ON FINITENESS OF THE NUMBER OF BOUNDARY

SLOPES OF IMMERSED SURFACES IN 3-MANIFOLDS

Joel Hass and Shicheng Wang and Qing Zhou

Abstract. For any hyperbolic 3-manifold M with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of ∂M or the volume of M is bounded above. When the volume is bounded above, then area of ∂M is bounded above and the length of closed geodesic on ∂M is bounded below.

We say that a proper immersion of a surface F into M is an essential surface if

it is incompressible and ∂-incompressible, meaning that the immersion induces an

injection of the fundamental group and relative fundamental group. Let c be an

essential simple loop on the boundary ∂M of a compact 3-manifold M. If there is

a proper immersion of an essential surface F into M such that each component of

∂F is homotopic to a multiple of c, we call c a boundary slope of F.

We are interested in the following two questions:

Questions.

(1) Given a compact 3-manifold M and a genus g, are there finitely many boundary

slopes for immersed essential surfaces with genus at most g?

The referee made many comments to enhance the paper, including found a mistake in the proof of Theorem 2, and suggested a replacement. We are grateful to him. This paper grew out of work begun while the first two authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996-97. In final version when the second author is visiting Max-Planck-Institut f. Mathematik at Bonn in 2000. Research at MSRI is supported in part by NSF grant DMS-9022140 and The first author was partially supported by NSF grant DMS-9704286. The research at MPIM is supported by MPIM. The second and third authors were partially supported by Outstanging Younth Fellowship of NSFC

Typeset by AMS-TEX

(2) Under what conditions is there a bound for the number of boundary slopes in (1) which is independent of the 3-manifold?

Many results in these directions have been obtained for various classes of 3manifolds:

(1) If ∂M is a torus and the surfaces are embedded, Hatcher [H] showed that there are only finitely many boundary slopes, without any genus restriction.

(2) When the surfaces are embedded punctured spheres or tori, explicit bounds are known on the number of boundary slopes. These bounds are based on highly developed combinatorial methods in knot theory and the theory of representations of knot groups. See the survey papers [Go], [Lu] and [Sh].

(3) When ∂M is a torus and the surfaces are immersed, a positive answer to Question (1) has been obtained recently in [HRW]. When M is hyperbolic, minimal surface theory is used to derive these bounds. For fixed genus g, these turn out to be quadratic functions of g, independent of M. See also recent work of Agol [Agol].

(4) If M is an irreducible, ∂-irreducible, acylindrical, atoroidal 3-manifold and the surfaces are embedded, Scharlemann and Wu [SW] gave a positive answer to Question 1 using combinatorial arguments.

(5) Suppose ∂M is a torus and the surfaces are immersed. Baker has given examples to show that the bounded genus assumption cannot be dropped [B]. Oertel, using branched surface theory, has found manifolds in which every slope is realized by the boundary of an immersed essential surface [Oe].

In this note we give a positive answer to Question (1), which extends (3) to the case where ∂M can contain high genus components and generalizes (4) from embedded to immersed surfaces.

Theorem 1. Suppose M is ∂-irreducible, acylindrical, atoroidal 3-manifold. Then for any g, there are only finitely many ∂-slopes for essential surfaces of genus g.

Next we consider the question of obtaining bounds for the number of possible slopes which are independent of the particular manifold we are studying. It turns out that only the genus of the boundary of M is relevant.

We define the genus of ∂M be the sum of the genus of the components of ∂M.

Theorem 2. There is a function n(g, g_{∂}) such that there are at most n(g, g_{∂}) ∂- slopes for essential surfaces of genus g in a ∂-irreducible, acylindrical, atoroi- dal 3-manifold whose boundary has genus equal to g_{∂}.

We can also obtain bounds on the number of boundary slopes in terms of hyperbolic geometry.

Definition. Let M(V) be the set of all hyperbolic 3-manifolds of totally geodesic boundary and with volume bounded above by V > 0.

Theorem 3. There is a function n_{1}(g, V) such that there are at most n_{1}(g, V) ∂-slopes for essential surfaces of genus g in a 3-manifold M ∈ M(V).

Theorem 3 follows from either Theorem 1 or Theorem 2, the fact that all maximum torus cusps have volumes > C > 0 [Ad], and the following

Theorem 4. There is an integer g^{∗} > 0 and a number L > 0 such that if M ∈ M(V), then

(1) the genus of ∂M is at most g^{∗}.

(2) the length of any closed geodesic on ∂M is at least L.

Remark on Theorem 4. Theorem 4 can be restated as follows: For hyperbolic 3-manifolds with totally geodesic boundary and bounded volume, the areas of their boundaries have an upper bound, and the lengths of simple closed geodesics on their boundary have a lower bound. Neither of those two assertions is true in dimension 2. Surfaces of given area can have geodesic boundaries of any length.

Proof of Theorem 1. If M has any 2-sphere boundary components, we can fill them in with balls without changing the number of boundary slopes. Since any essential surface can be homotoped off of a splitting 2-sphere, we can without loss of generality assume that M is irreducible. The number of boundary slopes of essential surfaces lying on a torus boundary component of M is finite by [HRW], so we restrict attention to surfaces with boundary on a higher genus component of ∂M. By Thurston’s Geometrization Theorem for Haken manifolds, M admits a complete hyperbolic structure of finite volume with totally geodesic boundary [T]. We assume that M is equipped with such a hyperbolic structure. The totally geodesic boundary components consist of the non-torus boundary components of M. Since ∂M have only finitely many components, to prove Theorem 1, we need only to show that for each component of ∂M there are finitely many boundary slopes of proper essential surfaces of genus at most g.

Suppose F is an incompressible, boundary incompressible proper immersion with ∂F consisting of n copies of a slope l. Let DM be the double of M along its totally geodesic boundary components. DM is Haken and atoroidal, and admits a hyperbolic structure obtained by doubling that of M. The double DF of F is incompressible, and therefore a theorem of Schoen-Yau and Sacks-Uhlenbeck shows that there is a least area representative of its homotopy class, denoted by DF^{∗} [SY]. The intersection of DF^{∗} with the incompressible least area (in fact totally geodesic) surface ∂M consists of curves essential on both DF^{∗} and ∂M. Since F is boundary incompressible in M, the intersection F^{∗} of DF^{∗} with M is a surface homotopic (rel boundary) in M to F. Since DM admits an isometry which is a reflection about ∂M, DF^{∗} is perpendicular to ∂M. If not, we could reflect DF^{∗}∩M and get a homotopic surface with lower area. So F^{∗} is properly homotopic to F, F^{∗} is perpendicular to ∂M and ∂F^{∗} is a (possibly multiply covered) geodesic.

Choosing geodesic orthogonal coordinates near the geodesic boundary of the surface F^{∗}, we have (line 7 of p.374, [BM])

(1)  ds^{2} = du^{2} + J^{2}(u, v)dv^{2},  (J(u, v) > 0 and J(0, v) = 1).

where the u-curves (those where v = constant) are geodesics perpendicular to the boundary and the v-curves lie on the boundary when u = 0.

The geodesic curvature in F^{∗} of a curve t ↦ (u(t), v(t)) is given by Formula 10.4.7.1 of [BM],

(2)  \frac{1}{\sqrt{Eu′^{2} + Gv′^{2}}} (\frac{dϕ}{dt} + \frac{1}{2\sqrt{EG}}(\frac{∂G}{∂u}v′ − \frac{∂E}{∂v} u′)),

where ϕ is the angle between the curve and the u-curves and the metric on F^{∗} is given by

Edu^{2} + Gdv^{2}.

When we consider the v-curves, we have u′ = 0, v′ = 1, ϕ = π/2, E = 1 and G = J^{2}. Substituting into (2), the geodesic curvature for a v-curve {u = c} oriented as the boundary of {0 ≤ u ≤ c} is given by:

(3)  k_{g} = \frac{1}{J} \frac{∂J}{∂u} .

Orienting the curve as the boundary of {u ≥ c} changes the sign and gives

(3’)  k_{g} = −\frac{1}{J} \frac{∂J}{∂u} .

The Gaussian curvature of the surface is Formula 10.5.3.3 of [BM]

(4)  K = −\frac{1}{J} \frac{∂^{2}J}{∂u^{2}} .

A direct computation shows that k_{g} satisfies the following equation

(5)  \frac{∂k_{g}}{∂u} = K + k_{g}^{2}.

Since M is of constant curvature −1, we have

K = k_{1}k_{2} − 1

by Gauss’s Formula (p.179 [Sp]), where k_{1} and k_{2} are the principle curvatures. Since F is a minimal surface, we have k_{1}k_{2} ≤ 0, and hence K ≤ −1. Then by (4) it follows that

(6)  \frac{∂^{2}J}{∂u^{2}} ≥ J.

Fixing v = v_{0}, by (6) we have

\frac{∂J}{∂u} (u, v_{0}) = \frac{∂J}{∂u} (u, v_{0}) − \frac{∂J}{∂u} (0, v_{0})

(7)  = ∫_{0}^{u} \frac{∂^{2}J}{∂u^{2}} (s, v_{0})ds ≥ ∫_{0}^{u} J(s, v_{0})ds ≥ 0.

(1), (3’) and (7) imply that k_{g} < 0, if u > 0. Now consider the function

(8)  h(u) = \frac{e^{−u} − e^{u}}{e^{−u} + e^{u}} .

which is the solution to the differential equation

(9)  \frac{dh}{du} = −1 + h^{2}

with the initial condition h(0) = 0. Note that h(u) < 0 when u > 0 and that the function k_{g} − h satisfies the differential inequality

\frac{d(k_{g} − h)}{du} = K + k_{g}^{2} + 1 − h^{2}

(10)  ≤ k_{g}^{2} − h^{2} = (k_{g} − h)(k_{g} + h)

by (5) and (9). We want to show that k_{g} − h ≤ 0.

Suppose on the contrary that on an interval [0, U], k_{g} − h is somewhere positive. Pick a u_{0} ∈ [0, U], such that k_{g} − h takes its positive maximum at u_{0}. We know u_{0} ≠ 0 since ∂F^{∗} is a geodesic and k_{g} = h = 0 at 0. Then

\frac{d(k_{g} − h)}{du}

is zero if u_{0} ∈ (0, u), and is ≥ 0 if u_{0} = U. Hence

\frac{d(k_{g} − h)}{du} ≥ 0

at u_{0}. Since both k_{g} and h are negative at u_{0}, we have

(k_{g} − h)(k_{g} + h) < 0.

This contradicts (10), and so k_{g} ≤ h.

For t > 0, let N_{t}(∂M) be the subset of M with distance ≤ t from the boundary. There is a b > 0 such that when t < b then N_{t}(∂M) is a collar of ∂M.

Choose U < b in the above and let N_{U}(∂F^{∗}) be the neighborhood of ∂F^{∗} with u coordinates at most U. Clearly N_{U}(∂F^{∗}) ⊂ N_{b}(∂M). Since N_{b}(∂M) is a collar of ∂M and the surface F^{∗} is ∂-incompressible, it follows that N_{U}(∂F^{∗}) is a collar of ∂F^{∗}. Letting

F_{U} = \overline{F^{∗} − N_{U}(∂F^{∗})},

each component of ∂F_{U} is in the same homotopy class in M as the slope l and

#∂F_{U} = #∂F^{∗} = #∂F = n.

By Gauss-Bonnet, we have that

∫_{F_{U}} KdA + ∫_{∂F_{U}} k_{g}ds = 2π(χ(F)) = 2π(2 − 2g − n).

Let d be the length of the geodesic in the homotopy class of the slope l. Then the length of each component of ∂F_{U} is larger than d. Since K ≤ −1 and k_{g} ≤ h < 0 at U, we have

nhd ≥ 2π(2 − 2g − n).

Then we have

(11)  d ≤ \frac{2π(2g + n − 2)}{−hn} ≤ \frac{2π(2g + 1)}{−h}.

Since g is given and h = h(U) < 0, d is bounded above. There are only finitely many homotopy class of essential closed curves in ∂M containing elements of length less than a given constant. Therefore for any fixed g there are only finitely many ∂-slopes for immersed incompressible, boundary incompressible surfaces of genus g. □

Proof of Theorem 2. By Kojima and Miyamoto [KM], and by Basmajian [Ba]. there is a lower bound U_{∗} to the width of a collar around ∂M for any hyperbolic metric on a manifold M in which ∂M is totally geodesic of genus ≤ g_{∂}.

Below we only consider 3-manifolds whose totally geodesic boundary of genus at most g_{∂}, g_{∂} is given. Then in the proof of Theorem 1 the boundarys have collars of width U bounded below by U_{∗}, and h = h(U) ≤ h(U_{∗}) = − tanh U_{∗} ≤ 0. Let L^{∗} = \frac{2π(2g+1)}{tanh U_{∗}}, by (11) we have

(12)  d ≤ L^{∗}.

We are going to show that the number of homotopy classes of essentail circles of length bounded by L∗ on the collar of ∂M is uniformly bounded. Moreover since the projection from a collar of the boundary of a hyperbolic manifold with totally geodesic boundary to the totally geodesic boundary is length decreasing, so it suffices to consider curves lying on the boundary ∂M.

We will divide the geodesics under consideration into two types:

(1) those with length less than L,

(2) those with length between L and L^{∗},

then we estimate the bound of the number of each tpye geodesics, where L < L^{∗} is a constant arising from the collar lemma ([Bu] and also Theorem 2.18 of [Mu]).

More precisely, let S(x) = sinh^{−1}(1/ sinh(x/2)). For a given simple closed geodesic c with length l_{c} on a hyperbolic surface, let N(c) = {x : d(x, c) ≤ S(l_{c})}. Then the collar lemma states that N(c) is a collar. Moreover if c_{1} and c_{2} are disjoint simple closed geodesics, then N(c_{1}) and N(c_{2}) are disjoint. There is an L such that if l ≤ L, then S(l) > l/2 and l > sinh(l/2); for example, we can choose L = 1.75; then S(l) > S(L) > 0.887 > 0.85 = L/2 ≥ l/2 and 1.76 ≤ L/ sinh(L/2) ≤ l/ sinh(l/2).

Then any two simple closed geodesics of length ≤ L are disjoint. Moreover the area of N(c) is

2l_{c} sinh(S(l_{c})) = 2l_{c}/ sinh(l_{c}/2) ≥ 2l_{c}/l_{c} = 2.

Let g(F) be the genus of F. Then the number of simple closed geodesics of length at most L is bounded above by

(13)  2π(2g(F) − 2)/2 = 2π(g(F) − 1).

Moreover we can choose a fixed L small enough so that L < L^{∗}, L < 1.75 and S(L) > L^{∗}. Below we simply use L-thick part and L-thin part to call the L-thick part and L-thin part of ∂M respectively. Then each geodesic circle meeting the Lthin part must has length either longer than L^{∗} or shorter than L. In other words, each geodesic circle of type (2) must stay in the L-thick part. Note the L-thick part may be not connected, but the number of compoennets is bounded |χ(∂M)|.

Since the area of the L-thick part is bounded by 2π(2g−2) above and the injective rediaus of each point is bounded by L below, The standard argument shows that each component of the L-thick part had a uniform bound on the diameter, (denoted diam). We can now bound the number of geodesics in L-thick part with length at most L^{∗}. Pick a point x in L-thick part. The lifts of this point are all at least L from one another. Let x^{∗} be a lift of x. A geodesic in L-thick part with length at most L^{∗} gives rise to a covering translation that moves x^{∗} at most 2diam + L^{∗}.

Let Γ_{x} be the lattice on the universal covering of the L-thick part given by the lift of x. Let A(R) be the area of D(R), the hyperbolic disc with radius R centered at x^{∗}. By the augument of last paragraph, the geodesic on the L-thick part with length at most L^{∗} is bounded by the number of vertices of Γ_{x} in D(2diam + L^{∗}), which is at most

(14)  n(g, L) = \frac{A(2diam + L^{∗})}{A(L)}.

For simplicity, we first assume that ∂M is connected. By (13) and (14) we have

(15)  n(g, g_{∂}) = \frac{A(2diam + L^{∗})}{A(L)} + 2π(g_{∂} − 1).

In general suppose ∂M consists of k torus components and l components of genus g_{i} > 1, i = 1, ..., l. Then g_{∂} = ∑_{i=1}^{l} g_{i} + k and there are at most ∑_{i=1}^{l} n(g, g_{i}) + kN(g) boundary slopes for proper essential surfaces of genus g, where N(g) is the uniform bound for the number of boundary slopes of proper essential surface of genus g on a torus boundary component given in [HRW], and n(g, g_{i}) is given by (15). One can verify that ∑_{i=1}^{l} n(g, g_{i}) + kN(g) ≤ n(g, g_{∂}). □

Remark on Theorem 2. By Lemma 3.1 of [Ba], we have the lower bound U^{∗} = \frac{1}{4}log \frac{g_{∂}+1}{g_{∂}−1}. Moreover A(R) = \frac{4π}{1 − tanh^{2} R/2}. It is also easy to get an upper bound for the diameter for L-thick part. We can get an explicit value for n(g, g_{∂}) by plugging in these functions, though this does not appear to give sharp values. Proof of Theorem 4. Pick any infinite sequence of totally geodesic hyperbolic 3manifolds {M_{n}} in M(V). Consider the sequence {D(M_{n})}, where D(M_{n}) is the double of each M_{n}. Then the volume of the closed hyperbolic 3-manifold D(M_{n}) is bounded by 2V. By passing to a subsequence, we can assume that D(M_{n}) has a Gromov limit M^{∗}. It is known that

(a) M^{∗} is a complete hyperbolic 3-manifold of finite volume, which can be viewed as the complement of a hyperbolic link L in a closed 3-manifold, and each D(M_{n}) is obtained by a Dehn surgery on M^{∗}.

(b) Since each D(M_{n}) admits a reflection r_{n} (isometry) about its geodesic boundary, so does M^{∗}. Hence M^{∗} = D(M_{∞}), where M_{∞} is a hyperbolic 3-manifold with totally geodesic boundary. Let r_{∞} be the reflection of D(M_{∞}) about ∂M_{∞}. We have not claimed as yet that there is no cusp at ∂M_{∞}.

(c) Let T H_{ϵ}(P) be the ϵ thick part of P for any hyperbolic 3-manifold P. Then for any ϵ > 0 and 1 − ϵ ≤ k ≤ 1, there is an integer N such that for n > N there is a homeomorphism h_{n} : T H_{ϵ}(D(M)) → T H_{ϵ}(D(M_{n})) which is a k-quasi-isometry. Moreover h_{n} can be chosen to commute with the reflections.

For the result on the Gromov limit of closed hyperbolic 3-manifolds of bounded volume, see Chapter 6 of [T1], or Chapter E of [BP]. For the fact about reflections, one can argue as follow: As in the case of closed hyperbolic 3-manifolds, any sequence of hyperbolic 3-manifolds with totally geodesic boundary and bounded volume V has a subsequence with Gromov limit M_{∞}, which is a complete hyperbolic 3-manifold with totally geodesic boundary. Then the double D(M_{∞}) will be the limit of the doubles.

Suppose there is a torus in ∂M, which must result in cusps in ∂M_{∞}. Let the torus T be the boundary component of T H_{ϵ}(M_{∞}) corresponding to the cusp C, and let c be a component of T ∩ ∂M. Then h_{n}(T) ⊂ D(M_{n}) is invariant under the reflection r_{n} about ∂M_{n}, and it follows that h_{n}(c) is a meridian of the Dehn filling solid torus on h_{n}(T), and therefore h_{n}(c) ⊂ ∂M_{n} is a trivial loop. However each cusp in ∂M_{∞} can only be a limit of essential loops, and this is a contradiction. Hence ∂M_{∞} contains no cusps.

Since ∂M_{∞} contains no cusps, for small ϵ, ∂M_{∞} is contained in the interior of the compact manifold T H_{ϵ}D(M_{∞}). Moreover as the fixed point set of the reflection r_{∞}|T H_{ϵ}(D(M_{∞})), ∂M_{∞} is compact, therefore it is closed. Since ∂M_{n} converges to ∂M_{∞} in the limit, it follows that

(1’) the genus of ∂M_{n} is stable when n is large enough,

(2’) the length of the shortest simple closed geodesic on ∂M_{n} cannot converge to zero (otherwise there will be a cusp in M_{∞}).

Now suppose (1) of Theorem 4 is not true. Then we can find a sequence {M_{n}} in M(V) such that the genus of ∂M_{n} is > n. The genus of any subsequence must also tend to infinity, which contradicts (1’); hence (1) of Theorem 4 is true. Similarly, if (2) of Theorem 4 is not true, then we can find a sequence {M_{n}} in M(V) such that the length of the shortest geodesic of ∂M_{n} is < 1/n, which contradicts (2’). This finishes the proof of Theorem 4. □

References.

[Ad] C. Adams, Volumes of n-cusped hyperbolic 3-manifolds, J. London Math. Soc. 1988, 38, 2, 555-565.

[Agol] I. Agol, Topology of Hyperbolic 3-manifolds, Ph.D. thesis, UCSD, 1998.

[B] M. Baker, On the boundary slopes of immersed incompressible surfaces, Ann. Inst Fourier (Grenoble) 46, 1443-1449, (1996).

[Ba] A. Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994), 207-225.

[BP] R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer-Verlag, (1991).

[Bu] P. Buser, The collar theorem and examples. Manuscripta Math. 25 (1978), 349–357.

[BG] M. Berger and B. Gostiaux, Differential Geometry: Manifolds, curves and surfaces, GTM 115, Springer Verlag, Berlin, New York.

[Go] C. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 631–642, Math. Soc. Japan, Tokyo, 1991.

[HRW] J. Hass, H. Rubinstein and S.C.Wang, Immersed surfaces in 3-manifolds, J. Differential. Geom. Vol. 52 303-325, (1999).

[Ha] A. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), 373-377.

[KM] S. Kojima and Y. Miyamoto, The smallest hyperbolic 3-manifolds with totally geodesic boundary. J. Differential Geom. 34 (1991), 175–192.

[Lu] J. Luecke, Dehn surgery on knots in S^{3}, Proc. ICM Vol 2 (Zurich, 1994), 585-594.

[Mu] C. McMullen, Complex dynamics and renormalization, Ann. Math. Study, No, 135, PUP, 1994.

[Oe] U. Oertel, Boundaries of π_{1}-injective surfaces, Topology Appl. 78, (1997), 215-234.

[SW] M. Scharlemann and Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Aust. Math. Soc. (Series A) 55 (1993), 72-89.

[Sh] P. Shalen, Representations of 3-manifold groups and its application to topology, Proc. ICM Berkeley, (1986), 607-614.

[Sp] M. Spivak, A comprehensive Introduction to Differential Geometry, Vol. 4, Publish or Perish, Inc. Berkeley 1979.

[SY] R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127-142.

[T] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS, Vol. 6, (1982) 357-388.

[T1] W. Thurston, Geometry and Topology of 3-manifolds, Princeton University Lecture Notes, 1978.

The University of California at Davis, CA, 95616, USA

Peking University, Beijing 100871, China

East China Normal University, Shanghai, 200062, China