-parenthesis alpha left-parenthesis upper I right-parenthesis right-parenthesis less-than-or-equal-to delta

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

dist left-parenthesis gamma comma gamma Superscript prime Baseline right-parenthesis greater-than delta

whenever gamma comma gamma prime are distinct components of normal upper Gamma . The supremum of such delta is called the weak separation constant of normal upper Gamma .

Definition 1.13 (Neighborhood and tube neighborhood).

Let r greater-than 0 be given. For a point x element-of upper M , we let upper N Subscript r Baseline left-parenthesis x right-parenthesis denote the closed ball of radius r about x , and let upper N Subscript r Baseline left-parenthesis x right-parenthesis comma partial-differential upper N Subscript r Baseline left-parenthesis x right-parenthesis denote, respectively, the interior and the boundary of upper N Subscript r Baseline left-parenthesis x right-parenthesis . For a closed geodesic gamma in upper M , we let upper N Subscript r Baseline left-parenthesis gamma right-parenthesis denote the closed tube of radius r about gamma , and let upper N Subscript r Baseline left-parenthesis gamma right-parenthesis comma partial-differential upper N Subscript r Baseline left-parenthesis gamma right-parenthesis denote, respectively, the interior and the boundary of upper N Subscript r Baseline left-parenthesis gamma right-parenthesis . If normal upper Gamma denotes a union of geodesics gamma Subscript i , then we use the shorthand notation

upper N Subscript r Baseline left-parenthesis normal upper Gamma right-parenthesis equals union Underscript gamma Subscript i Baseline Endscripts upper N Subscript r Baseline left-parenthesis gamma Subscript i Baseline right-parenthesis period

Remark 1.14

Topologically, partial-differential upper N Subscript r Baseline left-parenthesis x right-parenthesis is a sphere and partial-differential upper N Subscript r Baseline left-parenthesis gamma right-parenthesis is a torus, for sufficiently small r . Similarly, upper N Subscript r Baseline left-parenthesis x right-parenthesis is a closed ball, and upper N Subscript r Baseline left-parenthesis gamma right-parenthesis is a closed solid torus. If normal upper Gamma is delta -separated, then upper N Subscript delta slash 2 Baseline left-parenthesis normal upper Gamma right-parenthesis is a union of solid tori.

Lemma 1.15 (Bounded Diameter Lemma).

Let upper M be a complete hyperbolic 3 -manifold. Let normal upper Gamma be a disjoint collection of delta -separated embedded geodesics. Let epsilon greater-than 0 be a Margulis constant for dimension 3 , and let upper M Subscript epsilon denote the subset of upper M where the injectivity radius is at most epsilon . If upper S subset-of upper M minus normal upper Gamma is a 2 -incompressible normal upper Gamma -minimal surface, then there is a constant upper C equals upper C left-parenthesis chi left-parenthesis upper S right-parenthesis comma epsilon comma delta right-parenthesis element-of double-struck upper R and n equals n left-parenthesis chi left-parenthesis upper S right-parenthesis comma epsilon comma delta right-parenthesis element-of double-struck upper Z such that for each component upper S Subscript i of upper S intersection left-parenthesis upper M minus upper M Subscript epsilon Baseline right-parenthesis , we have

diam left-parenthesis upper S Subscript i Baseline right-parenthesis less-than-or-equal-to upper C period

Furthermore, upper S can only intersect at most n components of upper M Subscript epsilon .

Proof.

Since upper S is 2 -incompressible, any point x element-of upper S either lies in upper M Subscript epsilon or is the center of an embedded m -disk in upper S , where

m equals min left-parenthesis epsilon slash 2 comma delta slash 2 right-parenthesis period

Since upper S is CAT left-parenthesis negative 1 right-parenthesis , Gauss–Bonnet implies that the area of an embedded m -disk in upper S has area at least 2 pi left-parenthesis hyperbolic cosine left-parenthesis m right-parenthesis minus 1 right-parenthesis greater-than pi m squared .

This implies that if x element-of upper S intersection upper M minus upper M Subscript epsilon , then

a r e a left-parenthesis upper S intersection upper N Subscript m Baseline left-parenthesis x right-parenthesis right-parenthesis greater-than-or-equal-to pi m squared period

The proof now follows by a standard covering argument.

A surface upper S satisfying the conclusion of the Bounded Diameter Lemma is sometimes said to have diameter bounded by upper C modulo upper M Subscript epsilon .

Remark 1.16

Note that if epsilon is a Margulis constant, then upper M Subscript epsilon consists of Margulis tubes and cusps. Note that the same argument shows that, away from the thin part of upper M and an epsilon -neighborhood of normal upper Gamma , the diameter of upper S can be bounded by a constant depending only on chi left-parenthesis upper S right-parenthesis and epsilon .

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 upper S , subject to the constraint that the track of this isotopy does not cross normal upper Gamma . Unfortunately, upper M minus normal upper Gamma is not complete, so the prospects for doing minimal surface theory in this manifold are remote. To remedy this, we deform the metric on upper M in a neighborhood of normal upper Gamma 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 negative 1 less-than-or-equal-to upper K less-than-or-equal-to 0 , and which is totally Euclidean in a neighborhood of normal upper Gamma , 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 delta greater-than 0 be such that normal upper Gamma is delta -separated. Choose some small r with r less-than delta slash 2 . For t element-of left-bracket 0 comma 1 right-parenthesis we define a family of Riemannian metrics g Subscript t on upper M in the following manner. The metrics g Subscript t agree with the hyperbolic metric away from some fixed tubular neighborhood upper N Subscript r Baseline left-parenthesis normal upper Gamma right-parenthesis .

Let

h colon upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis right-arrow left-bracket 0 comma r left-parenthesis 1 minus t right-parenthesis right-bracket

be the function whose value at a point p is the hyperbolic distance from p to normal upper Gamma . We define a metric g Subscript t on upper M which agrees with the hyperbolic metric outside upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis , and on upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis is conformally equivalent to the hyperbolic metric, as follows. Let phi colon left-bracket 0 comma 1 right-bracket right-arrow left-bracket 0 comma 1 right-bracket be a upper C Superscript normal infinity bump function, which is equal to 1 on the interval left-bracket 1 slash 3 comma 2 slash 3 right-bracket , which is equal to 0 on the intervals left-bracket 0 comma 1 slash 4 right-bracket and left-bracket 3 slash 4 comma 1 right-bracket , and which is strictly increasing on left-bracket 1 slash 4 comma 1 slash 3 right-bracket and strictly decreasing on left-bracket 2 slash 3 comma 3 slash 4 right-bracket . Then define the ratio

StartFraction g Subscript t Baseline length element Over hyperbolic length element EndFraction equals 1 plus 2 phi left-parenthesis StartFraction h left-parenthesis p right-parenthesis Over r left-parenthesis 1 minus t right-parenthesis EndFraction right-parenthesis period

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

The deformed metrics g Subscript t have the following properties:

Lemma 1.18 (Metric properties).

The g Subscript t metric satisfies the following properties:

(1)

For each t there is an f left-parenthesis t right-parenthesis satisfying r left-parenthesis 1 minus t right-parenthesis slash 4 less-than f left-parenthesis t right-parenthesis less-than 3 r left-parenthesis 1 minus t right-parenthesis slash 4 such that the union of the tori partial-differential upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis is totally geodesic for the g Subscript t metric.

(2)

For each component gamma Subscript i and each t , the metric g Subscript t restricted to upper N Subscript r Baseline left-parenthesis gamma Subscript i Baseline right-parenthesis admits a family of isometries which preserve gamma Subscript i and acts transitively on the unit normal bundle (in upper M ) to gamma Subscript i .

(3)

The area of a disk cross section on upper N Subscript r left-parenthesis 1 minus t right-parenthesis is upper O left-parenthesis left-parenthesis 1 minus t right-parenthesis squared right-parenthesis .

(4)

The metric g Subscript t dominates the hyperbolic metric on 2 -planes. That is, for all 2 -vectors nu , the g Subscript t area of nu is at least as large as the hyperbolic area of nu .

Proof.

Statement (2) follows from the fact that the definition of g Subscript t has the desired symmetries. Statements (3) and (4) follow from the fact that the ratio of the g Subscript t metric to the hyperbolic metric is pinched between 1 and 3 . Now, a radially symmetric circle linking normal upper Gamma of radius s has length 2 pi hyperbolic cosine left-parenthesis s right-parenthesis in the hyperbolic metric, and therefore has length

2 pi hyperbolic cosine left-parenthesis s right-parenthesis left-parenthesis 1 plus 2 phi left-parenthesis StartFraction s Over r left-parenthesis 1 minus t right-parenthesis EndFraction right-parenthesis right-parenthesis

in the g Subscript t metric. For sufficiently small (but fixed) r , this function of s has a local minimum on the interval left-bracket r left-parenthesis 1 minus t right-parenthesis slash 4 comma 3 r left-parenthesis 1 minus t right-parenthesis slash 4 right-bracket . It follows that the family of radially symmetric tori linking a component of normal upper Gamma has a local minimum for area in the interval left-bracket r left-parenthesis 1 minus t right-parenthesis slash 4 comma 3 r left-parenthesis 1 minus t right-parenthesis slash 4 right-bracket . By property (2), such a torus must be totally geodesic for the g Subscript t metric.

Notation 1.19

We denote length of an arc alpha colon upper I right-arrow upper M with respect to the g Subscript t metric as length Subscript t Baseline left-parenthesis alpha left-parenthesis upper I right-parenthesis right-parenthesis , and area of a surface psi colon upper R right-arrow upper M with respect to the g Subscript t metric as area Subscript t Baseline left-parenthesis psi left-parenthesis upper R right-parenthesis right-parenthesis .

1.5. Constructing the homotopy

As a first approximation, we wish to construct surfaces in upper M minus normal upper Gamma which are globally least area with respect to the g Subscript t metric. There are various tools for constructing least area surfaces in Riemannian 3 -manifolds under various conditions, and subject to various constraints. Typically one works in closed 3 -manifolds, but if one wants to work in 3 -manifolds with boundary, the “correct” boundary condition to impose is mean convexity. A co-oriented surface in a Riemannian 3 -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 upper S 1 is a mean convex surface and upper S 2 is a minimal surface. Suppose furthermore that upper S 2 is on the negative side of upper S 1 . Then if upper S 2 and upper S 1 are tangent, they are equal. One should stress that this barrier property is local. See ReferenceMSY for a more thorough discussion of barrier surfaces.

Lemma 1.20 (Minimal surface exists).

Let upper M comma normal upper Gamma comma upper S be as in the statement of Theorem 1.10. Let f left-parenthesis t right-parenthesis be as in Lemma 1.18, so that partial-differential upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis is totally geodesic with respect to the g Subscript t metric. Then for each t , there exists an embedded surface upper S Subscript t isotopic in upper M minus upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis to upper S , and which is globally g Subscript t -least area among all such surfaces.

Proof.

Note that with respect to the g Subscript t metrics, the surfaces partial-differential upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis described in Lemma 1.18 are totally geodesic and therefore act as barrier surfaces. We remove the tubular neighborhoods of normal upper Gamma bounded by these totally geodesic surfaces and denote the result upper M minus upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis by upper M prime throughout the remainder of this proof. We assume, after a small isotopy if necessary, that upper S does not intersect upper N Subscript f left-parenthesis t right-parenthesis for any t , and therefore we can (and do) think of upper S as a surface in upper M prime . Notice that upper M prime is a complete Riemannian manifold with totally geodesic boundary. We will construct the surface upper S Subscript t in upper M prime , in the same isotopy class as upper S (also in upper M prime ).

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

First we show that these last two possibilities cannot occur. If upper S is nonseparating in upper M , then it intersects some essential loop beta with algebraic intersection number 1 . It follows that upper S cannot be homotoped off beta and does not bound an upper I -bundle. Similarly, if gamma 1 comma gamma 2 are distinct geodesics of normal upper Gamma separated from each other by upper S , then the gamma Subscript i ’s can be joined by an arc alpha which has algebraic intersection number 1 with the surface upper S . The same is true of any upper S prime homotopic to upper S ; it follows that upper S cannot be homotoped off the arc alpha , nor does it bound an upper I -bundle disjoint from normal upper Gamma , and therefore does not bound an upper I -bundle in upper M prime .

Now suppose that the injectivity radius on upper M prime is not bounded below. We use the following trick. Let g prime Subscript t be obtained from the metric g Subscript t by perturbing it on the complement of some enormous compact region upper E so that it has a flaring end there, and such that there is a barrier g prime Subscript t -minimal surface close to partial-differential upper E , separating the complement of upper E in upper M prime from upper S . Then by ReferenceMSY there is a globally g prime Subscript t least area surface upper S prime Subscript t , contained in the compact subset of upper M prime bounded by this barrier surface. Since upper S prime Subscript t must either intersect beta or alpha , by the Bounded Diameter Lemma 1.15, unless the hyperbolic area of upper S prime Subscript t Baseline intersection upper E is very large, the diameter of upper S prime Subscript t in upper E is much smaller than the distance from alpha or beta to partial-differential upper E . Since by hypothesis, upper S prime Subscript t is the least area for the g prime Subscript t metric, its restriction to upper E has hyperbolic area less than the hyperbolic area of upper S , and therefore there is an a priori upper bound on its diameter in upper E . By choosing upper E large enough, we see that upper S prime Subscript t is contained in the interior of upper E , where g Subscript t and g prime Subscript t agree. Thus upper S prime Subscript t is the globally least area for the g Subscript t metric in upper M prime , and therefore upper S Subscript t Baseline equals upper S prime Subscript t exists for any t .

The bounded diameter lemma easily implies the following:

Lemma 1.21 (Compact set).

There is a fixed compact set upper E subset-of upper M such that the surfaces upper S Subscript t constructed in Lemma 1.20 are all contained in upper E .

Proof.

Since the hyperbolic areas of the upper S Subscript t are all uniformly bounded (by e.g. the hyperbolic area of upper S ) and are 2 -incompressible rel. normal upper Gamma , they have uniformly bounded diameter away from normal upper Gamma outside of Margulis tubes. Since for homological reasons they must intersect the compact sets alpha or beta , they can intersect at most finitely many Margulis tubes. It follows that they are all contained in a fixed bounded neighborhood upper E of alpha or beta , containing normal upper Gamma .

To extract good limits of sequences of minimal surfaces, one generally needs a priori bounds on the area and the total curvature of the limiting surfaces. Here for a surface upper S , the total curvature of upper S is just the integral of the absolute value of the (Gauss) curvature over upper S . For minimal surfaces of a fixed topological type in a manifold with sectional curvature bounded above, a curvature bound follows from an area bound by Gauss–Bonnet. However, our surfaces upper S Subscript t are minimal with respect to the g Subscript t metrics, which have no uniform upper bound on their sectional curvature, so we must work slightly harder to show that the upper S Subscript t have uniformly bounded total curvature. More precisely, we show that their restrictions to the complement of any fixed tubular neighborhood upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis have uniformly bounded total curvature.

Lemma 1.22 (Finite total curvature).

Let upper S Subscript t be the surfaces constructed in Lemma 1.20. Fix some small, positive epsilon . Then the subsurfaces

upper S prime Subscript t Baseline colon equals upper S Subscript t intersection upper M minus upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis

have uniformly bounded total curvature.

Proof.

Having chosen epsilon , we choose t large enough so that r left-parenthesis 1 minus t right-parenthesis less-than epsilon slash 2 .

Observe firstly that each upper S Subscript t has g Subscript t area less than the g Subscript t area of upper S , and therefore hyperbolic area less than the hyperbolic area of upper S for sufficiently large t .

Let tau Subscript t comma s Baseline equals upper S Subscript t Baseline intersection partial-differential upper N Subscript s Baseline left-parenthesis normal upper Gamma right-parenthesis for small s . By the coarea formula (see ReferenceFed, ReferenceCM, p. 8) we can estimate

a r e a left-parenthesis upper S Subscript t Baseline intersection left-parenthesis upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis minus upper N Subscript epsilon slash 2 Baseline left-parenthesis normal upper Gamma right-parenthesis right-parenthesis right-parenthesis greater-than-or-equal-to integral Subscript epsilon slash 2 Superscript epsilon Baseline l e n g t h left-parenthesis tau Subscript t comma s Baseline right-parenthesis d s period

If the integral of geodesic curvature along a component sigma of tau Subscript t comma epsilon is large, then the length of the curves obtained by isotoping sigma into upper S Subscript t Baseline intersection upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis grows very rapidly, by the definition of geodesic curvature.

Since there is an a priori bound on the hyperbolic area of upper S Subscript t , it follows that there cannot be any long components of tau Subscript t comma s with big integral geodesic curvature. More precisely, consider a long component sigma of tau Subscript t comma s . For l element-of left-bracket 0 comma epsilon slash 2 right-bracket the boundary sigma Subscript l of the l -neighborhood of sigma in upper S Subscript t Baseline intersection upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis is contained in upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis minus upper N Subscript epsilon slash 2 Baseline left-parenthesis normal upper Gamma right-parenthesis . If the integral of the geodesic curvature along sigma Subscript l were sufficiently large for every l , then the derivative of the length of the sigma Subscript l would be large for every l , and therefore the lengths of the sigma Subscript l would be large for all l element-of left-bracket epsilon slash 4 comma epsilon slash 2 right-bracket . It follows that the hyperbolic area of the epsilon slash 2 collar neighborhood of sigma in upper S Subscript t would be very large, contrary to existence of an a priori upper bound on the total hyperbolic area of upper S Subscript t .

This contradiction implies that for some l , the integral of the geodesic curvature along sigma Subscript l can be bounded from above. To summarize, for each constant upper C 1 greater-than 0 there is a constant upper C 2 greater-than 0 , such that for each component sigma of tau Subscript t comma epsilon which has length upper C 1 there is a loop

sigma prime subset-of upper S Subscript t intersection left-parenthesis upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis minus upper N Subscript epsilon slash 2 Baseline left-parenthesis normal upper Gamma right-parenthesis right-parenthesis

isotopic to sigma by a short isotopy, satisfying

integral Underscript sigma Superscript prime Baseline Endscripts kappa d l less-than-or-equal-to upper C 2 period

On the other hand, since upper S Subscript t is g Subscript t minimal, there is a constant upper C 1 greater-than 0 such that each component sigma of tau Subscript t comma epsilon which has length upper C 1 bounds a hyperbolic globally least area disk which is contained in upper M minus upper N Subscript epsilon slash 2 Baseline left-parenthesis normal upper Gamma right-parenthesis . For t sufficiently close to 1 , such a disk is contained in upper M minus upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis and therefore must actually be a subdisk of upper S Subscript t .

By the coarea formula above, we can choose epsilon so that l e n g t h left-parenthesis tau Subscript t comma s Baseline right-parenthesis is a priori bounded. It follows that if upper S double-prime Subscript t is the subsurface of upper S Subscript t bounded by the components of tau Subscript t comma s of length upper C 1 , then we have a priori upper bounds on the area of upper S double-prime Subscript t , on integral Underscript partial-differential upper S double-prime Subscript t Endscripts kappa d l , and on minus chi left-parenthesis upper S double-prime Subscript t right-parenthesis . Moreover, upper S double-prime Subscript t is contained in upper M minus upper N Subscript r left-parenthesis 1 minus t right-parenthesis , where the metric g Subscript t agrees with the hyperbolic metric, so the curvature upper K of upper S double-prime Subscript t is bounded above by negative 1 pointwise, by Lemma 1.2. By the Gauss–Bonnet formula, this gives an a priori upper bound on the total curvature of upper S double-prime Subscript t and therefore on upper S prime Subscript t Baseline subset-of upper S double-prime Subscript t .

Remark 1.23

A more highbrow proof of Lemma 1.22 follows from Theorem 1 of ReferenceS, using the fact that the surfaces upper S prime Subscript t are locally least area for the hyperbolic metric, for t sufficiently close to 1 (depending on epsilon ).

Lemma 1.24 (Limit exists).

Let upper S Subscript t be the surfaces constructed in Lemma 1.20. Then there is an increasing sequence

0 less-than t 1 less-than t 2 less-than ellipsis

such that limit Underscript i right-arrow normal infinity Endscripts t Subscript i Baseline equals 1 , and the upper S Subscript t Sub Subscript i converge on compact subsets of upper M minus normal upper Gamma in the upper C Superscript normal infinity topology to some upper T prime subset-of upper M minus normal upper Gamma with closure upper T in upper M .

Proof.

By definition, the surfaces upper S Subscript t have g Subscript t area bounded above by the g Subscript t area of upper S . Moreover, since upper S is disjoint from normal upper Gamma , for sufficiently large t , the g Subscript t area of upper S is equal to the hyperbolic area of upper S . Since the g Subscript t area dominates the hyperbolic area, it follows that the upper S Subscript t have hyperbolic area bounded above, and by Lemma 1.22, for any epsilon , the restrictions of upper S Subscript t to upper M minus upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis have uniformly bounded finite total curvature.

Moreover, by Lemma 1.21, each upper S Subscript t is contained in a fixed compact subset of upper M . By standard compactness theorems (see, e.g., ReferenceCiSc) any infinite sequence upper S Subscript t Sub Subscript i contains a subsequence which converges on compact subsets of upper E minus normal upper Gamma , away from finitely many points where some subsurface with nontrivial topology might collapse. That is, there might be isolated points p such that for any neighborhood upper U of p , the intersection of upper S Subscript t Sub Subscript i with upper U contains loops which are essential in upper S Subscript t Sub Subscript i for all sufficiently large i .

But upper S is 2 -incompressible rel. normal upper Gamma , so in particular it is incompressible in upper M minus normal upper Gamma , and no such collapse can take place. So after passing to a subsequence, a limit upper T prime subset-of upper M minus normal upper Gamma exists (compare ReferenceMSY). Since each upper S Subscript t is a globally least area surface in upper M minus upper N Subscript f left-parenthesis t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis with respect to the g Subscript t metric, it is a locally least area surface with respect to the hyperbolic metric on upper M minus upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis . It follows that upper T prime is locally least area in the hyperbolic metric, properly embedded in upper M minus normal upper Gamma , and we can define upper T to be the closure of upper T prime in upper M .

Lemma 1.25 (Interpolating isotopy).

Let StartSet t Subscript i Baseline EndSet be the sequence as in Lemma 1.24. Then after possibly passing to a subsequence, there is an isotopy upper F colon upper S times left-bracket 0 comma 1 right-parenthesis right-arrow upper M minus normal upper Gamma such that

upper F left-parenthesis upper S comma t Subscript i Baseline right-parenthesis equals upper S Subscript t Sub Subscript i

and such that for each p element-of upper S the track of the isotopy upper F left-parenthesis p comma left-bracket 0 comma 1 right-parenthesis right-parenthesis either converges to some well-defined limit upper F left-parenthesis p comma 1 right-parenthesis element-of upper M minus normal upper Gamma or else it is eventually contained in upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis for any epsilon greater-than 0 .

Proof.

Fix some small epsilon . Outside upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis , the surfaces upper S Subscript t Sub Subscript i converge uniformly in the upper C Superscript normal infinity topology to upper T prime . It follows that for any epsilon , and for i sufficiently large (depending on epsilon ), the restrictions of upper S Subscript t Sub Subscript i and upper S Subscript t Sub Subscript i plus 1 to the complement of upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis are both sections of the exponentiated unit normal bundle of upper T prime minus upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis , and therefore we can isotope these subsets of upper S Subscript t Sub Subscript i to upper S Subscript t Sub Subscript i plus 1 along the fibers of the normal bundle. We wish to patch this partial isotopy together with a partial isotopy supported in a small neighborhood of upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis to define the correct isotopy from upper S Subscript t Sub Subscript i to upper S Subscript t Sub Subscript i plus 1 .

Let upper Z be obtained from upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis by isotoping it slightly into upper M minus upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis so that it is transverse to upper T , and therefore also to upper S Subscript t Sub Subscript i for i sufficiently large. For each i , we consider the intersection

tau Subscript i Baseline equals upper S Subscript t Sub Subscript i Baseline intersection partial-differential upper Z

and observe that the limit satisfies

limit Underscript i right-arrow normal infinity Endscripts tau Subscript i Baseline equals tau equals upper T intersection partial-differential upper Z period

Let sigma be a component of tau which is inessential in partial-differential upper Z . Then for large i , sigma can be approximated by sigma Subscript i Baseline subset-of tau Subscript i which are inessential in partial-differential upper Z . Since the upper S Subscript t Sub Subscript i are 2 -incompressible rel. normal upper Gamma , the loops sigma Subscript i must bound subdisks upper D Subscript i of upper S Subscript t Sub Subscript i . Since partial-differential upper Z is a convex surface with respect to the hyperbolic metric, and the g Subscript t metric agrees with the hyperbolic metric outside upper Z for large t , it follows that the disks upper D Subscript i are actually contained in upper Z minus normal upper Gamma for large i . It follows that upper D Subscript i and upper D Subscript i plus 1 are isotopic by an isotopy supported in upper Z minus normal upper Gamma , which restricts to a very small isotopy of sigma Subscript i to sigma Subscript i plus 1 in partial-differential upper Z .

Let sigma be a component of tau which is essential in partial-differential upper Z . Then so is sigma Subscript i for large i . Again, since upper S , and therefore upper S Subscript t Sub Subscript i is 2 -incompressible rel. normal upper Gamma , it follows that sigma Subscript i cannot be a meridian of partial-differential upper Z and must actually be a longitude. It follows that there is another essential curve sigma prime Subscript i in each tau Subscript i , such that the essential curves sigma prime Subscript i and sigma Subscript i cobound a subsurface upper A Subscript i in upper S Subscript t Sub Subscript i Baseline intersection upper Z minus normal upper Gamma . After passing to a diagonal subsequence, we can assume that the sigma prime Subscript i converge to some component sigma prime of tau .

By 2 -incompressibility, the surfaces upper A Subscript i are annuli. Note that there are two relative isotopy classes of such annuli. By passing to a further diagonal subsequence, we can assume upper A Subscript i and upper A Subscript i plus 1 are isotopic in upper Z minus normal upper Gamma by an isotopy which restricts to a very small isotopy of sigma Subscript i Baseline union sigma prime Subscript i to sigma Subscript i plus 1 Baseline union sigma prime Subscript i plus 1 in partial-differential upper Z .

We have shown that for any small epsilon and any sequence upper S Subscript t Sub Subscript i , there is an arbitrarily large index i and infinitely many indices j with i less-than j so that the surfaces upper S Subscript t Sub Subscript i and upper S Subscript t Sub Subscript j are isotopic, and the isotopy can be chosen to have the following properties:

(1)

The isotopy takes upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis intersection upper S Subscript t Sub Subscript i to upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis intersection upper S Subscript t Sub Subscript j by an isotopy supported in upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis .

(2)

Outside upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis , the tracks of the isotopy are contained in fibers of the exponentiated normal bundle of upper T prime minus upper N Subscript epsilon Baseline left-parenthesis normal upper Gamma right-parenthesis .

Choose a sequence epsilon Subscript i Baseline right-arrow 0 , and pick a subsequence of the upper S Subscript t Sub Subscript i ’s and relabel so that upper S Subscript t Sub Subscript i Subscript Baseline comma upper S Subscript t Sub Subscript i plus 1 Subscript Baseline satisfy the properties above with respect to upper N Subscript epsilon Sub Subscript i Baseline left-parenthesis normal upper Gamma right-parenthesis . Then the composition of this infinite sequence of isotopies is upper F .

Remark 1.26

The reason for the circumlocutions in the statement of Lemma 1.25 is that we have not yet proved that upper T is a limit of the upper S Subscript t as maps from upper S to upper M . This will follow in §1.6, where we analyze the structure of upper T near a point p element-of normal upper Gamma and show it has a well-defined tangent cone.

1.6. Existence of tangent cone

We have constructed upper T as a subset of upper M and have observed that away from normal upper Gamma , upper T is a minimal surface for the hyperbolic metric. We refer to the intersection upper T intersection normal upper Gamma as the coincidence set. In general, one cannot expect upper T to be smooth along the coincidence set. However, we show that it does have a well defined tangent cone in the sense of Gromov, and this tangent cone is in fact of a very special form. In particular, this is enough to imply that upper T exists as the image of a map from upper S to upper M , and we may extend the isotopy upper F colon upper S times left-bracket 0 comma 1 right-parenthesis right-arrow upper M to a homotopy upper F colon upper S times left-bracket 0 comma 1 right-bracket right-arrow upper M with upper T equals upper F left-parenthesis upper S comma 1 right-parenthesis .

By a tangent cone we mean the following: at each point p element-of upper T intersection normal upper Gamma , consider the pair of metric spaces left-parenthesis upper N Subscript s Baseline left-parenthesis p right-parenthesis comma upper T Subscript s Baseline left-parenthesis p right-parenthesis right-parenthesis where upper T Subscript s Baseline left-parenthesis p right-parenthesis is the intersection upper T Subscript s Baseline equals upper T intersection upper N Subscript s Baseline left-parenthesis p right-parenthesis . We rescale the metric on this pair by the factor 1 slash s . Then we claim that this sequence of (rescaled) pairs of metric spaces converges in the Gromov–Hausdorff sense to a limit left-parenthesis upper B comma upper C right-parenthesis where upper B is the unit ball in Euclidean 3 -space, and upper C is the cone (to the origin) over a great bigon in the unit sphere. Here by a great bigon we mean the union of two spherical geodesics joining antipodal points in the sphere. In fact we do not quite show that upper T has this structure, but rather that each local branch of upper T has this structure. Here we are thinking of the map upper F left-parenthesis dot comma 1 right-parenthesis colon upper S right-arrow upper M whose image is upper T , and by “local branch” we mean the image of a regular neigborhood of a point preimage.

Lemma 1.27 (Tangent cone).

Let upper T be as constructed in Lemma 1.24. Let p element-of upper T intersection normal upper Gamma . Then near p , upper T is a (topologically immersed) surface, each local branch of which has a well-defined tangent cone, which is the cone on a great bigon.

Proof.

We use what is essentially a curve-shortening argument. For each small s , define

upper T Subscript s Baseline equals partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis intersection upper T period

For each point q element-of upper T minus normal upper Gamma , we define alpha left-parenthesis q right-parenthesis to be the angle between the tangent space to upper T at q and the radial geodesic through q emanating from p . By the coarea formula, we can calculate

a r e a left-parenthesis upper T intersection upper N Subscript s Baseline left-parenthesis p right-parenthesis right-parenthesis equals integral Subscript 0 Superscript s Baseline integral Underscript upper T Subscript t Baseline Endscripts StartFraction 1 Over cosine left-parenthesis alpha right-parenthesis EndFraction d l d t greater-than-or-equal-to integral Subscript 0 Superscript s Baseline l e n g t h left-parenthesis upper T Subscript t Baseline right-parenthesis d t comma

where d l denotes the length element in each upper T Subscript t . Note that this estimate implies that upper T Subscript t is rectifiable for a.e. t . We choose s to be such a rectifiable value.

Now, each component tau of upper T Subscript s is a limit of components tau Subscript i of upper S Subscript t Sub Subscript i Baseline intersection partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis for large i . By 2 -incompressibility of the upper S Subscript t Sub Subscript i , each tau Subscript i is a loop bounding a subdisk upper D Subscript i of upper S Subscript t Sub Subscript i for large i .

Now, partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis is convex in the hyperbolic metric, though not necessarily in the g Subscript t metric. By cutting out the disks partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis intersection upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis and replacing them with the disks upper D Superscript plus-or-minus orthogonal to normal upper Gamma which are totally geodesic in both the g Subscript t and the hyperbolic metrics, we can approximate partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis by a surface partial-differential upper B bounding a ball upper B subset-of upper N Subscript s Baseline left-parenthesis p right-parenthesis which is convex in the g Subscript s metric for all s greater-than-or-equal-to t . The ball upper B is illustrated in Figure 1.

Note that after lifting upper B to the universal cover, there is a retraction onto upper B which is length nonincreasing, in both the g Subscript t and the hyperbolic metric. This retraction projects along the fibers of the product structure on upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis to upper D Superscript plus-or-minus , and outside upper N Subscript r left-parenthesis 1 minus t right-parenthesis Baseline left-parenthesis normal upper Gamma right-parenthesis , it is the nearest point projection to partial-differential upper B minus upper D Superscript plus-or-minus .

Let tau prime Subscript i be the component of upper S Subscript t Sub Subscript i Baseline intersection partial-differential upper B approximating tau Subscript i , and let upper D prime Subscript i be the subdisk of upper S Subscript t Sub Subscript i which it bounds.

Then the disk upper D prime Subscript i must be contained in upper B , or else we could decrease its g Subscript t and hyperbolic area by the retraction described above. The disks upper D prime Subscript i converge to the component upper D subset-of upper T prime bounded by tau , and the hyperbolic areas of the upper D prime Subscript i converge to the hyperbolic area of upper D .

Note that upper B as above is really shorthand for upper B Subscript t , since it depends on a choice of t . Similarly we have tau Subscript t and upper D Subscript t . Since the component upper D Subscript t Baseline subset-of upper T prime bounded by tau Subscript t is contained in upper B Subscript t for all t , the component upper D subset-of upper T prime bounded by tau subset-of partial-differential upper N Subscript s is contained in upper N Subscript s , since upper B Subscript t Baseline right-arrow upper N Subscript s as t right-arrow 1 . So we can, and do, work with upper N Subscript s Baseline left-parenthesis p right-parenthesis instead of upper B in the sequel.

Now, let upper D prime be the cone on tau to the point p . upper D prime can be perturbed an arbitrarily small amount to an embedded disk upper D double-prime , and therefore by comparing upper D double-prime with the upper D prime Subscript i , we see that the hyperbolic area of upper D prime must be at least as large as that of upper D . Note that this perturbation can be taken to move upper D prime off normal upper Gamma and can be approximated by perturbations which miss normal upper Gamma . Similar facts are true for all the perturbations we consider in the sequel.

Since this is true for each component tau of upper T Subscript s , by abuse of notation we can replace upper T by the component of upper T intersection upper N Subscript s Baseline left-parenthesis p right-parenthesis bounded by a single mapped in circle tau . This will be the local “branch” of the topologically immersed surface upper T . We use this notational convention for the remainder of the proof of the lemma. Note that the inequality above still holds. It follows that we must have

a r e a left-parenthesis upper T intersection upper N Subscript s Baseline left-parenthesis p right-parenthesis right-parenthesis less-than-or-equal-to integral Subscript 0 Superscript s Baseline l e n g t h left-parenthesis upper T Subscript s Baseline right-parenthesis StartFraction hyperbolic sine left-parenthesis t right-parenthesis Over hyperbolic sine left-parenthesis s right-parenthesis EndFraction d t equals a r e a left-parenthesis cone on upper T Subscript s Baseline right-parenthesis period

Now, for each sphere partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis , we let phi be the projection, along hyperbolic geodesics, to the unit sphere upper S squared in the tangent space at p . For each t element-of left-parenthesis 0 comma 1 right-bracket , define

double-vertical-bar upper T Subscript t Baseline double-vertical-bar equals l e n g t h left-parenthesis phi left-parenthesis upper T Subscript t Baseline right-parenthesis right-parenthesis equals StartFraction l e n g t h left-parenthesis upper T Subscript t Baseline right-parenthesis Over hyperbolic sine left-parenthesis t right-parenthesis EndFraction period

It follows from the inequalities above that for some intermediate s prime we must have

double-vertical-bar upper T Subscript s prime Baseline double-vertical-bar less-than-or-equal-to double-vertical-bar upper T Subscript s Baseline double-vertical-bar

with equality iff upper T intersection upper N Subscript s Baseline left-parenthesis p right-parenthesis is equal to the cone on upper T Subscript s .

Now, the cone on upper T Subscript s is not locally least area for the hyperbolic metric in upper N Subscript s Baseline left-parenthesis p right-parenthesis minus normal upper Gamma unless upper T Subscript s is a great circle or geodesic bigon in partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis (with endpoints on partial-differential upper N Subscript s Baseline left-parenthesis p right-parenthesis intersection normal upper Gamma ), in which case the lemma is proved. To see this, just observe that a cone has vanishing principal curvature in the radial direction, so its mean curvature vanishes iff it is totally geodesic away from normal upper Gamma .

So we may suppose that for any s there is some s prime less-than s such that double-vertical-bar upper T Subscript s prime Baseline double-vertical-bar less-than double-vertical-bar upper T Subscript s Baseline double-vertical-bar . Therefore we choose a sequence of values s Subscript i with s Subscript i Baseline right-arrow 0 such that double-vertical-bar upper T Subscript s Sub Subscript i Subscript Baseline double-vertical-bar greater-than double-vertical-bar upper T Subscript s Sub Subscript i plus 1 Subscript Baseline double-vertical-bar , such that double-vertical-bar upper T Subscript s Sub Subscript i Subscript Baseline double-vertical-bar converges to the infimal value of double-vertical-bar upper T Subscript t Baseline double-vertical-bar with t element-of left-parenthesis 0 comma s right-bracket , and such that double-vertical-bar upper T Subscript s Sub Subscript i Subscript Baseline double-vertical-bar is the minimal value of double-vertical-bar upper T Subscript t Baseline double-vertical-bar on the interval t element-of left-bracket s Subscript i Baseline comma 1 right-bracket . Note that for any small t , the cone on upper T Subscript t has area

a r e a left-parenthesis cone on upper T Subscript t Baseline right-parenthesis equals StartFraction t Over 2 EndFraction l e n g t h left-parenthesis upper T Subscript t Baseline right-parenthesis plus upper O left-parenthesis t cubed right-parenthesis equals StartFraction t squared Over 2 EndFraction double-vertical-bar upper T Subscript t Baseline double-vertical-bar plus upper O left-parenthesis t cubed right-parenthesis period

The set of loops in the sphere with length bounded above by some constant, parameterized by arclength, is compact, by the Arzela–Ascoli theorem, and so we can suppose that the phi left-parenthesis upper T Subscript s Sub Subscript i Subscript Baseline right-parenthesis converge in the Hausdorff sense to a loop upper C subset-of upper S squared .

Claim

upper C is a geodesic bigon.

Proof.

We suppose not and will obtain a contradiction.

We fix notation: for each i , let upper C Subscript i denote the inverse image phi Superscript negative 1 Baseline left-parenthesis upper C right-parenthesis under phi colon partial-differential upper N Subscript s Sub Subscript i Subscript Baseline left-parenthesis p right-parenthesis right-arrow upper S squared . So upper C Subscript i is a curve in partial-differential upper N Subscript s Sub Subscript i Baseline left-parenthesis p right-parenthesis . By the cone on upper C Subscript i we mean the union of the hyperbolic geodesic segments in upper N Subscript s Sub Subscript i Baseline left-parenthesis p right-parenthesis from upper C Subscript i to p . By the cone on upper C we mean the union of the geodesic segments in the unit ball in Euclidean 3 -space from upper C subset-of upper S squared to the origin. For each i , we have an estimate

a r e a left-parenthesis cone on upper C Subscript i Baseline right-parenthesis equals s Subscript i Superscript 2 Baseline a r e a left-parenthesis cone on upper C right-parenthesis plus upper O left-parenthesis s Subscript i Superscript 3 Baseline right-parenthesis period

For each i , let upper T Superscript i denote the surface obtained from upper T intersection upper N Subscript s Sub Subscript i Baseline left-parenthesis p right-parenthesis by rescaling metrically by 1 slash s Subscript i . Then upper T Superscript i is a surface with boundary contained in a ball of radius 1 in a space of constant curvature minus s Subscript i Superscript 2 . Moreover, it enjoys the same least area properties as upper T intersection upper N Subscript s Sub Subscript i Baseline left-parenthesis p right-parenthesis .

By the monotonicity property of the double-vertical-bar upper T Subscript s Sub Subscript i Subscript Baseline double-vertical-bar and the coarea formula, we have an inequality

limit Underscript i right-arrow normal infinity Endscripts a r e a left-parenthesis upper T Superscript i Baseline right-parenthesis greater-than-or-equal-to a r e a left-parenthesis cone on upper C right-parenthesis period

On the other hand, since each upper T Superscript i is least area, we have an estimate

one-half double-vertical-bar upper T Subscript s Sub Subscript i Subscript Baseline double-vertical-bar plus upper O left-parenthesis s Subscript i Baseline right-parenthesis equals StartFraction a r e a left-parenthesis cone on upper T Subscript s Sub Subscript i Subscript Baseline right-parenthesis Over s Subscript i Superscript 2 Baseline EndFraction greater-than-or-equal-to a r e a left-parenthesis upper T Superscript i Baseline right-parenthesis period

It follows that the limit of the area of the upper T Superscript i is actually equal to the area of the cone on upper C .

On the other hand, since the phi left-parenthesis upper T Subscript s Sub Subscript i Subscript Baseline right-parenthesis converge to upper C , for sufficiently large i we can find an immersed annulus upper A Subscript i in upper S squared with area kappa for any positive kappa , which is the track of a homotopy (in upper S squared ) from phi left-parenthesis upper T Subscript s Sub Subscript i Subscript Baseline right-parenthesis to upper C . We let phi Superscript negative 1 Baseline left-parenthesis upper A Subscript i Baseline right-parenthesis denote the corresponding annulus in