j. differential geometry 58 (2001) 113-149

THE SMALE CONJECTURE FOR HYPERBOLIC 3-MANIFOLDS: ISOM (M^{3}) ≃ Diff (M^{3})

DAVID GABAI

Abstract

The main result of this paper is, “If M is a closed hyperbolic 3-manifold, then the inclusion of the isometry group Isom(M) into the diffeomorphism group Diff(M) is a homotopy equivalence.”

1. Introduction

The main result of this paper is:

Theorem 1.1. If M is a closed hyperbolic 3-manifold, then the inclusion of the isometry group Isom(M) into the diffeomorphism group Diff(M) is a homotopy equivalence.

Theorem 1.1 had been proven for Haken manifolds in 1976 by Hatcher [9] and Ivanov [11, 12]. We showed in [6], [7] that π_{0}(Diff(M)) is canonically bijective with π_{0}(Isom(M)), which is well known to be finite. Thus, if Diff_{0}(M) denotes the path component of Diff(M) containing id_{M}, then this result is equivalent to:

Theorem 1.2. If M is a closed hyperbolic 3-manifold, then Diff_{0}(M), is contractible.

The proof of Theorem 1.1 follows along the same lines as the proof of Theorem 0.1 ii) [6]. Namely use the contractibility of the space of Riemannian metrics on M in conjunction with the insulator theory of [6] to reduce to the Haken case.

Partially supported by National Science Foundation grants DMS-9505253, DMS-0071852.

Received April 5, 2001.

The main technical innovations of this paper are the Canonical Solid Torus Theorem, Theorem 3.4, together with the Non-Encroachment Lemma, Lemma 4.3. Roughly speaking the Canonical Solid Torus Theorem asserts that if δ is a special type of oriented simple closed geodesic in a closed hyperbolic 3-manifold M, then associated to any Riemannian metric r on M there exists a canonically immersed solid torus V_{r} whose interior V^{∘} _{r} is embedded. Furthermore, if γ is a core of V^{∘} _{r}, then it is canonically oriented and with that orientation is isotopic to δ. Finally, if r is a hyperbolic metric and δ_{r} is the oriented geodesic in M freely homotopic to δ, then δ_{r} is a core of V^{∘} _{r} and its orientation agrees with the canonical orientation given to cores of V^{∘} _{r}. While these immersed tori may not vary continuously with r, the Non-Encroachment Lemma provides sufficient control to approximately pass from one V_{r} to another V_{r′}, when r′ is close to r. (More or less, the worst that can happen is that a sequence of solid tori gets a 3-ball pinched off in the limit.)

Remarks 1.3. The canonical solid torus depends on an a priori chosen non-coalescable insulator family for the geodesic δ.

A discussion of the difference between the insulator construction of [6] and the full insulator construction developed in this paper is given in §3.

The Mostow Rigidity theorem [16] together with the results in [7] allow us to equate Diff_{0}(M) with Hyp(M), the space of hyperbolic metrics on M. Thus we obtain:

Theorem 7.3. The space Hyp(M) of hyperbolic metrics on a complete hyperbolic 3-manifold M of finite volume is contractible.

This paper is organized as follows. In §2 we present a detailed outline of the proof of Theorem 1.1. In particular we show how Theorem 1.1 is deduced from the Coarse Torus Isotopy Theorem 4.6 and the Local Contractibility Theorem 6.3. In §3 we give the proof of Theorem 3.4. In §4 we give the proof of the Coarse Torus Isotopy Theorem. Section 5 gives a new formulation of Hatcher’s theorem (the Smale Conjecture). The proof of the Local Contractibility Theorem is given in §6 and applications are presented in §7.

Isom (M^{3}) ≃ Diff (M^{3}) 115

Definition 1.4. Diff(M) will denote the space of diffeomorphisms of M with the C^{∞} topology. Diff_{0}(M) will denote the path component of Diff(M) containing id_{M}, i.e., Diff_{0}(M) is the set of diffeomorphisms isotopic to id_{M}. If X, Y are smooth manifolds then Emb(X, Y) is the space of smooth embeddings of X into Y. If X is a subspace of Y, then Emb_{0}(X, Y) is the subspace consisting of embeddings isotopic to the standard inclusion.

If E ⊂ Y, then N(k, E) = {y ∈ Y |d(y, E) ≤ k}. Similarly if x ∈ Y, then B(k, x) = {y ∈ Y |d(y, x) ≤ k}. The symbol ρ will always represent the standard hyperbolic metric on ℍ^{3}. The particular model of hyperbolic space will be clear from context. We will assume that ρ is induced from a metric also called ρ on the closed hyperbolic 3manifold M. We will use notations such as N_{ρ}(k, E) or d_{r}(x, y) when the metric ρ or r is not clear from context. |E| will denote the number

of components of E and E^{∘} will denote the interior of E. If X ⊂ Y then Bd(X) = X − X^{∘}. If Δ is a cellulation, then Δ^{k} denotes its k-skeleton.

Acknowledgements. The author is grateful to Hyam Rubinstein, Darryl McCullugh, Allen Hatcher, Toby Colding and Valentin Poenaru for long conversations. Hyam Rubinstein graciously and enthusiastically suggested that my work on π_{0}(Diff(M)) should generalize to all the higher homotopy groups and told me about [9], Hatcher’s “multiparameter” version of Waldhausen’s theorem. Allen Hatcher replaced my clumsy proof of Theorem 5.1 by a concise one. He also suggested Theorem 7.5. Darryl McCullough taught me techniques from high dimensional manifold theory. Toby Colding provided minimal surface advice and Po taught me about the work of Cerf and many others. The referee provided many expository suggestions as well as Remark 2.1 i).

2. The proof of Theorem 1.1

By [6] and [7] the inclusion Isom(M) → Diff(M) induces a bijection of Isom(M) with π_{0}(Diff(M)). Therefore to prove Theorem 1.1 it remains to show that Diff_{0}(M) is contractible. By [18] Diff(M) is an ANR and by [14, p. 35] an ANR has the homotopy type of a CW complex. Thus by J. H. C. Whitehead’s theorem it suffices to show that all the homotopy groups of Diff_{0}(M) are trivial.

In this section we explain how Theorem 1.1 follows directly from the following results which are established in §4 and §7 respectively.

Coarse Torus Isotopy Theorem 4.6. Let M be a closed orientable hyperbolic 3-manifold with geodesic δ satisfying the insulator condition. If f : S^{n} → Diff_{0}(M), then there exists a cellulation Δ of B^{n+1} and a function which associates to each cell σ ∈ Δ an embedded solid torus V_{σ} such that:

i) If κ is a proper face of σ, then V_{κ} ⊂ V^{∘} _{σ} and V_{κ} is isotopic to the standard embedding in V_{σ}.

ii) If x ∈ σ ∩ S^{n}, then f_{x}(δ) is a core of V^{∘} _{σ}.

Idea of the proof. Associated to x ∈ S^{n}, there is the push forward hyperbolic metric (f(x))_{∗}ρ. This gives rise to a map h : S^{n} → RM(M) the space of Riemannian metrics on M. The contractibility of RM(M) enables us to extend h to B^{n+1} → RM(M). By Theorem 3.4 associated to each x ∈ B^{n+1}, there is a canonical solid torus V_{x} immersed in M, with embedded interior. Furthermore, the curve δ is isotopic to a core of V_{x}. While the various V_{x}’s do not vary continuously in x, the Non-Encroachment Lemma 4.3 enables us to maintain sufficient control to obtain a cellulation Δ^{∗} and find embedded solid tori by shrinking various V_{x}’s which together satisfy the conclusions of Theorem 4.6 except that, in Conclusion i) the statement V_{κ} ⊂ V^{∘} _{σ} is replaced by the condition

V_{σ} ⊂ V^{∘} _{κ}. Roughly speaking our desired cellulation Δ is the cellulation dual to Δ^{∗}, with V_{σ} being the solid torus V_{σ^{∗}}, where σ^{∗} is the cell dual to σ.

Local Contractibility Theorem 6.3. Let δ be an oriented simple geodesic in the closed hyperbolic 3-manifold M and V a solid torus embedded in M. If H : S^{n} → Diff_{0}(M) is such that H_{t}(δ) ⊂ V^{∘} for each t ∈ S^{n}, then H extends to a map G : B^{n+1} → Diff_{0}(M) such that

G_{s}(δ) ⊂ V^{∘} for each s ∈ B^{n+1}.

Idea of the proof. Suppose that that n > 0, V is a closed regular neighborhood N(δ) of δ and the restriction of H_{t} to N(δ) is the identity

for all t ∈ B^{n+1}. In that case M − N^{∘}(δ) is Haken, and Theorem 6.3 follows by the Hatcher, Ivanov theorem [9], [11], [12] for Haken manifolds. (Actually they proved this in the PL category, but as noted in [9], the proof can be promoted to Diff using [10].) In general one can control the maps H_{t}|N(δ), since Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) ≃ S^{1} × S^{1} (an equivalent formulation of the Smale conjecture), and thereby reduce to the previous case.

The proof that Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) ≃ S^{1} × S^{1} is carried out in §5 and the proof of Theorem 6.3, including the case n = 0, is given in §6.

Proof of Theorem 1.1. Given f : S^{n} → Diff(M^{3}) construct a cellulation Δ of B^{n+1} and solid tori V_{σ} as in Theorem 4.6. That theorem implies that for each σ ∈ Δ, δ is isotopic to a core of V^{∘} _{σ}. Therefore to each x ∈ Δ^{0} − S^{n} there exists an f_{x} ∈ Diff_{0}(M) such that f_{x}(δ) is a core of V_{x}. Assume by induction that f has been extended to S^{n} ∪ Δ^{i} such that if x ∈ σ a k-cell, 0 ≤ k ≤ i, then f_{x}(δ) is a core of V^{∘} _{σ}. By Theorems 4.6 and 6.3 we can extend f to S^{n} ∪ Δ^{i+1} so that if x ∈ σ a k-cell, 0 ≤ k ≤ i + 1, then f_{x}(δ) is a core of V^{∘} _{σ}. The proof is complete when i = n. □

Remarks 2.1. i) Here is another view of the structure of the proof of Theorem 1.1 offered by the referee. Consider the fibration p : Diff_{0}(M) → Emb_{0}(V, M) where V is as above. By Hatcher and Ivanov [9], [11], [12], each component of Diff(M, V), the subspace of Diff(M) which fixes V pointwise, is contractible. Thus, using Lemma 6.1 it suffices to show that the induced map p_{∗} in the long exact homotopy sequence is trivial. Theorem 4.6 can be viewed as an approximate statement that p_{∗} is trivial, with Theorem 6.3 providing the necessary refinement.

ii) The proof of Theorem 6.3 does not rely on the insulator technology.

3. The Canonical Solid Torus Theorem

The reader should be familiar with the notions of noncoalescable insulator family and trilinking (see 0.4-0.5 [6] as well as the minimal surface theory developed in §3 [6].

In what follows the set {λ_{ij}} will denote a (π_{1}(M), {∂δ_{i}}) noncoalescable insulator family where δ is a simple closed oriented geodesic and δ_{0}, δ_{1}, δ_{2}, … denote the lifts of δ to ℍ^{3}. Let g ∈ π_{1}(M) denote the primitive element which fixes δ_{0}, such that if δ_{0} is oriented from the repelling fixed point of ∂δ_{0} to the attracting one, then δ_{0} inherits the orientation induced from δ.

Definition 3.1. By an immersed solid torus V ⊂ M we mean that V is the image of an immersion f : D^{2} × S^{1} → M. By the interior of

V or V^{∘} we mean f(D^{2}^{∘} × S^{1}). By ∂V we mean f(∂D^{2} × S^{1}).

Review of the insulator construction [6] 3.2. Let r be any Riemannian metric on M and let r also denote the induced metric on M̃ = ℍ^{3}. Let {σ_{ij}} be a family of π_{1}(M) equivariant r-least area D^{2}- limit laminations which span the {λ_{ij}}. To each σ_{0i} let H_{0i} denote the component of ℍ^{3} − σ_{0i} whose closure contains ∂δ_{0}. The set H_{0} = ∩_{i}H_{0i} contains a unique component that is invariant by the maximal cyclic group ⟨g⟩ which fixes δ_{0} and projects to an open solid torus V^{∘} which is the interior of an immersed solid torus V. The orientation on δ induces an orientation on δ_{0} and hence on ∂δ_{0} and hence on each core of V^{∘}. Any positively oriented core of V^{∘} is isotopic in M to δ.

To first approximation, one should think of σ_{0i} as a properly embedded r-least area plane spanning the smooth circle λ_{0i} and H_{0i} as the open half space of ℍ^{3} − σ_{0i} which contains ∂δ_{0}. Finally H_{0} is the intersection of these half spaces and a component projects down to the open solid torus V^{∘}.

Given M, δ and the insulator family {λ_{ij}}, the construction of V depends only on the Riemannian metric r and the choice of spanning laminations. The main result of this chapter eliminates dependence on the spanning laminations.

Definition 3.3. Let r be the Riemannian metric on ℍ^{3} induced from the Riemannian metric r on M. For each λ_{ij}, let {σ_{ij}^{α}}_{α∈J} denote the collection of r-least area D^{2}-limit laminations which span λ_{ij}. Let H_{ij}^{α} denote the component of ℍ^{3} − σ_{ij}^{α} which contains the ends of δ_{i}. Let H_{ij} = ∩_{α}H_{ij}^{α}, H_{i} = ∩_{j}H_{ij}, Σ_{ij} = ∪_{α}σ_{ij}^{α} and Σ_{i} = ∪_{j}Σ_{ij}.

We will show that H_{0} contains a unique D^{2}^{∘} × ℝ component which projects to the interior V^{∘} _{r} of an immersed solid torus V_{r} in M. This V_{r} is said to arise from the full insulator construction. The adjective full stresses the fact that all possible spanning laminations of each λ_{0j} are used in the construction. This should be contrasted with the insulator construction of [6], reviewed above, where a single spanning lamination is used.

Canonical Solid Torus Theorem 3.4. Let δ be an oriented simple closed geodesic in the closed orientable hyperbolic 3-manifold M possessing a non-coalescable insulator family {λ_{ij}}. Given any Riemannian metric r on M the full insulator construction gives rise to a canonically immersed solid torus V_{r} having the following properties:

i) V^{∘} _{r} is embedded. ii) If γ is a core of V^{∘} _{r}, then it is canonically oriented and with that

orientation is isotopic to δ.

iii) If r is a hyperbolic metric and δ_{r} is the oriented geodesic in M

freely homotopic to δ, then δ_{r} is a core of V^{∘} _{r}. Its orientation agrees with the canonical orientation given to cores of V^{∘} _{r}.

Lemma 3.5. For all i, j ∈ ℕ, Σ_{0i} ∩ Σ_{0j} ≠ ∅ implies λ_{0i} ∩ λ_{0j} ≠ ∅.

Proof. This follows from Lemma 3.5 viii) [6]. q.e.d.

Lemma 3.6.

i) After reordering the λ_{0j}’s there exists a finite set {λ_{01}, λ_{02}, …, λ_{0m}} which are representatives of the outermost ⟨g⟩-orbits of {λ_{0j}}_{j∈ℕ}. I.e., given any λ_{0k}, either λ_{0k} = g^{q}(λ_{0i}) for some q ∈ ℤ and 1 ≤ i ≤ m or there exists an n ∈ ℤ and j ∈ {1, …, m} such that λ_{0k} ⊂ Y (g^{n}(λ_{0j})), where Y (g^{n}(λ_{0j})) is the component of S_{∞}^{2} − g^{n}(λ_{0j}) which does not contain ∂δ_{0}.

ii) H_{0} = ∩_{i=1}^{m} ∩_{n∈ℤ} g^{n}(H_{0j}). In words, to construct H_{0} we need only consider the collection of spanning laminations corresponding to {λ_{01}, λ_{02}, …, λ_{0m}} and their ⟨g⟩-translates.

Proof. This follows as on p. 63 [6]. q.e.d.

Lemma 3.7.

i) If h ∈ ⟨g⟩, then h(H_{0}) = H_{0} and h(Σ_{0}) = Σ_{0}.

ii) H_{i} ∩ H_{j} ≠ ∅ if and only if i = j.

iii) There exists an a > 0 (which depends on r) so the H_{0} ⊂ N_{ρ}(a, δ_{0}).

Proof. Again these facts follow as in Step 2, p. 63 [6]. q.e.d.

Definition 3.8. If t > 0 let Σ^{t} = N^{∘}(t, δ_{0}) ∩ Σ_{0}. We say that L is a leaf of Σ_{0} (resp. Σ^{t}), if L is a leaf of some σ_{0j}^{α} (resp. L is a component

of R ∩ N^{∘}(t, δ_{0}) ∩ Σ_{0}, where R is a leaf of some σ_{0j}^{α}).

This paper makes extensive use of the notion of convergence of sequences of embedded surfaces or laminations in Riemannian 3-manifolds. The reader is advised to read Definition 3.2 [6] for the definition of the word converges.

Correction 3.9. In this paper and [6] all convergence takes place in the C^{k}-topology, all k < ∞ rather than the C^{∞}-topology.

Lemma 3.10. Let w ∈ ℍ^{3} and L_{1}, L_{2}, … a sequence of leaves of the laminations σ_{0i_{j}}^{α_{j}} such that Lim d_{ρ}(L_{i}, w) → 0. After passing to subsequence, each L_{j} is a leaf of σ_{0i}^{α_{j}} for some fixed i and the L_{j}’s converge to an r-least area D^{2}-limit lamination σ_{0i}^{α_{∞}} which contains w.

Lemma 3.10 can be thought of as saying that Σ_{0i} and Σ_{0} are closed in the C^{k}-topology, all k < ∞.

Proof. By the local finiteness of {λ_{0j}}, the last sentence of Proposition 3.9 [6] and the fact that each L_{j} meets a fixed neighborhood of w, we can pass to a subsequence so that each L_{i} is a leaf of a lamination spanning the same insulator λ_{0i}. By passing to another subsequence, and invoking Proposition 3.10 [6] we conclude that these leaves converge to a r-least area D^{2}-limit lamination σ_{0i}^{α_{∞}}. q.e.d.

Corollary 3.11. H_{0} and each H_{0i} are open sets.

Lemma 3.12.

i) Each leaf L of Σ^{t} is properly embedded in N^{∘}(t, δ_{0}).

ii) If t > a, L separates N^{∘}(t, δ_{0}) and one component of L − N^{∘}(t, δ_{0}) contains both ends of δ_{0}.

iii) The leaves of Σ^{t} have uniformly bounded area.

iv) (Leaves of Σ^{t} are closed in the C^{k}-topology, all k < ∞.) If L_{1}, L_{2}, … are leaves of Σ^{t} respectively containing points x_{1}, x_{2}, … and Lim x_{i} → x ∈ N^{∘}(t, δ_{0}), then there exists a leaf L_{x} of Σ^{t} containing x and a subsequence of {L_{i}} such that the following holds. If K is any compact subsurface in L_{x} containing x, then there exist embeddings f_{i} : K → L_{i} such that f_{i} → id_{K} in the C^{k}-topology all k < ∞ and for all i, x_{i} ∈ f_{i}(K).

Proof. i) The last sentence of Proposition 3.9 [6] together with the fact that L is contained in a leaf of a D^{2}-limit lamination implies that there exists an embedded disc D ⊂ ℍ^{3} such that ∂D ∩ N^{∘}(t, δ_{0}) = ∅ and

L is a component of D ∩ N^{∘}(t, δ_{0}).

ii) Apply Lemma 3.7 iii).

iii) By the local finiteness of λ_{0i} and the last sentence of Proposi-

tion 3.9 [6] only finitely many ⟨g⟩-orbits of Σ_{0i} can intersect N^{∘}(t, δ_{0}). Applying Proposition 3.9 [6] to these orbits we conclude that there exists a C > 0 such that if L is a leaf of Σ^{t}, then L ⊂ B_{ρ}(C, z) for some z ∈ δ_{0}. The disc D in the proof of i) can be chosen to have boundary in ∂B_{ρ}(2C, z) or some 2-sphere which is an arbitrarily close perturbation of it. Thus for some uniform constant C_{1} > 0, area_{r}(L) < C_{1} area_{ρ}(L) ≤ C_{1} area_{ρ}(D) ≤ C_{1} area_{ρ}(∂B_{ρ}(2C, z)).

iv) This follows from Lemma 3.10 and Definition 3.2 [6]. q.e.d.

Definition 3.13. A leaf L_{x} of Σ^{t} satisfying the conclusion of Lemma 3.12 iv) is said to be obtained as a limit of leaves L_{1}, L_{2}, … of Σ^{t}.

Lemma 3.14. Let U denote B^{∘}_{ρ}(η, x) or B^{∘}_{r}(η, x). For every t > 0 there exists ϵ > 0 so that if η ≤ ϵ, d_{ρ}(x, δ_{0}) ≤ .9t and L leaf of Σ^{t}, then L ∩ U is empty or a properly embedded disc D whose closure in U is a properly embedded disc transverse to ∂U.

Proof. Since the leaves of Σ_{0} are stable minimal surfaces with respect to a metric induced from a compact manifold, it follows by Schoen [19] that the leaves have uniformly bounded normal curvature. There-

fore there exists ϵ_{0} > 0 such that if ϵ_{1} ≤ ϵ_{0} and either U = B^{∘}_{r}(ϵ_{1}, x)

or B^{∘}_{ρ}(ϵ_{1}, x), then each component D of Σ_{0}|U is a properly embedded disc whose closure in U is a properly embedded disc transverse to ∂U. Furthermore if U′ = B^{∘}_{ρ}(ϵ′, x) or B^{∘}_{r}(ϵ′, x) and U′ ⊂ U, then D ∩ U′ is connected or empty.

Suppose that M_{1}, M_{2}, … is a sequence of leaves of Σ^{t} which hit progressively smaller ρ-balls about y_{1}, y_{2}, … in multiple components, where Lim y_{i} → x ∈ N_{ρ}(.9t, δ_{0}). Let L_{1}, L_{2}, … denote the leaves of Σ^{2t} which respectively contain the M_{1}, M_{2}, . . .. Let L_{x} denote a limit of L_{1}, L_{2}, … which contains x. By Lemma 3.12 and the previous paragraph there exists ϵ_{2} < ϵ_{1} so that L_{x} ∩ B_{ρ}(ϵ_{2}, x) is connected and transverse to ∂B_{ρ}(ϵ_{2}, x). Let K ⊂ L_{x} be a compact subsurface containing x, such that d_{ρ}(∂K, δ_{0}) > 1.5t. Let f_{i}(K) ⊂ L_{i} be as in Lemma 3.12 iv). Since f_{i}(K) → K, it follows that for i sufficiently large, d_{ρ}(∂f_{i}(K), δ_{0}) > t and hence M_{i} ⊂ f_{i}(K). Therefore by Schoen’s theorem if d_{ρ}(y_{j}, x) ≤ ϵ_{2} − ϵ′,

then B^{∘}_{ρ}(ϵ′, y_{j}) ∩ M_{i} is connected or empty for all i sufficiently large. q.e.d.

Notation 3.15. In what follows, J will denote a component of H_{0}, f = 2a (see Lemma 3.7) and m will be as in Lemma 3.6. Taking t = a, fix ϵ to satisfy the conclusion of Lemma 3.14.

Our next goal, carried out in sections 3.16–3.36, is to show that J is a simply connected manifold with boundary.

Definition 3.16. Given a component J of H_{0}, normally orient each leaf L of Σ^{f} to point into the component of N^{∘}_{ρ}(f, δ) − L which contains J. This orientation is called the J-normal orientation. The + side of L is the side facing J.

Warning 3.17. J-normal orientations may not in general induce a consistent transverse orientation on leaves of Σ_{0}.

Lemma 3.18.

i) If K_{L} denotes the component of N^{∘}(f, δ_{0}) − L which contains J, then J is a component of ∩_{L leaf of Σ^{f}} K_{L}. If J′ is another component of ∩_{L leaf of Σ^{f}} K_{L}, then J′ is a component of H_{0}.

ii) (Continuity of J-normal orientation). If the leaf L ∈ Σ^{f} is a limit of leaves L_{1}, L_{2}, … of Σ^{f} and J ∩ L ≠ ∅ then the J-normal orientation on L induces the J-normal orientation on L_{i} for i sufficiently large.

Proof. i) The first assertion is immediate. The hypothesis of the second implies that for each σ_{0j}^{α}, J and J′ lie in the same component of

N^{∘}(f, δ_{0}) − σ_{0j}^{α} and hence in the same component of ℍ^{3} − σ_{0j}^{α}.

ii) Let U = B^{∘}_{ρ}(ϵ, x) where x ∈ J ∩L. If D_{i} = L_{i}∩U, then D_{1}, D_{2}, …

is a sequence of discs converging in the C^{k}-topology, all k < ∞, to D. For i sufficiently large D_{i} and D nearly coincide, hence if their J-normal orientations are opposite, then J ∩ U would be confined to the region “between” D_{i} and D. Thus, if for i sufficiently large all such J-normal orientations were opposite, then J ∩ U = ∅. q.e.d.

Definition 3.19. If two leaves L_{1}, L_{2} of Σ^{f} are tangent at x with opposite J-normal orientations then we say that L_{1}, L_{2} are antitangent at x and x is an antitangential point. If U is a sufficiently small ηneighborhood of x, then the components of U ∩L_{1} and U ∩L_{2} containing x are discs which meet along a saddle or multi-saddle tangency. Call the closure of a component of U − L_{1} ∪ L_{2} lying on the + side of both L_{1} and L_{2} a wedge of L_{1}, L_{2} at x.

Two minimal surfaces S and T which are tangent at y either coincide or meet at y along a multi-saddle. We say that the multi-saddle is of multiplicity m(y), if locally S ∩ T consists of 2m(y) + 2 arcs which meet at y. If x is an antitangential point, then define the m(x), the multiplicity of x, to be the supremum of multiplicity of pairs of leaves that are antitangent at x.

We say that x ∈ Bd(J) is a spike point if:

i) There exist a triple of leaves A, B, C of Σ^{f} having a common tangent vector at x, and no pair of A, B, C are tangent at x.

ii) There exists no v ∈ T_{x}(ℍ^{3}) which is transverse to A, B, C and whose direction agrees with the three J-normal orientations at x.

If x is a spike point and U is a neighborhood of x which intersects each A, B, C in discs, then a spike region is the closure of a component of U − A ∪ B ∪ C which lies on the + side of A, B, C and limits on x.

Example 3.20. A spike point and spike region can be found in Figure 4.1 [6]. Note that if the normal orientation of one of the leaves was reversed, then these leaves would not define a spike point. To understand spike regions, consider very small spheres centered at x, and the possibly empty tiny triangles in these spheres which lie on the + sides of the leaves. A spike region is the closure of a continuum of such triangles which limit on x.

Lemma 3.21.

i) If L_{1} and L_{2} are antitangent at x, then J lies in at most one wedge of L_{1} and L_{2} at x.

ii) If leaves A, B, C define a spike point x, then at most one spike region of A, B, C at x intersects J.

Proof. i) Let σ_{0i}^{α} and σ_{0j}^{β} be respectively laminations whose leaves contain L_{1} and L_{2}. If J′ is the component of ℍ^{3} − σ_{0j}^{α} ∪ σ_{0i}^{β} which contains J, then Lemma 4.1 [6] implies that J′ can contain at most one wedge of L_{1} and L_{2} at x.

ii) Repeat the above argument using the laminations σ_{0a}^{α_{a}}, σ_{0b}^{α_{b}}, σ_{0c}^{α_{c}} which respectively contain leaves A, B and C. q.e.d.

Lemma 3.22.

i) There exists an M < ∞ such that every antitangential point of Bd(J) has multiplicity ≤ M.

ii) If x_{1}, x_{2}, … → x is a sequence of antitangential points of Bd(J), then Lim sup_{i→∞}m(x_{i}) < m(x).

iii) The points of Bd(J) which are spike points but not antitangential points is a discrete set. Any limit point is an antitangential point.

Proof. Suppose that x_{1}, x_{2}, … are a sequence of antitangential points of Bd(J) which limit on x. Suppose that A_{1}, B_{1}; A_{2}, B_{2}; … are leaves of Σ^{f} with antitangencies respectively at x_{1}, x_{2}, . . .. By passing to subsequence, restricting to an η ≤ ϵ open ball U about x, and letting D_{i} (resp. E_{i}) denote A_{i} ∩ U (resp. B_{i}), then by Lemmas 3.10 and 3.12 we can assume that these D_{i}’s and E_{i}’s are discs containing x_{i} which converge in the C^{k}-topology, all k < ∞, to antitangent discs D, E at x. Note that D and E cannot coincide in a neighborhood of x, else they would coincide in U and therefore J ∩ U = ∅.

If we give U Euclidean coordinates and let Q (resp. Q_{i}) denote the tangent plane to D (resp. D_{i}) at x (resp. x_{i}), then D and E (resp. D_{i} and E_{i}) are the graphs of functions on Q (resp. Q_{i}) and the difference function w (resp. w_{i}) satisfies the following properties. After changing coordinates on Q by a linear transformation T, w(z) = p(z)+q(z) where p is a linear homogenous harmonic polynomial of degree d, 2 ≤ d < ∞ and |q(z)| + |z||∇q(z)| + ⋯ + |z|^{d}|∇^{d}q(z)| ≤ C|z|^{d+1}. Here z = (z_{1}, z_{2}) denotes a point in ℝ^{2} and |∇^{n}q(z)| denotes the Euclidean norm of the vector of n^{2} degree-n partial derivatives evaluated at z. This is the well known local description of tangent minimal surfaces, which is powered by the Bers - Vekua theorem on general continuation [1], [20]. A proof of the above result is given in Colding-Minicozzi II [5] and we acknowledge here the use of their notation and error term which appears sharper than that found in the literature.

i) Since D_{i} → D in the C^{k}-topology, k < ∞, it follows that

Lim sup_{i→∞}m(x_{i}) ≤ m(x).

Using the compactness of Bd(J)/⟨g⟩ we obtain conclusion i).

ii) Now suppose that for all i, m(x_{i}) = m(x) = d − 1. It follows from [5] that after passing to subsequence, and changing coordinates on Q_{i} by a linear transformation T_{i}, with T_{i} → T, that w_{i} = p_{i} + q_{i} where p_{i} is a linear homogenous harmonic polynomial of degree d, p_{i} → p, and |q_{i}(z)| + |z||∇q_{i}(z)| + ⋯ + |z|^{d}|∇^{d}q_{i}(z)| ≤ C|z|^{d+1}. Here C is the same uniform constant and the original coordinates on Q are related to those of Q_{i} via an orthogonal change of coordinates. In words, around a uniformly sized neighborhood of x, these (multi)-saddles are geometrically extremely close.

Since each of A_{i} and B_{i} meets U in a single disc it follows that J ∩U must lie on the + side of each of D, E and each D_{i} and E_{i}. Therefore for each i, J ∩ U lies in W_{i} ∩ U (resp. W ∩ U) where W_{i} (resp. W) is a unique wedge of D_{i} ∪ E_{i} emanating from x_{i}. This implies that for all i and j, x_{i} ∈ W_{j} ∩ U and x_{j} ∈ W_{i} ∩ U and finally {x_{1}, x_{2}, … } ∈ W ∩ U.

Using the above equations it follows that there exists η > 0, N < ∞ so that if U = B_{ρ}(η, x) and i ≥ N, then W (resp. W_{i}) is contained in K ∩ U (resp. K_{i} ∩ U) where K is a cone based at x (resp. x_{i}) of uniform angle c < π. Thus {x, x_{1}, x_{2}, … } ⊂ K ∩_{i≥N} K_{i} which is evidently impossible.

iii) If x_{1}, x_{2}, … is a sequence of distinct spike points limiting on x, then after passing to subsequence there exist triples of leaves R_{i}, S_{i}, T_{i} of Σ^{f} defining the spikes which converge to a triple R, S, T which have a common tangent vector at x. If no pair of R, S and T are tangent at x, then using Schoen’s normal curvature lemma it follows that there exists η > 0, N < ∞ so that if i ≥ N and U = B^{∘}_{ρ}(η, x) then J ∩ U is contained in K ∩ U (resp. K_{i} ∩ U) where K is a cone based at x (resp. x_{i}) of uniform angle c < π. Thus one obtains a contradiction as above.

If say R and S are tangent at x but not antitangent, then T must be antitangent to both. Otherwise for i sufficiently large the triple R_{i}, S_{i}, T_{i} would not satisfy the normal orientation requirement of Definition 3.19.

q.e.d.

Definition 3.23 (Linear Model near a boundary point x). Suppose that x ∈ Bd(J). Let T_{x}^{1} denote the sphere of unit vectors in T (ℍ^{3}) through x. To each leaf L of Σ^{f} through x, let A_{L} denote the closed hemisphere in T_{x}^{1} bounded by the tangent plane through L and lying on the positive side of that plane. Let C_{L} = ∂A_{L}.

It follows from Lemma 3.10 that the collection of such C_{L}’s is closed (in the space of great circles). Thus we obtain:

Lemma 3.24. A_{x} =^{def} ∩A_{L} is either

i) a closed disc,

ii) a closed interval,

iii) one or two points, or

iv) empty.

Lemma 3.25. If Lim x_{i} → x where x_{i} ∈ J and x ∈ Bd(J) and the x_{i}’s approach x asymptotically along a tangential direction v ∈ T_{x}^{1}, then v ∈ A_{L}, for all leaves of Σ^{f} through x.

Proof. If not let E ⊂ T_{x}^{1} −A_{L} be a small round disc such that v ∈ E^{∘}. If y is sufficiently close to x and in a small cone based at x defined by the directions in E, then y lies on the non + side of L. This implies that y ∉ J. q.e.d.

Definition 3.26. A point x ∈ Bd(J) is of Type I if either there exists a vector u based at x which is transverse to each leaf of Σ^{f} passing through x or x is a spike point which is not an antitangent point. Otherwise we say x ∈ Bd(J) is of Type 0. Let O denote the collection of Type 0 points of Bd(J).

Lemma 3.27. If x is of Type 0, then x is an antitangent point.

Proof. We need to show that if ∪_{L leaf through x}C_{L} = T_{x}^{1}, then x is either a spike point or an antitangential point. Under this hypothesis A_{x} is not a disc and by Lemma 3.25, A_{x} ≠ ∅.

If A_{x} is an interval, then let w ∈ A^{∘}_{x} and [−1, 1] ⊂ T_{x}^{1} a small geodesic interval orthogonal to A_{x} and passing through w at 0. We will show that x is an antitangential point by showing that there exist leaves R and S such that A_{x} ⊂ C_{R} (resp. A_{x} ⊂ C_{S}) and the + side of R (resp. S) points towards −1 ∈ [−1, 1] (resp. +1). If not we derive a contradiction as follows. If t > 0, and L_{t} is a leaf with t ∉ A_{L_{t}}, then C_{L_{t}} hits [−1, 1] at some unique point in (0, t) with normal pointing towards 0. Since C_{L_{t}} is

disjoint from A^{∘}_{x} it follows that as t → 0, C_{L_{t}} becomes nearly tangent to A_{x}. Using Lemmas 3.12, 3.18, we conclude that a limit corresponds to a leaf R such that C_{R} contains A_{x} with normal pointing towards −1. In a similar manner one constructs a leaf S through x so that C_{S} contains A_{x} with normal pointing to +1.

If w ∈ A_{x} and A_{x} is one or two points, then let T ⊂ T_{x}^{1} be the circle of vectors orthogonal to w. Using a limiting argument as above, for each t ∈ T, there exists a leaf L_{t} through x such that t ∉ A^{∘}_{L_{t}} but w ∈ C_{L_{t}}. Just consider a limit of L_{t_{s}}’s where t_{s} → w and t_{s} lies on the geodesic arc from t to w and t_{s} ∉ A_{L_{ts}}. This implies that either

∪_{t∈T} A^{∘}_{L_{t}} = S_{∞}^{2} − {w, −w} or S_{∞}^{2} − γ where γ lies in a great semi-circle from w to −w. In the latter case a limiting argument implies that x is an antitangent point. In the former case a compactness argument

implies that for a finite set {t_{1}, t_{2}, …, t_{n}}, ∪_{i=1}^{n}A^{∘}_{L_{t_{i}}} = S_{∞}^{2} − {w, −w}.

If no pair A_{t_{i}}, A_{t_{j}} are antitangent at x, then a combinatorial argument implies n = 3 suffices and hence x is a spike point. q.e.d.

Corollary 3.28. Type 0 boundary points of J of multiplicity 1 are a closed discrete subset of ℍ^{3}.

Definition 3.29. If X ⊂ Y, then we say that X is 0-LC at x ∈ Bd(X) if for each open set U of Y containing x, there exists an open set V of Y about x such that if y, z ∈ X ∩ V, then there exists a path from y to z lying in X ∩ U.

Lemma 3.30. If x ∈ Bd(J) is Type I, then J is a manifold near x.

Proof. The point of this paragraph and the next is to show that a Type I spike point can be treated just like any Type I point which satisfies the first sentence of Definition 3.26. Suppose the leaves A, B, and C of Σ^{f} define a spike at x with S the germ of the spike region emanating from x and v a unit vector tangent to each of the three leaves. Let P denote the union of A, B and C together with the collection of leaves of Σ^{f} which pass through x and which are limits of leaves L_{1}, L_{2}, … of Σ^{f} such that for all i, there exists x_{i} ∈ L_{i} with x_{i} ∈ J − x and Lim x_{i} = x. As η → 0, B_{ρ}(η, x) ∩ S is contained in narrower and narrower hyperbolic cones based at x. Therefore v must be tangent to every leaf in P. By Lemmas 3.12 iv) and 3.18 ii) it follows that either x is an antitangential point or there exists a nonzero vector u based at x and transverse to each leaf in P.

If x is Type I because there exists a vector u transverse to each leaf of Σ^{f} through x, then define Σ to be Σ^{f}. Otherwise x is not antitangential and there exists a u as in the preceding paragraph. Let U = N^{∘}_{ρ}(η, x) and define Σ to be the collection of leaves of Σ^{f} which hit J ∩ U − x together with all limits of such leaves which pass through x. Choose η ≤ ϵ sufficiently small so that if L is a leaf of Σ passing through x, then L is transverse to the vector u constructed in the previous paragraph. Such an η exists, else one can find a sequence L_{1}, L_{2}, … of leaves containing x which hit Bd(J) in points closer and closer to x and have u as a tangent vector. A limit of such leaves lies in P and is tangent to u, a contradiction.

Since x is of Type I, and leaves have bounded normal curvature, there exists a closed neighborhood of the form V = D^{2} × [−1, 1], where x is identified with the origin, each z ×[−1, 1] is transverse to Σ and if L is a leaf of Σ|V, and L∩J ≠ ∅, then L is a closed disc properly embedded in D^{2}×(−1, 1). Call a leaf L of Σ, + if its normal vector points up and − otherwise. If no + leaf meets x, then by restricting the size of V we can assume that the set of + leaves is empty. Similarly for the − leaves. If there exist + leaves (resp. − leaves) define the function A : D^{2} → [−1, 1] (resp. B : D^{2} → [−1, 1]) by A(z) = max{t| some + leaf hits (z, t)} (resp. B(z) = min{t| some − leaf hits z × t}). That Σ is closed in the C^{k}-topology implies such maxima and minima exist.

Claim 1. A and B are continuous.

Proof. We prove continuity of A. Let y_{1}, y_{2}, … → y be a convergent sequence in D^{2}. By Lemma 3.18 ii) and 3.12 iv) it follows that Lim sup A(y_{i}) ≤ A(y). Since there exists a + leaf L through (y, A(y)) and transverse to y × [−1, 1] it follows that A(y) ≤ Lim inf {A(y_{i})}.

q.e.d.

If there exists no − leaves, then from the continuity of A one readily deduces that J is a manifold near x.

From now on we will assume that there exist both + and − leaves. Applying the standard results for intersections of least area surfaces, the Meeks-Yau exchange roundoff trick and the fact that J ∩ V ≠ ∅ to our setting we have:

Claim 2. A + leaf and a − leaf cannot coincide. If L_{1} and L_{2} are distinct leaves of Σ, then L_{1} is transverse to L_{2} except at finitely many (multi)-saddle tangencies. L_{1} ∩ L_{2} contains no embedded simple closed curve. q.e.d.

Let π : J ∩ V → D^{2} =^{def} D^{2} × 0 be the projection onto the D^{2}-factor. Let J_{1} be a component of V ∩ J and E_{1} =^{def} π(J_{1}) ⊂ D^{2}.

Claim 3. E^{∘}_{1} = π(J_{1} ∩ V^{∘}) and E^{∘}_{1} is a disc.

Proof. The failure of the either claim would give rise to plus and minus leaves L_{1}, L_{2} of Σ violating Claim 2. q.e.d.

Claim 4. E^{∘}_{1} is 0-LC at every point x of Bd(E^{∘}_{1}).

Proof. It follows directly from the next paragraph that given κ > 0 each component of B_{D^{2}}(κ/2, x)∩E^{∘}_{1} is contained in one of finitely many

components of B_{D^{2}}(κ, x)∩E^{∘}_{1}. The subsequent paragraph will show that

for some η < κ/2 at most one of the components of B_{D^{2}}(κ, x) ∩ E^{∘}_{1} can meet B_{D^{2}}(η, x). These facts imply Claim 4.

We will assume that x ∈ D^{2}^{∘} and d_{D^{2}}(x, ∂D^{2}) > κ, for the general case is similar but notationaly messier. We first show that if ϵ_{1} < ϵ_{2} ≤ κ, and A is the annulus spanning the circles S_{1}, S_{2} of radius ϵ_{1}, ϵ_{2} about x, then only finitely many distinct components of E^{∘}_{1} ∩ A can hit both S_{1} and S_{2}. Otherwise let α_{1}, α_{2}, … be properly embedded arcs lying in distinct components of E^{∘}_{1} ∩ A whose endpoints meet both S_{1} and S_{2}. By passing to subsequence, reordering and choosing a correct basepoint and orientation on S_{1} we can assume that the α_{i}’s are ordered according the linear ordering of the points {α_{i} ∩S_{1}}. Let S_{3} be the circle of radius ϵ_{3} about x, where ϵ_{1} < ϵ_{3} < ϵ_{2} and let R_{i} ⊂ A denote the region between α_{i} and α_{i+1}. Finally for all i, let s_{i} ∈ S_{3} ∩ R_{i} be such that s_{i} ∩ π(J) = ∅ and s a limit point of s_{1}, s_{2}, . . .. Let L_{i}^{+} (resp. L_{i}^{−}) denote a + leaf (resp. − leaf) through (s_{i}, A(s_{i})) (resp. (s_{i}, B(s_{i})). Let A_{i} = {z ∈ R_{i}|L_{i}^{+} is above L_{i}^{−} at z}. Since least area surfaces intersect transversely or in multi-saddles or coincide it follows that A_{i} ≠ ∅. The A_{i}’s being disjoint implies that Lim area(A_{i}) → 0. Let i_{1} < i_{2} < … be a sequence such that L_{i_{1}}^{+}, L_{i_{2}}^{+}, … limits to L^{+}, L_{i_{1}}^{−}, L_{i_{2}}^{−}, … limits to L^{−} and L^{+}, L^{−} respectively pass through (s, A(s)) = (s, B(s)). If Z = {z ∈ A|L^{+} is above L_{−} at z} and C_{1} > 0 is the area of the smallest

(of the finitely many) components of Z^{∘} which limit on s, then for i sufficiently large, area(A_{i}) > \frac{1}{2}C_{1} > 0.

The previous paragraph implies that if η ≤ κ and F is a component of B(η, x) ∩ E^{∘}_{1} which limits on x, then for y ∈ F there exists an embedded path from y to x which, except for x, lies in F. If two such components F_{1}, F_{2} limit on x, then there exists a simple closed curve α in E_{1} which intersects Bd(E^{∘}_{1}) only at x. If D_{α} was the disc bounded by

α, then D^{∘}_{α} ∩ Bd(E^{∘}_{1}) ≠ ∅. Let z ∈ D^{∘}_{α} be such that A(z) = B(z). If L_{1} and L_{2} are respectively plus and minus leaves passing through (z, A(z)), then L_{1} ∩ L_{2} gives rise to a contradiction to Claim 2. q.e.d.

Claim 5. For every κ > 0, E^{∘}_{1} is the union of finitely many connected subsets of of diameter less than κ.

Proof Use Claim 4 and the compactness of Bd(E^{∘}) to find an η > 0

so that N^{∘}_{D^{2}}(η, Bd(E^{∘})) ∩ E^{∘} is contained in the union of finitely many

connected subsets of E^{∘} of diameter < κ. Any maximal collection {y_{i}} of points in E^{∘} − N^{∘}_{D^{2}}(η, Bd(E^{∘})), with the property i ≠ j implies d_{D^{2}}(y_{i}, y_{j}) ≥ η gives rise to a finite set F of open η-discs centered at

the points of F which contain E^{∘} − N^{∘}_{D^{2}}(η, Bd(E^{∘})). q.e.d.

Claim 6. E_{1} is a disc.

Proof. Using techniques similar to those of [3, pp. 26–32] and Claim 4 it follows that E is the image of an immersed disc whose interior maps to E^{∘}. Actually it is an embedding otherwise one obtains a contradiction to Claim 2. What follows is a formal argument.

In the classical language [22] a metric space satisfying the conclusion

of Claim 5 is said to satisfy Property S. By Theorem 4.2 [22], Bd(E^{∘}_{1}) is

a locally connected compact continuum such that each point of Bd(E^{∘}_{1})

is accessible from all sides from E^{∘}_{1}.

If Bd(E^{∘}_{1}) had a cut point y, then being accessible from all sides implies that there exists a simple closed curve α in E_{1} which meets Bd(E^{∘}_{1}) only at y and the disc bounded by α contains points of Bd(E^{∘}_{1}) in it’s interior. This leads to a contradiction to Claim 2.

Thus Bd(E^{∘}_{1}) is a Peano continuum without cut points, p. 76 [23],

and so by Corollary 3.32a [23] through each point of Bd(E^{∘}_{1}) there exists

a simple closed curve lying in Bd(E^{∘}_{1}).

Bd(E^{∘}_{1}) must be a simple closed curve, else Bd(E^{∘}_{1}) contains an

embedded graph θ of Euler characteristic −1. Being connected, E^{∘} lies in only one component of D^{2} − θ, contradicting the fact that each point of Bd(E^{∘}_{1}) is a limit point of E^{∘}_{1}. By the Schoenflies theorem Bd(E^{∘}_{1}) bounds a disc which is evidently E_{1}. q.e.d.

Claim 7. π(x) has a neighborhood that intersects exactly one component of π(J ∩ V).

Proof. First suppose that there exist two components F_{1}, F_{2} of π(J ∩ V) with closures E_{1}, E_{2} which contain x. By Claim 6 these E_{i}’s are discs and the connectivity of J implies that both must intersect ∂D^{2}. Let y_{1}, y_{2} ∈ ∂D^{2} be such that A(y_{i}) = B(y_{i}) and y_{1}, y_{2} separate F_{1} ∩ ∂D^{2} from F_{2} ∩ ∂D^{2}. Let L_{i}^{+}, L_{i}^{−}, i = 1, 2 be leaves of Σ respectively of plus and minus type which pass through (y_{i}, A(y_{i})). Finally consider the (at most 4) laminations σ_{0k}^{α_{i}} which contain these leaves. If J′ is the component of ℍ^{3} − ∪σ_{0k}^{α_{i}} which contains J, then J′ is either non-simply connected or J^{∗} the closure of J′ with respect to the induced path metric is not injectively immersed. Either situation contradicts Lemma 4.1 [6].

If every neighborhood of π(x) hits infinitely many components of π(J ∩ V), then we obtain a contradiction in a manner similar to that of the second paragraph of the proof of Claim 4. q.e.d.

Claims 6 and 7 imply that after reducing the size of the D^{2} and reparametrization π(J ∩ V) = {(x, y) ∈ D^{2}|y > 0}. Therefore J ∩ V = {(x, y, t)|y ≥ 0 and A(x, y) ≤ t ≤ B(x, y)} which is a tamely embedded 3-cell, thereby completing the proof. q.e.d.

Lemma 3.31. J is simply connected.

Proof. Let F be the foliation of ℍ^{3} by totally geodesic hyperbolic planes orthogonal to δ_{0}. Let α ⊂ J be a simple closed curve homotopically nontrivial in J. Choose α to be transverse to F except at finitely many points and assume that α has been chosen to minimize this number. Our α may lie completely in a leaf of F.

Case 1. α bounds an embedded disc D in ℍ^{3}.

Proof. Let N ⊂ J be a closed tubular neighborhood of α and N_{1} a smaller tubular neighborhood with N_{1} ⊂ N^{∘}. Modify the metric r to r_{1} so that r|ℍ^{3} − N = r_{1}|ℍ^{3} − N and ℍ^{3} − N^{∘}_{1} has strictly convex

boundary with respect to metric r_{1}. Furthermore if E ⊂ ℍ^{3} − N^{∘}_{1} is a disc with ∂E ∩ N = ∅, then area_{r_{1}}(E) ≥ area_{r}(E). By [15] there exists

an essential properly embedded r_{1}-least area disc D ⊂ ℍ^{3} − N^{∘}_{1} with ∂D ⊂ ∂N_{1}. Furthermore D is least area among all essential immersed discs with boundary ∂D on ∂N_{1}. If D ∩ L ≠ ∅ where L is a leaf of some σ_{0j}^{α}, then since L is a leaf of a D^{2}-limit lamination, there exists an embedded disc F ⊂ L, with ∂F ∩ D = ∅ and F ∩ D ≠ ∅. Since F is an r-least area disc and D is locally r-least area near D∩F, D is transverse to F except at finitely many tangencies of the standard saddle or multisaddle type. A simple closed curve in D ∩ F bounds subdiscs D′ ⊂ D, F′ ⊂ F with ∂D′ = ∂F′. If area_{r_{1}}D′ ≥ area_{r_{1}} F′, then the Meeks-Yau exchange roundoff technique gives rise to a disc contradicting the r_{1}-area minimality of D. If area_{r_{1}} D′ < area_{r_{1}} F′, then the exchange roundoff technique gives rise to a disc contradicting the r-area minimality of F, since by construction area_{r} D′ ≤ area_{r_{1}} D′.

q.e.d.

Case 2. α is homotopically nontrivial in J.

Proof. We first show that α is homologically trivial in J. Let r and r_{1} be Riemannian metrics as in Case 1. By [15] there exists an

oriented genus minimizing, properly embedded surface S ⊂ ℍ^{3} − N^{∘}_{1} with connected boundary which is an r_{1}-least area surface among all immersed surfaces properly homotopic to S rel ∂S in ℍ^{3} − N^{∘}_{1}. (S can be thought of as a minimal genus Seifert surface for α.) Again if S ∩σ_{0j}^{α} ≠ ∅, there exists a disc F lying in a leaf of σ_{0j}^{α} which nontrivially intersects S and ∂F ∩ S = ∅. Let F′ ⊂ F be a disc such that ∂F′ ⊂ S. Since S is π_{1}-injective in ℍ^{3} − N^{∘}_{1} it follows that ∂F′ bounds a disc D′ ⊂ S. We now obtain a contradiction as in Case 1.

Therefore if T is a (connected) leaf of F|J and T is transverse to α, then |T ∩ α| ≥ 2. If for all leaves T of F|J, |T ∩ α| ≤ 2, then α is unknotted in ℍ^{3}. Finally, if |T ∩ α| > 2, and x, y ∈ T ∩ α are such that the oriented α points “up” at both x and y, then we obtain a homotopically nontrivial curve in J with fewer points of tangency with F via the following procedure. Concatenate the appropriate component of α − {x, y} with an arc in T with endpoints x, y and isotope slightly.

q.e.d.

Definition 3.32. Given x ∈ Bd(J), η < ϵ and L a leaf of Σ^{f} we say that L is x_{η} − relevant if L ∩ J ∩ B_{ρ}(η, x) ≠ ∅.

Lemma 3.33. For each x ∈ Bd(J) there exists ϵ_{x} ≤ ϵ such that if η ≤ ϵ_{x} and L is x_{η}-relevant, then L ∩ B_{ρ}(η, x) is a disc transverse to ∂B_{ρ}(η, x) and each component of ∂B_{ρ}(η, x) − L has area at least .40 area(∂B_{ρ}(η, x))).

Proof. It follows from Schoen’s bounded normal curvature lemma [19] that there exists a constant C such that if η is sufficiently small, β < Cη and L any leaf of Σ^{f}|B_{ρ}(η, x) which intersects B_{ρ}(β, x), then L satisfies the conclusion of the lemma. Since the metric r is induced from a metric on a closed hyperbolic 3-manifold, C can be chosen independent of x.

Suppose that L_{1}, L_{2}, … were a sequence of x_{η_{i}}-relevant leaves which failed the conclusion of the lemma, where η_{i} → 0. Let y_{i} be the ρ-closest point of L_{i} to x. By passing to subsequence we can assume that y_{i} approaches x asymptotically along a ray. Since d_{ρ}(L_{i}, x) > Cη it follows again by Schoen, that for i sufficiently large and j sufficiently larger L_{j} ∩L_{i} ∩B_{ρ}(η_{i}, x) = ∅ and L_{j} ∩B_{ρ}(η_{i}, x) separates x from L_{i} ∩B_{ρ}(η_{i}, x). Since x lies on the + side of L_{j} it follows that J ∩ L_{i} ∩ B_{ρ}(η_{i}, x) = ∅, a contradiction. q.e.d.

Lemma 3.34. If x ∈ J is a Type 0 point, then J is a manifold near x.

Proof. The proof will follow by induction on m(x).

Step 1. m(x) = 1.

Proof of Step 1. By Lemma 3.22 there exists ϵ_{1} such that 0 < ϵ_{1} < ϵ_{x} and B_{ρ}(ϵ_{1}, x) contains a unique Type 0 point. Let B_{t} denote B_{ρ}(t, x).

Claim 1. If 0 < t ≤ ϵ_{1}, then there exists t_{1} > 0 so that for each component X of B_{t} ∩ J either x ∈ X or d_{ρ}(X, x) > t_{1}.

Proof. If not then B_{t} ∩ J has infinitely many components which hit B_{t/2}. Let y_{1}, y_{2}, … a sequence of points lying in distinct components of B_{t} ∩ J, with y_{i} ∈ ∂B_{t/2}. If y is a limit point of {y_{i}}, then y ∈ J and is of Type I. But this contradicts the fact that J is a manifold near y.

q.e.d.

Claim 2. There exists a unique component J_{i} of B_{ϵ_{1}} ∩ J which limits on x.

Proof. It follows from Claim 1 (with t = ϵ_{1}) that at least one component limits on x.

Suppose that two components J_{1} and J_{2} limit on x. For i = 1, 2 let E^{∘}_{i} be the component of ∂B_{ϵ_{1}} ∩ J_{i} such that there exists a path

σ ⊂ (J − B^{∘}_{ϵ_{1}}) from E_{1} to E_{2}. Each E^{∘}_{i} is an open disc else:

i) There exists a leaf L tangent to ∂B_{η}, η ≤ ϵ_{1} at a point y ∈ J. This contradicts Lemma 3.33.

ii) There exist leaves L_{1} and L_{2} such that L_{1} ∩ L_{2} ≠ ∅ but L_{1} ∩ ∂B_{ϵ_{1}} and L_{2} ∩ ∂B_{ϵ_{1}} lay in disjoint components of ∂B_{ϵ_{1}} − E_{i}. This implies that L_{1} ∩ L_{2} contains a simple closed curve, which is a contradiction.

A proof similar to that of Lemma 3.30 shows that each E_{i} is a closed disc.

To complete the proof it suffices to show that there exists a finite set of leaves of Σ^{f}|B_{ϵ_{1}} whose boundaries separate E^{∘}_{1} from E^{∘}_{2}. Indeed, if σ_{0i_{1}}^{α_{1}}, …, σ_{0i_{n}}^{α_{n}} were the laminations containing these leaves and Ĵ the component of ℍ^{3} − ∪_{k=1}^{n}σ_{0i_{k}}^{α_{k}} which contained J, then Ĵ^{∗} is either not simply connected or not injectively immersed in ℍ^{3}. (Here Ĵ^{∗} is the closure of Ĵ with respect the induced path metric.) This violates Lemma 4.1 [6].

Since x is an antitangency point there exists a wedge W containing B_{ϵ_{2}} ∩ J. By reducing the size of ϵ_{1}, if necessary, we can assume that area_{ρ}(D_{W}) < .01 area ∂B_{ϵ_{1}} where D_{W} = W ∩ ∂B_{ϵ_{1}}. Suppose that W is defined by the antitangent leaves A, B. We will show that n ≤ 4 where two of the leaves are A and B. The other one or two leaves will intersect E_{1}.

Parametrize ∂E_{1} by [0, 1] mod 1. (From the observer standing on B_{ϵ_{1}}, choose the parametrization to correspond to a clockwise path about ∂E_{1}.) If t ∈ ∂E_{1}, then L_{t} will denote a leaf of Σ^{f} ∩ B_{ϵ_{1}} through t. L_{t} is not in general unique. Since E_{1} ∪ E_{2} ⊂ D_{W} it follows from the previous paragraph and Lemma 3.33, that for any t, ∂L_{t} ∩ ∂D_{W} ≠ ∅.

If there exists a leaf L_{0} such that ∂L_{0} ∪ ∂D_{W} separates E^{∘}_{1} from E^{∘}_{2} then L_{0}, A, B are our desired leaves. Otherwise fix a L_{0}. Next consider a L_{t}. If ∂L_{0} ∪ ∂L_{t} ∪ ∂D_{W} separate E^{∘}_{1} from E^{∘}_{2} we are done, otherwise there exists a path α from E^{∘}_{2} to E^{∘}_{1} missing ∂L_{0} ∪ ∂L_{t} ∪ ∂D_{W}. We say L_{t} is to the left of 0 if α can be chosen to miss (0, t). Otherwise we say that L_{t} is to the right of 0. If there exists leaves L and L′ both of which pass through t with one to the left of 0 and the other to the right of 0, then ∂L ∪ ∂L′ ∪ ∂D_{W} separate E^{∘}_{1} and E^{∘}_{2}. Again we are done if there is a L_{t} to the left (resp. right) and a L_{t′} to the right (resp. left) with 0 < t′ ≤ t (resp. 0 < t ≤ t′). If L_{t_{1}}, L_{t_{2}}, … limits to L_{s} and each L_{t_{i}} is to the left (resp. right) of 0, then either L_{s} is to the left (resp. right) of 0 or ∂L_{s} ∪ ∂L_{0} ∪ ∂D_{W} separate. If L_{t}, t ≠ 0 is to the left and L′ is a limit of leaves L_{t}, t ∈ (0, 1), t → 1, then ∂L_{0} ∪ ∂D_{W} ∪ ∂L′ separate.

q.e.d.

By Claims 1 and 2 there exists an embedded path α : [0, 1] → ℍ^{3} such that α([0, 1)) ⊂ J and α(1) = x. We can assume α is transverse to

∪_{i=1}^{∞}∂B_{s_{i}}, where s_{i} = ϵ_{1}/i. Also if F^{∘} is a component of ∂B_{s_{i}} ∩ J, then

|F^{∘} ∩ α| ≤ 1. Let F^{∘}_{1}, F^{∘}_{2}, … denote the collection of such discs ordered by how they are hit by α. Let t_{i} ∈ {s_{i}} denote that value so that F_{i} ⊂ ∂B_{t_{i}}. If j > i and j, i ∈ ℕ, then let R_{(i,j)} denote the component

of J between F^{∘}_{i} and F^{∘}_{j}. Let R_{i} denote R_{(i,i+1)}.

The region R_{i} is simply connected since each of F^{∘}_{i}, F^{∘}_{i+1} and J are simply connected. The continuity of α and Claim 1 imply that for each t ≤ ϵ_{1} there exists N_{t} > 0 such that R_{(i,j)} ⊂ B_{t} for j > i ≥ N_{t}. We now show that for i > N_{ϵ_{1}}, R_{i} is a 3-ball. First, each point of Bd(R_{i})

not on F^{∘}_{i}, is a Type I point of Bd(J). Second, by Lemma 3.33 ∂B_{t_{i}} is transverse to each x_{t_{i}}-relevant leaf. Thus the argument of Lemma 3.30 shows that R_{i} is a manifold near each point of Bd(F_{i}) ∪ Bd(F_{i+1}). Thus R_{i} is an irreducible compact simply connected manifold and hence is a 3-ball. A similar argument shows that for j > i, R_{(i,j)} is a 3-ball. By the second sentence of this paragraph the sequence of balls R_{j}, limit only on x. Thus, if X is the component of J −F_{i} which limits on x, then X is homeomorphic to ∪_{j≥i}R_{j} ∪ x where the latter space is topologized with the 1-point compactification with x being the point at infinity. Thus X is topologically a 3-ball. This completes the proof of Step 1. q.e.d.

Step 2. 1 < m(x) ≤ M.

Proof of Step 2. Assume by induction that the theorem is true for {x|m > m(x) ≥ 1}. We will prove it for m(x) = m. The proof of Step 2 follows exactly like the proof of Step 1 with the following minor modifications. By Lemma 3.22, for almost all s, ∂B_{s} ∩ O = ∅. Choose ϵ_{1} so that ∂B_{ϵ_{1}} ∩ O = ∅. In the proof of Claim 1 use only {t|(∂B_{t} ∪ ∂B_{t/2}) ∩ O} = ∅. In the paragraph after the proof of Claim 2, choose {s_{i}} so that ϵ_{1} > s_{1} > s_{2} > … → 0 and for each i, ∂B_{s_{i}} ∩ O = ∅.

q.e.d.

Remark 3.35. It follows immediately from the proof of Lemma 3.30 that ∂J is tamely embedded in ℍ^{3} at each Type I point of ∂J. A more extensive argument shows that ∂J is tamely embedded at all the Type 0 points. Therefore by Bing [3] and Moise [Me] ∂J is tamely embedded in ℍ^{3}.

Proposition 3.36. If J is a component of H_{0}, then J is a simply connected manifold with boundary whose interior is J.

Proof. Apply Lemmas 3.30, 3.31 and 3.34. q.e.d.

A collection of smooth simple closed curves λ̂_{i} in S_{∞}^{2} is locally finite if in the unit ball model, for every η > 0 there are only finitely many λ̂_{i}’s of diameter > η.

Lemma 3.37. Let x, y ∈ S_{∞}^{2} and {λ̂_{i}} be a locally finite collection of smoothly embedded simple closed curves in S_{∞}^{2} − {x, y}, such that no λ̂_{i} separates x from y. Let r be a Riemannian metric induced from a closed hyperbolic 3-manifold. To each λ̂_{i}, let Σ̂_{i} be the union of r-least area D^{2}-limit laminations which span λ̂_{i}. Let Σ̂ = ∪Σ̂_{i}. Let Ĥ = ℍ^{3}−Σ̂.

If Ĵ is a component of Ĥ, then Ĵ is open and simply connected.

Proof. The proof of Lemma 3.10 shows that Ĥ is an open set. Simple connectivity follows from the proof of Lemma 3.31. q.e.d.

Lemma 3.38. Let x, y ∈ S_{∞}^{2}, r a Riemannian metric on ℍ^{3} induced from the closed 3-manifold M and λ̂_{1}, …, λ̂_{m} smooth simple closed curves in S_{∞}^{2}−{x, y} such that no λ̂_{i} separates x from y. For each i, let Σ̂_{i} be the union of r-least area D^{2}-limit laminations which span λ̂_{i} and let H_{i} be the complementary region of 𝔹^{3} − Σ̂_{i} which contains x, y. Then one of the following must hold:

i) x, y lie in the same component H of ∩_{i=1}^{m}H_{i}.

ii) There exist λ̂_{i}, λ̂_{j}, λ̂_{k} such that λ̂_{i} ∪ λ̂_{j} ∪ λ̂_{k} separate x from y in S_{∞}^{2}.

Proof. This is Lemma 4.3 [6] with Σ̂_{i} in place of σ_{i}. If m ≤ 3 and ii) does not hold, then x, y lie in the same component of S_{∞}^{2} − ∪λ̂_{i}, so i) follows by Proposition 3.9 [6]. Assuming inductively that the lemma is true for m < p, we will establish it for cardinality p. Therefore either ii) holds or

(∗) for every j ≤ p, x and y lie in the same component of ∩_{i≠j} H_{i}.

We show that if (∗) holds, then either i) holds or for each j and k, λ̂_{j} ∩ λ̂_{k} ≠ ∅. Let τ_{j} ⊂ ∩_{i≠j}H_{i} (resp. τ_{k} ⊂ ∩_{i≠k}H_{i}) be a properly embedded path from x to y. By Lemma 3.37, each component of int(∩_{i∉{j,k}}H_{i}) is simply connected. Thus, there exists h : I × I → ∩_{i∉{j,k}}H_{i} a homotopy from τ_{j} to τ_{k}. Either Σ̂_{k} ∩ Σ̂_{j} ≠ ∅ and hence τ_{k} ∩ τ_{j} ≠ ∅ by the proof of Lemma 3.5, or h^{−1}(Σ̂_{k}) and h^{−1}(Σ̂_{j}) are disjoint closed sets which are disjoint from ∂I × I. Also Σ̂_{k} (resp. Σ̂_{j}) is disjoint from I × 0 (resp. I × 1). Thus, τ_{k} ∩ τ_{j} = ∅ implies that there exists an embedded path τ from 0×I to 1×I disjoint from h^{−1}(Σ̂_{j} ∪Σ̂_{k}). Finally, h∘τ, is a path from x to y in ∩_{i=1}^{p}H_{i} and so conclusion i) holds. (To construct τ, first engulf h^{−1}(Σ̂_{j}) and h^{−1}(Σ̂_{k}) in two disjoint families, each of which is a finite union of closed smooth regions disjoint from ∂I × I and I × 0 (for h^{−1}(Σ̂_{k})) or I × 1 (for h^{−1}(Σ̂_{j})).)

Now argue as in the last paragraph of the proof of Lemma 4.3 [6]. q.e.d.

Lemma 3.39. There exists a unique unbounded component of H_{0}. There exists a uniform bound on the diameter of the bounded regions.

Proof. By Proposition 3.9 [6] each σ_{0j}^{α} lies in an e-neighborhood of the ρ-convex hull of λ_{0j} and by Lemma 3.6 there are only finitely many outermost ⟨g⟩-orbits of λ_{0j}’s. Thus there exists N_{0} > 0 such the image under ρ-orthogonal projection of any σ_{0j}^{α} into δ_{0} has ρ-diameter < N_{0}/2 and hence the image under orthogonal projection of any Σ_{0j} has ρ-diameter < N_{0}. By Lemma 3.7 iii), H_{0} ⊂ N_{ρ}(a, δ_{0}). Let N_{1} = 2(length_{ρ}(δ) + N_{0} + a). Parametrize 𝔹^{3} − ∂δ_{0} by D^{2} × ℝ so that in these coordinates g(x, t) = (x, t + L), where L = length(δ) and 0 × ℝ parametrizes δ_{0} by arclength. Let p : ℍ^{3} → δ_{0} denote ρ-orthogonal projection.

In the next two paragraphs we show that there exists at most one component of H_{0} with diameter ≥ N_{1} and that any component of diameter ≥ N_{1} is unbounded. If J_{1} and J_{2} are distinct components of H_{0} such that [−N_{0}, N_{0}] ⊂ p(J_{1})∩p(J_{2}) then let β_{1}, β_{2} be paths respectively in J_{1}, J_{2} whose orthogonal projections are paths from −N_{0} to N_{0}. Let Σ^{∗} = ∪{Σ_{0j}|0 ∈ p(Σ_{0j})}. By choice of N_{0}, p(Σ^{∗}) ⊂ (−N_{0}, N_{0}) and so J_{1} and J_{2} lie in the same component C of ℍ^{3} − Σ^{∗}. Indeed there exists a closed loop β ⊂ ℍ^{3}−Σ^{∗} obtained by concatenating β_{1}, β_{2} with arcs lying in D^{2} × −N_{0} and D^{2} × N_{0}. By construction D^{2} × 0 ∩ Σ^{∗} = D^{2} × 0 ∩ Σ_{0}. Thus some component of (D^{2} × 0) ∩ C is hit algebraically nonzero times by β. This contradicts Lemma 3.37.

If J_{3} is a component of H_{0} of diameter ≥ N_{1} such that p(J_{3}) ≠ δ_{0}, then there exist distinct ⟨g⟩-translates J_{1}, J_{2} of J_{3} such that [−N_{0}, N_{0}] ⊂ p(J_{1}) ∩ p(J_{2}), which contradicts the previous paragraph. Thus if J_{1}, J_{2} are distinct components of diameter ≥ N_{1}, then they both must project to all of δ_{0} and one thereby obtains a contradiction as in the previous paragraph.

To complete the proof we need to show that there exists a component of H_{0} of diameter ≥ N_{1}. Let Z^{∗} = ∪{Σ_{0j}|p(Σ_{0j}) ∩ [−N_{1}, N_{1}] ≠ ∅ and λ_{0j} = h(λ_{0i}) for some i ∈ {1, 2, …, m} and h ∈ ⟨g⟩}. (Recall Lemma 3.6.) So Z^{∗} = Σ_{0j_{1}}, …, Σ_{0j_{k}}. It follows from Lemma 3.6 that H_{0}∩D^{2}×[−N_{1}, N_{1}] = ∩_{i=1}^{k}H_{0j_{i}} ∩D^{2}×[−N_{1}, N_{1}]. By Lemma 3.38, there exists a component J_{1} of 𝔹^{3} − Z^{∗} which contains a properly embedded path β connecting ∂δ_{0}. For each i, β must lie in H_{0j_{i}}. Thus β ∩ D^{2} × [−N_{1}, N_{1}] lies in some component of H_{0} and diam_{ρ}(β∩D^{2}×[−N_{1}, N_{1}]) ≥ 2N_{1}. q.e.d.

Remark 3.40. This proof together with the usual convergence of laminations results implies that if r lies in a compact region K of the space of Riemannian metrics on M, then there exists a uniform bound N(K) for the ρ-diameter of bounded regions of H_{0}.

Lemma 3.41. If P′ : ℍ^{3} → M_{δ} = D^{2}^{∘} × S^{1} is the quotient map under the action of ⟨g⟩, then P′(H_{0}) is the union of open balls and one

open solid torus denoted T^{∘}_{r}. The closure T^{∘}_{r} is a solid torus denoted T_{r}. Therefore H_{0} ⊂ ℍ^{3} is a union of uniformly bounded open balls and one component U^{∘}_{r} whose ℍ^{3}-closure is a D^{2} ×ℝ denoted U_{r} whose ends limit on ∂δ_{0}.

Proof. Each component Z of P′(H_{0}) has π_{1}(Z) ∈ {1, ℤ} since it is covered by a component of H_{0}, which by Lemma 3.31 is simply connected, with covering translations contained in ⟨g⟩. If Z is covered by a bounded region J, then by Proposition 3.36 and Lemma 3.39 J is a uniformly bounded ball and hence Z is a uniformly bounded embedded open ball. If Z is covered by the unique unbounded region U^{∘}_{r}, then the uniqueness of this manifold together with its ⟨g⟩-invariance and the fact that its closure U_{r} is a manifold implies that Z is a compact manifold with boundary. Z is irreducible since it is covered by an irreducible manifold. Therefore Z is a solid torus and hence U_{r} is a D^{2} × ℝ whose ends limit on ∂δ_{0}. q.e.d.

Remark 3.42. By further generalizing the argument of Step 3, p. 63 [6] it is not difficult to show that if Z is an open ball, then Z is a closed ball.

Lemma 3.43. Every core of T_{r} is a core of M_{δ}.

Proof. Let {σ_{ij}^{α_{k}}} be a π_{1}(M)-invariant collection of r-least area D^{2}-limit laminations such that for 1 ≤ i ≤ m, each λ_{0i} is spanned by 0 < n_{i} < ∞ laminations. Since only finitely many ⟨g⟩-orbits of laminations are involved, the technology of [6] applies and we obtain

an immersed solid torus V_{α} in M. Also if T^{∘}_{α} is the injective lift of V^{∘} _{α} to M_{δ}, then its closure T_{α} is an embedded solid torus. Similarly if U^{∘}_{α} denotes the lift of V^{∘} _{α} to ℍ^{3} which contains U^{∘}_{r}, then U_{r} the closure of U^{∘}_{r}, is an embedded D^{2} × ℝ limiting on ∂δ_{0}. Because δ lifts to a core of M_{δ}, C_{α} lifts to a core C̃_{α} ⊂ T^{∘}_{α} of M_{δ}.

Using the local structure of ∂U_{r} it is not difficult to show that given η > 0 there exists a collection of laminations {σ_{ij}^{α_{k}}} as above so that U_{r} ⊂ U_{α} ⊂ N_{ρ}(η, U_{r}) and hence T_{r} ⊂ T_{α} ⊂ N_{ρ}(η, T_{r}).

Let C_{r} be a core of V^{∘} _{r} with C̃_{r} its closed lift to M_{δ}. To show that C_{r} is a core of M_{δ} it suffices to show that π_{1}(M_{δ} − C̃_{r}) = ℤ ⊕ ℤ. This is equivalent to showing that if T is a solid torus standardly embedded in M_{δ} and T_{r} ⊂ T, then every loop κ ⊂ M_{δ} − C̃_{r} is homotopic in M_{δ} − C̃_{r} to a loop in M_{δ} − T.

Let κ be a loop in M_{δ} − C̃_{r}. Since C̃_{r} is a core of T_{r}, κ is homotopic in M_{δ} − C̃_{r} to a loop κ_{1} in M_{δ} − T_{r}. Let η < d_{ρ}(κ_{1}, T_{r}) and choose T_{α} so that T_{α} ⊂ N_{ρ}(η, T_{r}). As in the proof of Lemma 4.1 [6], each point of ∂U_{α} is of Type I and hence ∂T_{α} is tamely embedded in M_{δ}. Since C̃_{α} is isotopically standardly embedded in M_{δ} this implies that T_{α} is isotopically standard in M_{δ}. Therefore κ_{1} (resp. κ) can be homotoped off of any compact set in M_{δ} via a homotopy disjoint from T_{α} (resp. C_{r}).

q.e.d.

Proof of the Solid Torus Theorem: i) Our desired canonical immersion of a D^{2} × S^{1} (up to reparametrization) into M is given by q|T_{r}, where q : M_{δ} → M is the covering projection associated to the subgroup ⟨g⟩ ⊂ π_{1}(M). Note that q|T^{∘}_{r} is an embedding since T^{∘}_{r} ⊂ T^{∘}_{α} and by [6]

q|T^{∘}_{α} is an embedding. Here T^{∘}_{α} is any D^{2}^{∘} × S^{1} obtained by applying the insulator construction of [6]. Let V_{r} denote the immersed solid torus

q|T_{r} and V^{∘} _{r} = q(T^{∘}_{r}). (Recall Definition 3.1.)

ii) By Lemma 3.43 if C_{r} is a core of V^{∘} _{r}, then C_{r} lifts to a core C̃_{r}

of M_{δ}. Let V^{∘} _{α} ⊂ M be obtained from the insulator construction of [6]

and C_{α} a core of V^{∘} _{α}. Since C_{α} is isotopic to δ and C_{r} ⊂ C_{α} it suffices to show that C_{r} is a core of V^{∘} _{α}. This in turn is equivalent to showing that C̃_{r} is a core of T^{∘}_{α}. But as noted in the proof of Lemma 3.43, T_{α} is isotopically a standard solid torus in M_{δ}. Therefore C̃_{r} is isotopic to C̃_{α} in T^{∘}_{α} if and only if is isotopic in M_{δ}. By Lemma 3.43 they are both isotopic to a core of M_{δ}.

As indicated in 3.2, the orientation on δ induces an orientation on the core C_{r} of V^{∘} _{r}. This orientation has the property that if γ_{0} is the lift of C_{r} with ends limiting on ∂δ_{0}, then γ_{0} is oriented from the negative to the positive endpoint. If γ_{0} and δ_{0} are viewed as properly embedded arcs in 𝔹^{3}, then any isotopy of C_{r} to δ lifts to a proper isotopy of γ_{0} to δ_{0}. This implies that C_{r} and δ are isotopic as oriented curves.

iii) By the convexity property of Definition 0.4 [6] there exists, for each j, a totally geodesic plane P_{j} such that ∂P_{j} separates ∂δ_{0} from λ_{0j}. Therefore P_{j} separates δ_{0} from any ρ-least area D^{2}-limit lamination σ_{0j}

spanned by λ_{0j}. Thus δ_{0} ⊂ U^{∘}_{ρ} and δ ⊂ V^{∘} _{ρ}. Reversing the proof of ii) we

see that δ is a core of V^{∘} _{ρ}. That argument also shows that the orientation on δ, viewed as a core of V_{ρ} coincides with the given orientation on δ. q.e.d.

4. Proof of the Coarse Torus Isotopy Theorem

Notation 4.1. Let RM(M) denote the space of Riemannian metrics on M and f : B^{n} → RM(M) continuous. Let δ be an oriented simple closed geodesic in the closed orientable hyperbolic 3-manifold M possessing a non-coalescable insulator family {λ_{ij}}. Let V_{x} denote the canonical immersed solid torus in M associated to the Riemannian metric f(x). If x ∈ B^{n}, then let d_{x}(p, q) denote distance in M measured by the Riemannian metric f(x) and let d_{B^{n}}(x, y) denote distance measured in the standard metric on B^{n}. Let W_{x}^{ϵ} = {y ∈ V_{x}|d_{x}(y, ∂V_{x}) ≥ ϵ}.

Lemma 4.2. Let f : B^{n} → RM(M) be continuous, where M is a closed hyperbolic 3-manifold. There exists e > 0 such that if x ∈ B^{n}, λ a smooth simple closed curve in S_{∞}^{2} and σ an x-least area D^{2}- limit lamination spanning λ, then σ ⊂ N_{ρ}(e, C(λ)) where C(λ) is the hyperbolic convex hull of λ.

Proof. This statement is exactly the last sentence of Proposition 3.9 [6], except the e is uniform over all of B^{n}. The proof of Proposition 3.9 [6] shows that an e will work for a fixed metric x, provided that e satisfy the conclusion of Lemma 3.7 [6]. Lemma 3.7 states that if g : B^{1} → RM(M), then there exists a uniform e which works for all the metrics g(x). The proof of Lemma 3.7 works equally well for B^{n} as it does for B^{1}, the essential point being the compactness of the parameter space. q.e.d.

While the V_{x}’s may not vary continuously in x (see Remark 4.2 [6]) we do have the following key result.

Non-Encroachment Lemma 4.3. Let M be a closed oriented hyperbolic 3-manifold, δ a simple closed geodesic possessing the (π_{1}(M), {∂δ_{i}}) non-coalescable insulator family {λ_{ij}} and f : B^{n} → RM(M) a continuous map.

If x ∈ B^{n} and ϵ > 0, then there exists η > 0 such that W_{x}^{ϵ} ⊂ W_{y}^{ϵ/2} if d_{B^{n}}(x, y) < η.

Proof. For t ∈ B^{n}, let U_{t} denote the lift of V_{t} to ℍ^{3} whose ends limit on ∂δ_{0}. We need to show that if z ∈ U_{x} and d_{ρ}(z, ∂U_{x}) ≥ ϵ, then for t sufficiently close to x, z ∈ U_{t} and d_{ρ}(z, ∂U_{t}) ≥ ϵ/2.

If this is false, then by the compactness of U_{t}/⟨g⟩, there exists x_{1}, x_{2}, … converging to x and a z ∈ U_{x} such that d_{ρ}(z, ∂U_{x}) ≥ ϵ and for all i either z ∉ U_{x_{i}} or d_{ρ}(z, ∂U_{x_{i}}) ≤ ϵ/2. Therefore we can assume one of the following holds:

i) For all i, z ∈ U_{x_{i}}.

ii) For all i, z ∉ U_{x_{i}}.

If i) occurs, then by Lemmas 3.6 and 4.2 after passing to subsequence we can find a fixed λ_{0k} such that for all i ∈ ℕ there exists a x_{i} − D^{2}-limit lamination σ_{0k}^{i} spanning λ_{0k} and y_{i} ∈ σ_{0k}^{i} such that d_{ρ}(y_{i}, z) ≤ ϵ/2. Again passing to subsequence we can assume that y_{i} → y, where d_{ρ}(y, z) ≤ ϵ/2 and σ_{0k}^{i} → σ_{0k} a x-least area D^{2}-limit lamination spanning λ_{0k}. (This uses Proposition 3.10 [6] which is stated for metrics parametrized by B^{1} rather than B^{n}, but the proof works for the more general setting.) By definition of convergence, y ∈ σ_{0k}, which implies the contradiction d_{ρ}(x, ∂U_{x}) ≤ ϵ/2.

We now show that ii) cannot occur. Since z ∈ U^{∘}_{x}, q(z) ∈ V^{∘} _{x} and

hence there exists a core γ of V^{∘} _{x} passing through q(z). Here q : ℍ^{3} → M is the universal covering space projection. The lift γ̃ ⊂ U_{x} passing through z limits on ∂δ_{0}. Arguing as in the previous paragraph it follows that if y ∈ B^{n} is sufficiently close to x, then γ̃ ∩ σ_{0k}^{α} = ∅ for all y-least area D^{2}-limit laminations σ_{0k}^{α}. Since γ̃ lies in the component of 𝔹^{3} −σ_{0k}^{α} containing ∂δ_{0}, for every σ_{0k}^{α}, it follows that γ̃ ⊂ U_{y} and hence z ∈ U_{y}.

q.e.d.

Remark 4.4. The η of Lemma 4.3 is probably not uniform in x. Indeed, the regions U_{x_{i}} may limit to a disconnected region, one component of which is U_{x}. In the limit a piece of the U_{x_{i}}’s may get pinched off at an antitangential point or a spike point.

Lemma 4.5. Let M be a closed hyperbolic 3-manifold with geodesic δ satisfying the insulator condition. If f : S^{n} → Diff_{0}(M), then there exists a cellulation Δ^{∗} of B^{n+1} such that for each cell σ of Δ^{∗}, there exists a solid torus T_{σ} such that:

i) If κ is a proper face of σ, then T_{σ} is a core of T^{∘}_{κ}.

ii) If x ∈ σ ∩ S^{n}, then f_{x}(δ) is a core of T^{∘}_{σ}.

Proof. Define h : S^{n} → RM(M) by h(x) = (f_{x})_{∗}(ρ), the push forward of ρ. By the contractibility of the space of Riemannian metrics on a closed 3-manifold, there exists g : B^{n+1} → RM(M) extending h. Fix a non-coalescable insulator family for δ. Thus we can define the canonical solid tori V_{x} and the sets W_{x}^{ϵ} ⊂ V^{∘} _{x} for each x ∈ B^{n+1}.

For each x ∈ B^{n+1} let T_{x}^{0} ⊂ V^{∘} _{x} be a D^{2} × S^{1} unknotted in V^{∘} _{x}.

If x ∈ S^{n}, then choose T_{x}^{0} so that f_{x}(δ) ⊂ T^{0}^{∘}_{x} (Note that f_{x}(δ) is the geodesic homotopic to δ in M with the hyperbolic metric (f_{x})_{∗}(ρ).) Thus for each x ∈ B^{n+1}, there exists ϵ_{x}^{0} > 0 so that T_{x}^{0} ∈ W_{x}^{ϵ_{x}^{0}} and by the non-encroachment lemma for y ∈ B^{n+1} sufficiently close to x, say y in the open set A_{x}^{0}, then T_{x}^{0} ⊂ W_{y}^{ϵ_{x}/2}. For x ∈ S^{n} choose A_{x}^{0} to have

the additional property that if y ∈ A_{x}^{0} ∩ S^{n}, then f_{y}(δ) ⊂ T^{0}^{∘}_{x}.

Fix a finite cover of B^{n+1} by elements of {A_{x}^{0}}, say using {x_{1}, …, x_{p}} =^{def} X^{0} so that S^{n} is covered by a subset {A_{x_{i_{k}}}^{0} }, with each x_{i_{k}} ∈ S^{n}∩X^{0}. Let Δ_{0} be any piecewise smooth cell division such that if σ ∈ Δ_{0}, then σ ⊂ A_{x_{i}}^{0} for some x_{i} ∈ X^{0}. Furthermore if σ ∩ S^{n} ≠ ∅, then σ ⊂ A_{x_{i}}^{0} for some x_{i} ∈ S^{n} ∩ X^{0}. Finally, define T_{σ}^{0} = T_{x_{i}}^{0} for some x_{i} as above. If possible, choose x_{i} ∈ S^{n}.

For each x ∈ Δ_{0}^{n} construct a solid torus T_{x}^{1} unknotted in V^{∘} _{x} satisfying the following property. If σ_{1}, …, σ_{r} are the cells of Δ_{0} which

contain x, then T_{σ_{1}}^{0} ∪ … ∪ T_{σ_{r}}^{0} ⊂ T^{1}^{∘}_{x} ⊂ T_{x}^{1} ⊂ V^{∘} _{x}. For x ∈ Δ_{0}^{n} let ϵ_{x}^{1} be

such that T_{x}^{1} ⊂ W_{x}^{ϵ^{1}}. By the non-encroachment lemma, there exists a neighborhood A_{x}^{1} of x such that if y ∈ A_{x}^{1}, then T_{x}^{1} ⊂ W_{y}^{(ϵ_{x}^{1})/2}.

Fix a finite cover of Δ_{0}^{n} by elements of {A_{x}^{1}}, say using {x_{1}^{1}, …, x_{q}^{1}}

=^{def} X^{1}. Let Δ_{1} be a piecewise smooth subdivision of Δ_{0} that only nontrivially subdivides Δ^{n}, such that if σ ∈ Δ_{1} has dimension ≤ n, then σ ⊂ A_{x_{i}^{1}}^{1} for some x_{i}^{1} ∈ X^{1}. Furthermore if in addition σ ∩ S^{n} ≠ ∅, then σ ⊂ A_{x_{i}^{1}}^{1} for some x_{i}^{1} ∈ S^{n} ∩ X^{1}. Finally, define T_{σ}^{1} = T_{x_{i}^{1}} for some x_{i}^{1} as above. If possible, choose x_{i}^{1} ∈ S^{n}. If dim(σ) = n + 1, then

T_{σ}^{1} =^{def} T_{σ}^{0}.

In a similar way construct Δ_{2}, …, Δ_{n+1}. Here Δ_{i} is obtained from Δ_{i−1} by subdividing only the (n − i + 1)-skeleton, and has the feature

that if σ is a proper face of τ and dim(τ) ≥ n − i + 2, then T_{τ}^{i} ⊂ T^{i}^{∘}_{σ}. Finally take Δ^{∗} = Δ_{n+1} and T_{σ} = T_{σ}^{n+1}. q.e.d.

Theorem 4.6 (Coarse Torus Isotopy Theorem). Let M be a closed orientable hyperbolic 3-manifold with geodesic δ satisfying the insulator condition. If f : S^{n} → Diff_{0}(M), then there exists a cellulation Δ of B^{n+1} and a function which associates to each cell σ ∈ Δ a solid torus V_{σ} such that:

i) If κ is a proper face of σ, then V_{κ} ⊂ V^{∘} _{σ} and V_{κ} is isotopic to the standard embedding in V_{σ}.

ii) If x ∈ σ ∩ S^{n}, then f_{x}(δ) is a core of V^{∘} _{σ}.

Proof. Let Δ^{∗} be the cell structure arising from Lemma 4.5 and let Δ′ be the relative cell structure of B^{n+1} dual to Δ^{∗}. Given a cell σ^{∗} ∈ Δ^{∗}, let σ denote the dual cell. This induces a natural 1-1 correspondence between cells of Δ^{∗} and Δ′. The restriction of Δ′ to S^{n} gives rise to a regular cell division Σ of S^{n}. The union of cells of Δ′ and Σ gives a regular cell division of B^{n+1} which we call Δ.

For σ ∈ Δ′, define V′_{σ} = T_{σ^{∗}}. If σ ∈ Σ is a k-cell, then σ = τ ∩S^{n} for a unique (k + 1)-cell τ ∈ Δ′. Define V′_{σ} = V′_{τ}. The collection {V′_{σ}}_{σ∈Δ′} satisfies all our conclusions except that if σ = τ ∩ S^{n}, then V′_{σ} = V′_{τ}

(rather than V′_{σ} ⊂ V′^{∘}_{τ}.) By appropriately shrinking the {V′_{σ}|σ ∈ Σ} and maintaining the {V′_{τ}|τ ∈ Δ′} we obtain the desired collection of D^{2} × S^{1}’s which we denote {V_{σ}}_{σ∈Δ}. q.e.d.

5. Another Formulation Of Hatcher’s Theorem

Let Emb(D^{2} × S^{1}, ℝ^{2} × S^{1}) denote the space of smooth embeddings with the C^{∞} topology which take the core of each D^{2} × S^{1} to a curve isotopic to 0 × S^{1}, the core of ℝ^{2} × S^{1}. Let Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) denote the smooth embeddings isotopic to the standard one.

In [10], Hatcher listed 17 equivalent formulations of the Smale Conjecture. Here is another along the same lines.

Theorem 5.1. Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) ≃ S^{1} × S^{1}.

Proof. View Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) as Emb_{0}(\frac{1}{2}D^{2} × S^{1}, D^{2}^{∘} × S^{1}),

where \frac{1}{2}D^{2} × S^{1} is the concentric solid torus of radius \frac{1}{2}. The map

Diff_{0}(D^{2} × S^{1}) → Emb_{0}(\frac{1}{2}D^{2} × S^{1}, D^{2}^{∘} × S^{1}) defined by restricting to \frac{1}{2}D^{2} × S^{1} is surjective and is a fibration by [17], [4]. The fiber is all diffeomorphisms which restrict to the identity on \frac{1}{2}D^{2} × S^{1}. By [9] or [11, 12] the fiber is contractible and hence Emb_{0}(D^{2} × S^{1}, ℝ^{2} × S^{1}) is homotopy equivalent to Diff_{0}(D^{2}×S^{1}). (Actually they proved this in the PL category, but as noted in [9], the proof can be promoted to Diff using [10].) Again restriction is a fibration Diff_{0}(D^{2} × S^{1}) → Diff_{0}(S^{1} × S^{1}) with fiber Diff(D^{2} × S^{1}, ∂D^{2} × S^{1}), the space which fixes ∂D^{2} × S^{1} pointwise. By [10] Diff(D^{2} × S^{1}, ∂D^{2} × S^{1}) is contractible and the contractibility of that space is equivalent to the Smale conjecture. Thus Diff_{0}(D^{2}×S^{1}) is homotopy equivalent to Diff_{0}(T^{2}). Fix a basepoint t_{0} ⊂ T^{2}. There is a fibration Diff_{0}(T^{2}) → Emb(t_{0}, T^{2}) = T^{2} by restriction. The fiber Diff_{0}(T^{2}, t_{0}) consists of the basepoint preserving maps of T^{2} isotopic to id_{T^{2}}. Since the latter space is contractible [8] the proof of Theorem 5.1 is complete. q.e.d.

Remark 5.2. By tracing back this homotopy equivalence we see that the S^{1} × S^{1} corresponds to maps of Diff_{0}(D^{2} × S^{1}) which are compositions of shifts and rolls. Shifts are maps of the form (z, t) → (z, e^{iθ}t) and rolls are of the form (z, t) → (e^{iθ}z, t), where z ∈ D^{2} and θ, t ∈ S^{1}.

A proof similar to that of Theorem 5.1 yields:

Theorem 5.3. Emb(D^{2} × S^{1}, ℝ^{2} × S^{1}) ≃ O(2) × O(2) × Ω(SO(2)).

Remark 5.4. The O(2)’s in Theorem 5.3 allow for reflections in the core and cocore directions of D^{2}×S^{1}. Finally given an element ϕ of ΩS^{1}, the loop space of S^{1}, one obtains a diffeomorphism h : D^{2}×S^{1} → D^{2}×S^{1} by h(z, t) = (e^{iϕ(t)}z, t).

6. The Local Contractibility Theorem

Lemma 6.1. Let δ be a simple closed geodesic in the closed hyperbolic 3-manifold M. If f : M → M is a diffeomorphism homotopic to id_{M} such that f|δ = id_{δ}, then f is isotopic to g : M → M such that:

(1) g|N(δ) = id_{N(δ)}.

(2) g|(M − N^{∘}(δ)) is isotopic to id rel ∂N(δ).

Furthermore, given any neighborhood J of δ there exists a g as above and a N(δ) ⊂ J such that the isotopy from f to g can be supported in J.

Proof. Given a neighborhood J of δ, let N_{1}(δ) be a small closed regular neighborhood such that f(N_{1}(δ)) ⊂ J. After an isotopy supported in J we can assume that f fixes N_{1}(δ) setwise. A priori f|N_{1}(δ) is isotopic to a finite number of Dehn twists about a meridian, however M −N^{∘}_{1}(δ) is atoroidal, anannular and Haken and so by [13] has a finite mapping class group. This implies that f|N_{1}(δ) is isotopic to id_{N_{1}(δ)} and hence we can assume that after another isotopy supported in J, that f|N_{1}(δ) = id.

Let q : ℍ^{3} → M be the universal covering projection and W^{1} =^{def} q^{−1}(N_{1}(δ)) with {W_{i}^{1}} the components of W^{1}. Since f is homotopic to id_{M} there exists a lift f̃ such that f̃|S_{∞}^{2} = id. Since f|N_{1}(δ) = id, for all i f̃|W_{i}^{1} = W_{i}^{1} and is a translation by some n ∈ ℤ fundamental

domain units, n being independent of i. Choose N(δ) ⊂ N^{∘}_{1}(δ) and let f_{1} be the map T^{−n}f where T is the following Dehn twist about a torus. T |N(δ) ∪ (M − N^{∘}_{1}(δ))) = id, where the restriction to each concentric torus about N(δ) in N_{1}(δ) is a shift, the θ (of Remark 5.2) varying from 0 to 2π as one goes from ∂N(δ) to ∂N_{1}(δ). Note that T is isotopic to id_{M} via an isotopy supported in N_{1}(δ) and if f̃_{1} is the lift of f_{1} isotopic to f̃, then f̃_{1}|W = id_{W}. Here W = q^{−1}(N(δ)). (This paragraph replaces the inaccurate second sentence of second paragraph of p. 48 [6].)

To complete the proof, proceed as in the rest of the second paragraph of p. 48 [6]. q.e.d.

Lemma 6.2. If δ is a simple closed oriented geodesic in the hyperbolic 3-manifold M and δ can be isotoped into the solid torus V ⊂ M, then the isotoped δ is a core of V. If δ_{1} and δ_{2} are two isotoped images of δ both lying in V, then δ_{1} and δ_{2} are isotopic (as oriented curves) in V.

Proof. Let γ be a curve in V isotopic to δ. Since δ represents a primitive element of π_{1}(M), γ represents a generator of π_{1}(V). Thus δ, γ and V lift to the covering space M_{δ} with fundamental group ⟨δ⟩. In M_{δ} we have γ ⊂ V ⊂ M_{δ} = D^{2}^{∘} × S^{1} and δ, hence γ, is a core of M_{δ}. This implies that γ is a core of V and V is unknotted in M_{δ}, thereby proving the first assertion.

The failure of the second assertion implies that δ_{1} and δ_{2} represent oppositely oriented cores. That in turn would imply that the geodesic δ is isotopic to itself oppositely oriented, an impossibility in a hyperbolic 3-manifold. q.e.d.

Theorem 6.3 (Local Contractibility Theorem). Let δ be an oriented simple geodesic in the closed hyperbolic 3-manifold M and V a solid torus embedded in M. If H : S^{n} → Diff_{0}(M) such that H_{t}(δ) ⊂ V^{∘} for each t ∈ S^{n}, then H extends to a map G : B^{n+1} → Diff_{0}(M) such

that G_{s}(δ) ⊂ V^{∘} for each s ∈ B^{n+1}.

Proof. Fix t_{0} ∈ S^{n}. After replacing H by H_{t_{0}}^{−1}H and V by H_{t_{0}}^{−1}(V),

we can assume that H_{t_{0}} = id_{M} and V^{∘} is a neighborhood of δ.

We start with the case n = 0 where t_{0} = 1 ∈ S^{0}. Use Lemma 6.2 to extend H to [−1, 0] ∪ {1} so that H_{0}|δ = H_{1}|δ = id_{δ}. Next use Lemma 6.1 to extend H to [−1, \frac{1}{2}] ∪ \{1\} so that H_{\frac{1}{2}} |N(δ) = H_{1}|N(δ) =

id_{N(δ)}, and H_{\frac{1}{2}} |(M − N^{∘}(δ)) is homotopic to id rel ∂N(δ). Finally apply [21] to extend H to [−1, 1] so that the isotopy H|[\frac{1}{2}, 1] is an isotopy fixed on N(δ). Since, by Lemmas 6.1 and 6.2, these constructions can

be carried out so that H_{t}(δ) ⊂ V^{∘} for all t ∈ [−1, 1], the case of n = 0 is complete.

Claim. If n > 0, then there exists a solid torus regular neighborhood X of δ and K : B^{n+1} → Emb_{0}(X, V^{∘}) such that H_{t}|X = K_{t} for each t ∈ S^{n}.

Proof of Claim. Choose X so that H_{t}(X) ⊂ V^{∘} for all t ∈ S^{n}. Note

that V and X can be parametrized so that H_{t_{0}}|X ∈ Emb_{0}(X, V^{∘}) is the standard inclusion. If n > 1, then the Claim follows directly from Theorem 5.1. Now assume that n = 1 and H|X : S^{1} → Emb_{0}(X, V^{∘})

represents a nontrivial element of π_{1}(Emb_{0}(X, V^{∘})). Theorem 5.1 and

Remark 5.2 imply that there exists a map L : S^{1} × I → Emb_{0}(X, V^{∘}) such that for t ∈ S^{1}, L_{(t,0)} = H_{t}|X; for s ∈ I, L_{(t_{0},s)} = id_{X}; for t ∈ S^{1}, L_{(t,1)} is a composition of twists and rolls and (L|S^{1} × 1)|∂X represents a nontrivial element of π_{1}(Diff_{0}(∂X)). By the Palais-Cerf covering isotopy theorem [17], [4], L extends to a map K : S^{1} × I → Diff_{0}(M) such that K|S^{1} ×0 = H and K_{(t_{0},s)} = id_{M} if s ∈ I. Therefore,

if N = M − X^{∘}, then B_{t} =^{def} K_{(t,1)}|N, t ∈ S^{1}, represents a loop in Diff_{0}(N), based at the identity, which restricts to a nontrivial loop in

Diff_{0}(∂N). This contradicts the fact that N^{∘} is a hyperbolic 3-manifold. (The loop B_{t}, t ∈ S^{1} lifts to a path of maps B̃_{t}, t ∈ [0, 1] of ℍ^{3} starting at the identity. The end of this path must be a nontrivial covering transformation. This contradicts the fact that the diameters of the homotopy tracks {B̃_{t}(x)|t ∈ [0, 1]} are uniformly bounded.) q.e.d.

By the covering isotopy theorem there exists a map J : B^{n+1} → Diff_{0}(M) which extends K and satisfies J_{t_{0}} = id_{M}. The map E : S^{n} → Diff_{0}(M) defined by E_{t} = J_{t}^{−1}H_{t} satisfies E_{t_{0}} = id_{M} and E_{t}|X = id_{X}

for all t ∈ S^{n}. Since M − X^{∘} is Haken it follows by [9] or [11, 12] that E extends to E^{∗} : B^{n+1} → Diff_{0}(M) such that for z ∈B^{n+1} E_{z}^{∗} fixes X pointwise. Define G : B^{n+1} → Diff_{0}(M) by G_{z} = J_{z}E_{z}^{∗}. If t ∈ S^{n}, G_{t} = J_{t}E_{t}^{∗} = J_{t}E_{t} = J_{t}J_{t}^{−1}H_{t} = H_{t} and if z ∈ B^{n+1}, then

G_{z}(X) = J_{z}E_{z}^{∗}(X) = J_{z}(X) = K_{z}(X) ⊂ V^{∘} . q.e.d.

7. Applications

Definition 7.1. Let Hyp(M) denote the subspace of the space of Riemannian metrics on M consisting of metrics of constant curvature −1.

Lemma 7.2. If M is a complete hyperbolic 3-manifold, then Hyp(M) is homeomorphic to Diff_{0}(M).

Proof. We will show that ϕ : Diff_{0}(M) → Hyp(M) by f → f_{∗}(ρ) is bijective. Since id_{M} is the only isometry of M (with respect to a fixed hyperbolic metric) which is homotopic to id_{M} it follows that ϕ is injective. Conversely if ρ′ is a hyperbolic metric on M, then by Mostow there exists an isometry h : M_{ρ} → M_{ρ′} such that h is homotopic to id_{M}. By [7] h ∈ Diff_{0}(M). q.e.d.

Theorem 7.3. The space Hyp(M) of hyperbolic metrics on a complete hyperbolic 3-manifold of finite volume M is contractible.

Proof. If M is closed, then the contractibility of Diff_{0}(M) follows from Theorem 1.1. If M is noncompact, then the contractibility of Diff_{0}(M) follows from [9] or [11, 12]. Now apply Lemma 7.2. q.e.d.

Definition 7.4. If δ is a smooth oriented simple closed curve in the manifold M, then let Emb_{δ}(S^{1}, M) denote the space of smooth embeddings of an oriented S^{1} into M whose image is isotopic, as oriented curves, to δ.

The following application was suggested by Allen Hatcher.

Theorem 7.5. Let δ be an oriented simple closed curve in the closed hyperbolic 3-manifold M. If M − δ is atoroidal, then

Emb_{δ}(S^{1}, M) ≃ S^{1}.

Proof. The fibration Diff_{0}(M) → Emb_{δ}(S^{1}, M) defined by restricting to δ has fiber Diff_{0}(M, δ), the subspace of Diff_{0}(M) that fixes δ pointwise. The path components of this space are naturally parametrized by ℤ, the various components corresponding to the 2πn shifts of δ. If Y ⊂ M, then let Diff_{00}(M, Y) denote the subspace of Diff_{0}(M) consisting of all maps isotopic to the identity, via an isotopy fixing Y pointwise. The theorem follows from the long exact homotopy sequence and the fact [9], [11, 12] that Diff_{00}(M, δ) is contractible. (Actually it follows directly from [9],[11, 12] that Diff_{00}(M, N(δ)) is contractible, but that result can be promoted to the desired one via standard differential topology techniques.) q.e.d.

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