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.
During the period 1960–1980, Ahlfors, Bers, Kra, Marden, Maskit, Sullivan, Thurston and many others developed the theory of geometrically finite ends of hyperbolic It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite. -manifolds.
Around 1978 William Thurston gave a conjectural description of geometrically infinite ends of complete hyperbolic An example of a geometrically infinite end is given by an infinite cyclic covering space of a closed hyperbolic 3-manifold which fibers over the circle. Such an end has cross sections of uniformly bounded area. By contrast, the area of sections of geometrically finite ends grows exponentially in the distance from the convex core. -manifolds.
For the sake of clarity we will assume throughout this introduction that where is parabolic free. Precise statements of the parabolic case will be given in §7.
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 .ReferenceBH 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 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.
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:
Let be a complete hyperbolic with finitely generated fundamental group. Then every end of -manifold 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 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. -manifold.
If is a complete hyperbolic with finitely generated fundamental group, then -manifold is topologically tame.
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 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 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.
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 ReferenceCa. It follows from Canary that the various results of ReferenceCa, §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 ReferenceMi and ReferenceBCM.
Theorem 0.6 (Classification Theorem).
If is a complete hyperbolic with finitely generated fundamental group, then -manifold 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 with finitely generated fundamental group, then -manifold 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 ReferenceT, ReferenceBo, ReferenceCa and ReferenceCaM.
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 and let -manifold, be a finite collection of pairwise disjoint simple closed geodesics in Furthermore, let . be a closed embedded surface rel. -incompressible which is either nonseparating in or separates some component of from another. Then is homotopic to a surface via a homotopy
is an embedding disjoint from for ,
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 (to be defined in §1), which implies in particular that it is intrinsically -minimal.
Here is the main technical result of this paper.
Let be an end of the complete orientable hyperbolic -manifold with finitely generated fundamental group. Let be a compact core of -dimensional , 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 .
The proof of Theorem 0.9 blends elementary aspects of minimal surface theory, hyperbolic geometry, and 3-manifold topology. The method will be demonstrated in §4 where we give a proof of Canary’s theorem. The first-time reader is urged to begin with that section.
This paper is organized as follows. In §1 and §2 we establish the shrinkwrapping technique for finding surfaces in hyperbolic 3-manifolds. In §3 we prove the existence of simple geodesics exiting the end of parabolic free manifolds. In §4 we prove Canary’s theorem. This proof will model the proof of the general case. The general strategy will be outlined at the end of that section. In §5 we develop the topological theory of end reductions in 3-manifolds. In §6 we give the proofs of our main results. In §7 we give the necessary embellishments of our methods to state and prove our results in the case of manifolds with parabolic cusps. -separated
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 .
In this section, we introduce a new technical tool for finding surfaces in hyperbolic called shrinkwrapping. Roughly speaking, given a collection of simple closed geodesics -manifolds, in a hyperbolic -manifold and an embedded surface a surface , is obtained from by shrinkwrapping rel. if it is homotopic to can be approximated by an isotopy from , supported in and is the least area subject to these constraints. ,
Given mild topological conditions on (namely to be defined below) the shrinkwrapped surface exists, and is -incompressibility, with respect to the path metric induced by the Riemannian metric on .
We use some basic analytical tools throughout this section, including the Gauss–Bonnet formula, the coarea formula, and the Arzela–Ascoli theorem. At a number of points we must invoke results from the literature to establish existence of minimal surfaces (ReferenceMSY), existence of limits with area and curvature control (ReferenceCiSc), and regularity of the shrinkwrapped surfaces along (ReferenceRi, ReferenceFre). General references are ReferenceCM, ReferenceJs, ReferenceFed and ReferenceB.
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:
A smooth surface in a Riemannian 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. -manifold
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 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 ReferenceJs.
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 .
For basic elements of the theory of comparison geometry, see ReferenceBH.
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 of constant sectional curvature -manifold 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
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 triangle. -comparison
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 ReferenceB 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., .ReferenceRe, §8, pp. 135–140 for more details and a general discussion of metric surfaces with (integral) curvature bounds.
Definition 1.8 ( surfaces). -minimal
Let be given. Let be a complete Riemannian with sectional curvature bounded above by -manifold 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 , surface. -minimal
Definition 1.9 ( -incompressibility).
An embedded surface in a -manifold disjoint from a collection of simple closed curves is said to be rel. -incompressible 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 and let -manifold, be a finite collection of pairwise disjoint simple closed geodesics in Furthermore, let . be a closed embedded surface rel. -incompressible which is either nonseparating in or separates some component of from another. Then is homotopic to a surface -minimal via a homotopy
is an embedding disjoint from for ,
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.
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.
Definition 1.12 ( -separation).
Let be a collection of disjoint simple geodesics in a Riemannian manifold The collection . is if any path -separated with endpoints on and satisfying
is homotopic rel. endpoints into The supremum of such . is called the separation constant of The collection . is weakly if -separated
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 ,
Topologically, is a sphere and is a torus, for sufficiently small Similarly, . is a closed ball, and is a closed solid torus. If is then -separated, is a union of solid tori.
Lemma 1.15 (Bounded Diameter Lemma).
Let be a complete hyperbolic Let -manifold. be a disjoint collection of embedded geodesics. Let -separated be a Margulis constant for dimension and let , denote the subset of where the injectivity radius is at most If . is a -incompressible surface, then there is a constant -minimal and such that for each component of we have ,
Furthermore, can only intersect at most components of .
Since is any point -incompressible, either lies in or is the center of an embedded in -disk where ,
Since is Gauss–Bonnet implies that the area of an embedded , in -disk 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 .
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 of -neighborhood 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 Choose some small -separated. 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 .
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:
For each there is an satisfying such that the union of the tori is totally geodesic for the metric.
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 ).
The area of a disk cross section on is .
The metric dominates the hyperbolic metric on That is, for all -planes. -vectors the , area of is at least as large as the hyperbolic area of .
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.
We denote length of an arc with respect to the metric as and area of a surface , with respect to the metric as .
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 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 -manifolds 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 -manifold 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 ReferenceMSY 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 , area among all such surfaces. -least
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 ReferenceMSY implies that either such a globally least area surface can be found, or is the boundary of a twisted over a closed surface in -bundle 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 Similarly, if -bundle. are distinct geodesics of separated from each other by then the , can be joined by an arc ’s 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 , disjoint from -bundle and therefore does not bound an , in -bundle.
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 surface close to -minimal separating the complement of , in from Then by .ReferenceMSY there is a globally least area surface contained in the compact subset of , bounded by this barrier surface. Since must either intersect or by the Bounded Diameter Lemma ,1.15, unless the hyperbolic area of is very large, the diameter of in is much smaller than the distance from or to Since by hypothesis, . is the least area for the metric, its restriction to has hyperbolic area less than the hyperbolic area of and therefore there is an a priori upper bound on its diameter in , By choosing . large enough, we see that is contained in the interior of where , and agree. Thus is the globally least area for the metric in and therefore , exists for any .
The bounded diameter lemma easily implies the following:
Lemma 1.21 (Compact set).
There is a fixed compact set such that the surfaces constructed in Lemma 1.20 are all contained in .
Since the hyperbolic areas of the are all uniformly bounded (by e.g. the hyperbolic area of and are ) rel. -incompressible they have uniformly bounded diameter away from , outside of Margulis tubes. Since for homological reasons they must intersect the compact sets or they can intersect at most finitely many Margulis tubes. It follows that they are all contained in a fixed bounded neighborhood , of or containing ,.
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 the total curvature of , is just the integral of the absolute value of the (Gauss) curvature over 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 . are minimal with respect to the metrics, which have no uniform upper bound on their sectional curvature, so we must work slightly harder to show that the have uniformly bounded total curvature. More precisely, we show that their restrictions to the complement of any fixed tubular neighborhood have uniformly bounded total curvature.
Lemma 1.22 (Finite total curvature).
Let be the surfaces constructed in Lemma 1.20. Fix some small, positive Then the subsurfaces .
have uniformly bounded total curvature.
Having chosen we choose , large enough so that .
Observe firstly that each has area less than the area of and therefore hyperbolic area less than the hyperbolic area of , for sufficiently large .
If the integral of geodesic curvature along a component of is large, then the length of the curves obtained by isotoping into grows very rapidly, by the definition of geodesic curvature.
Since there is an a priori bound on the hyperbolic area of it follows that there cannot be any long components of , with big integral geodesic curvature. More precisely, consider a long component of For . the boundary of the of -neighborhood in is contained in If the integral of the geodesic curvature along . were sufficiently large for every then the derivative of the length of the , would be large for every and therefore the lengths of the , would be large for all It follows that the hyperbolic area of the . collar neighborhood of in would be very large, contrary to existence of an a priori upper bound on the total hyperbolic area of .
This contradiction implies that for some the integral of the geodesic curvature along , can be bounded from above. To summarize, for each constant there is a constant such that for each component , of which has length there is a loop
isotopic to by a short isotopy, satisfying
On the other hand, since is minimal, there is a constant such that each component of which has length bounds a hyperbolic globally least area disk which is contained in For . sufficiently close to such a disk is contained in , and therefore must actually be a subdisk of .
By the coarea formula above, we can choose so that is a priori bounded. It follows that if is the subsurface of bounded by the components of of length then we have a priori upper bounds on the area of , on , and on , Moreover, . is contained in where the metric , agrees with the hyperbolic metric, so the curvature of is bounded above by pointwise, by Lemma 1.2. By the Gauss–Bonnet formula, this gives an a priori upper bound on the total curvature of and therefore on .
Lemma 1.24 (Limit exists).
Let be the surfaces constructed in Lemma 1.20. Then there is an increasing sequence
such that and the , converge on compact subsets of in the topology to some with closure in .
By definition, the surfaces have area bounded above by the area of Moreover, since . is disjoint from for sufficiently large , the , area of is equal to the hyperbolic area of Since the . area dominates the hyperbolic area, it follows that the have hyperbolic area bounded above, and by Lemma 1.22, for any the restrictions of , to have uniformly bounded finite total curvature.
Moreover, by Lemma 1.21, each is contained in a fixed compact subset of By standard compactness theorems (see, e.g., .ReferenceCiSc) any infinite sequence contains a subsequence which converges on compact subsets of away from finitely many points where some subsurface with nontrivial topology might collapse. That is, there might be isolated points , such that for any neighborhood of the intersection of , with contains loops which are essential in for all sufficiently large .
But is rel. -incompressible so in particular it is incompressible in , and no such collapse can take place. So after passing to a subsequence, a limit , exists (compare ReferenceMSY). Since each is a globally least area surface in with respect to the metric, it is a locally least area surface with respect to the hyperbolic metric on It follows that . is locally least area in the hyperbolic metric, properly embedded in and we can define , to be the closure of in .
Lemma 1.25 (Interpolating isotopy).
Let be the sequence as in Lemma 1.24. Then after possibly passing to a subsequence, there is an isotopy such that
and such that for each the track of the isotopy either converges to some well-defined limit or else it is eventually contained in for any .
Fix some small Outside . the surfaces , converge uniformly in the topology to It follows that for any . and for , sufficiently large (depending on the restrictions of ), and to the complement of are both sections of the exponentiated unit normal bundle of and therefore we can isotope these subsets of , to 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 to define the correct isotopy from to .
Let be obtained from by isotoping it slightly into so that it is transverse to and therefore also to , for sufficiently large. For each we consider the intersection ,
and observe that the limit satisfies
Let be a component of which is inessential in Then for large . , can be approximated by which are inessential in Since the . are rel. -incompressible the loops , must bound subdisks of Since . is a convex surface with respect to the hyperbolic metric, and the metric agrees with the hyperbolic metric outside for large it follows that the disks , are actually contained in for large It follows that . and are isotopic by an isotopy supported in which restricts to a very small isotopy of , to in .
Let be a component of which is essential in Then so is . for large Again, since . and therefore , is rel. -incompressible it follows that , cannot be a meridian of and must actually be a longitude. It follows that there is another essential curve in each such that the essential curves , and cobound a subsurface in After passing to a diagonal subsequence, we can assume that the . converge to some component of .
By the surfaces -incompressibility, are annuli. Note that there are two relative isotopy classes of such annuli. By passing to a further diagonal subsequence, we can assume and are isotopic in by an isotopy which restricts to a very small isotopy of to in .
We have shown that for any small and any sequence there is an arbitrarily large index , and infinitely many indices with so that the surfaces and are isotopic, and the isotopy can be chosen to have the following properties:
The isotopy takes to by an isotopy supported in .
Outside the tracks of the isotopy are contained in fibers of the exponentiated normal bundle of ,.
Choose a sequence and pick a subsequence of the , and relabel so that ’s satisfy the properties above with respect to Then the composition of this infinite sequence of isotopies is ..
The reason for the circumlocutions in the statement of Lemma 1.25 is that we have not yet proved that is a limit of the as maps from to This will follow in § .1.6, where we analyze the structure of near a point and show it has a well-defined tangent cone.
We have constructed as a subset of and have observed that away from , is a minimal surface for the hyperbolic metric. We refer to the intersection as the coincidence set. In general, one cannot expect 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 exists as the image of a map from to and we may extend the isotopy , to a homotopy with .
By a tangent cone we mean the following: at each point consider the pair of metric spaces , where is the intersection We rescale the metric on this pair by the factor . Then we claim that this sequence of (rescaled) pairs of metric spaces converges in the Gromov–Hausdorff sense to a limit . where is the unit ball in Euclidean and -space, 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 has this structure, but rather that each local branch of has this structure. Here we are thinking of the map whose image is and by “local branch” we mean the image of a regular neigborhood of a point preimage. ,
Lemma 1.27 (Tangent cone).
Let be as constructed in Lemma 1.24. Let Then near . , is a (topologically immersed) surface, each local branch of which has a well-defined tangent cone, which is the cone on a great bigon.
We use what is essentially a curve-shortening argument. For each small define ,
For each point we define , to be the angle between the tangent space to at and the radial geodesic through emanating from By the coarea formula, we can calculate .
where denotes the length element in each Note that this estimate implies that . is rectifiable for a.e. We choose . to be such a rectifiable value.
Now, each component of is a limit of components of for large By . of the -incompressibility each , is a loop bounding a subdisk of for large .
Now, is convex in the hyperbolic metric, though not necessarily in the metric. By cutting out the disks and replacing them with the disks orthogonal to which are totally geodesic in both the and the hyperbolic metrics, we can approximate by a surface bounding a ball which is convex in the metric for all The ball . is illustrated in Figure 1.
Note that after lifting to the universal cover, there is a retraction onto which is length nonincreasing, in both the and the hyperbolic metric. This retraction projects along the fibers of the product structure on to and outside , it is the nearest point projection to ,.
Let be the component of approximating and let , be the subdisk of which it bounds.
Then the disk must be contained in or else we could decrease its , and hyperbolic area by the retraction described above. The disks converge to the component bounded by and the hyperbolic areas of the , converge to the hyperbolic area of .
Note that as above is really shorthand for since it depends on a choice of , Similarly we have . and Since the component . bounded by is contained in for all the component , bounded by is contained in since , as So we can, and do, work with . instead of in the sequel.
Now, let be the cone on to the point . can be perturbed an arbitrarily small amount to an embedded disk and therefore by comparing , with the we see that the hyperbolic area of , must be at least as large as that of Note that this perturbation can be taken to move . off and can be approximated by perturbations which miss Similar facts are true for all the perturbations we consider in the sequel. .
Since this is true for each component of by abuse of notation we can replace , by the component of bounded by a single mapped in circle This will be the local “branch” of the topologically immersed surface . 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 .
Now, for each sphere we let , be the projection, along hyperbolic geodesics, to the unit sphere in the tangent space at For each . define ,
It follows from the inequalities above that for some intermediate we must have
with equality iff is equal to the cone on .
Now, the cone on is not locally least area for the hyperbolic metric in unless is a great circle or geodesic bigon in (with endpoints on 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 ),.
So we may suppose that for any there is some such that Therefore we choose a sequence of values . with such that such that , converges to the infimal value of with and such that , is the minimal value of on the interval Note that for any small . the cone on , has area
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 converge in the Hausdorff sense to a loop .
is a geodesic bigon.
We suppose not and will obtain a contradiction.
We fix notation: for each let , denote the inverse image under So . is a curve in By the cone on . we mean the union of the hyperbolic geodesic segments in from to By the cone on . we mean the union of the geodesic segments in the unit ball in Euclidean from -space to the origin. For each we have an estimate ,
For each let , denote the surface obtained from by rescaling metrically by Then . is a surface with boundary contained in a ball of radius in a space of constant curvature Moreover, it enjoys the same least area properties as ..
By the monotonicity property of the and the coarea formula, we have an inequality
On the other hand, since each is least area, we have an estimate
It follows that the limit of the area of the is actually equal to the area of the cone on .
On the other hand, since the converge to for sufficiently large , we can find an immersed annulus in with area for any positive which is the track of a homotopy (in , from ) to We let . denote the corresponding annulus in