MINIMAL SURFACES IN GEOMETRIC 3-MANIFOLDS J. HYAM RUBINSTEIN
Abstract. In these notes, we study the existence and topology of closed minimal surfaces in 3-manifolds with geometric structures. In some cases, it is convenient to consider wider classes of metrics, as similar results hold for such classes. Also we briefly diverge to consider embedded minimal 3-manifolds in 4-manifolds with positive Ricci curvature, extending an argument of Lawson to this case. In the topology of 3-manifolds, surfaces of two types play special roles, namely Heegaard splittings and incompressible surfaces. We will see that such surfaces also occur naturally as minimal surfaces, and often are essentially the only types of embedded minimal surfaces which can occur. In the most important class of 3-manifolds with complete hyperbolic metrics of finite volume, we will show that the existence of many different types of minimal surfaces is closely related to the thick/thin decomposition of Margulis. Dehn surgery along knots and links in the 3-sphere generically gives such hyperbolic 3-manifolds and when the surgery coefficients are large enough, we find that the resulting Margulis tubes about short closed geodesics give a type of barrier when forming minimal surfaces. Applications include new lower bounds for the genus of irreducible and strongly irreducible Heegaard splittings and of incompressible surfaces in terms of the injectivity radius of the thick part of the manifold. Also we construct ‘knotted’ minimal surfaces inside handlebodies with minimal boundary in hyperbolic 3-manifolds, answering a question of S.T. Yau. Another question of Yau is partially answered - we give infinitely many distinct closed minimal immersions in any hyperbolic 3-manifold with a complete metric of finite volume. Previous applications include describing the boundary slopes of properly immersed incompressible surfaces in complete finite volume hyperbolic 3-manifolds with cusps and giving the overall structure of Heegaard splittings of bounded genus in sequences of 3-manifolds obtained by all Dehn surgeries on a fixed link. Finally we discuss briefly ideas of B. Andrews, C. Epstein, K. Uhlenbeck, W. Thurston on minimal surfaces with small principal curvature. This includes some joint work with I. Aitchison and S. Matsumoto on a new method to explicitly construct such surfaces in some knot and link complements.
1. Introduction A good general reference on minimal surfaces is the recent set of notes [8]. General existence results for embedded and immersed closed surfaces in Riemannian 3-manifolds are given in [49], [31], [19], [36], [37]. We quickly summarise this material as it is crucial in Date: February 21, 2004. 1991 Mathematics Subject Classification. Primary 57M25, 57N10. Key words and phrases. Minimal surfaces, geometric 3-manifolds, incompressible surfaces, Heegaard splittings, Dehn surgery. I would like to thank I. Agol, I. Aitchison, J. Hass, J. Schultens and S.T. Yau for helpful conversations. J. Pitts is a joint author for much of the background material for these notes. 1
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all subsequent discussions. We also quickly review some basic topics from the topology of 3-manifolds - cf [20], [21], [50]. For simplicity, all 3-manifolds M will be orientable and either compact or the interior of a compact manifold. We will work in the smooth category. All 3-manifolds will be irreducible unless stated otherwise, ie every embedded 2-sphere bounds a 3-ball. (By results of Kneser and Milnor, compact 3-manifolds can be decomposed essentially uniquely into connected sums of irreducible 3-manifolds.) Definition 1.1. A mapping φ of a closed surface S different from the 2-sphere or projective plane into M is called incompressible or π1 -injective, if the induced map π1 (φ) : π1 (S) → π1 (M ) is one-to-one. A basic result of [49] and [11] ( cf also [31], [19]) is the following; Theorem 1.2. Given an incompressible surface φ : S → M in a Riemannian 3-manifold, there is a least area map ρ in the homotopy class of φ which is a minimal immersion. Moreover if φ is an embedding, so is ρ. In fact, one only needs that the map is injective on simple loops, ie there are no simple closed essential curves in the kernel of the induced map of φ on π1 . The opposite type of surface in a 3-manifold to an incompressible one is a ‘completely compressible’ one. The most important type of the latter is a Heegaard splitting. We consider the cases of closed, compact and the interior of compact manifolds. Definition 1.3. If M is closed, a Heegaard splitting is a decomposition of M into the union of two handlebodies H+ , H− glued along their common boundary S = ∂H+ = ∂H− . S is called a Heegaard surface for M . If M is compact with non-empty boundary or is the interior of a compact manifold with boundary, then a Heegaard splitting is similarly a representation of M as a union of compression bodies. Compression bodies are obtained by attaching one handles to a product L × [0, 1] or L × (0, 1] respectively, along the surface L × {1}. The common boundary is again called a Heegaard surface for M . A closely related concept is of a one-sided Heegaard splitting ([41]). Definition 1.4. A one-sided Heegaard splitting for a closed M is a decomposition into the union of a handlebody and a twisted I bundle over a non-orientable surface S 0 , along the common boundary surface. A similar notion occurs also for compact manifolds and interiors of compact manifolds, using compression bodies instead of handlebodies. The surface S 0 is called a one-sided Heegaard surface. One-sided splittings occur naturally by taking a geometrically incompressible non-orientable surface S 0 representing a non-zero element of H2 (M, Z2 ) in an irreducible 3-manifold with
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no embedded orientable incompressible surfaces. A surface S 0 is geometrically incompressible, if there is no embedded disk D with D ∩ S 0 = ∂D is a non-contractible loop on S 0 . (cf [41]). We need to consider some key properties and results about Heegaard decompositions. Definition 1.5. A meridian disk for a handlebody H is a properly embedded disk with ∂D a non-contractible loop in ∂H. A splitting (S, H+ , H− ) for M is called irreducible, if there is no pair of meridian disks D+ , D− for H+ , H− which meet transversely in a single point. If every meridian disk for H+ meets every disk for H− then the splitting is called strongly irreducible. One often calls a reducible splitting one where there is a pair of meridian disks D+ , D− with ∂D+ = ∂D− . It is easy to show that a strongly irreducible splitting is either irreducible, or else the manifold is S 3 and the splitting is the standard genus one decomposition. Trivial handles can be added to Heegaard splittings by taking an embedded arc ` in one handlebody, say H+ , which is boundary parallel ( ie there is an embedded disk D with ` ⊂ ∂D and ∂D \ ` included in the Heegaard surface S) and adding a small regular neighbourhood of ` to H+ . This increases the genus of the handlebody by one. On the other hand, it is easy to see the result of removing the open neighbourhood of ` from H− also gives a handlebody with one extra handle, since D gives a meridian disk for this handle. This process is called stabilisation. We can also say that irreducible splittings are not stabilised, since the condition of having a pair of disks D+ , D− meeting at one point is precisely the same as having added a neighbourhood of `, which can be viewed as a neighbourhood of say D− . Certainly minimal genus Heegaard splittings of 3-manifolds are always irreducible, since a stabilised splitting can be reduced in genus by throwing away the trivial handle. Haken proved that if M is a non-trivial connected sum, then any Heegaard splitting of M can also be decomposed into splittings of the factors. So it suffices to restrict attention to irreducible M . The concept and existence of strongly irreducible splittings were introduced by Casson and Gordon ([6]) in the following important result. Theorem 1.6. If M is closed and irreducible, then any irreducible Heegaard splitting is strongly irreducible or else M has an embedded incompressible surface. Definition 1.7. A closed irreducible 3-manifold is called Haken if it contains an embedded incompressible surface. Otherwise it is called non-Haken. The minimax technique for finding (unstable) minimal surfaces in 3-manifolds was introduced in [35], [47], [36], [37]. The key situation is when we are dealing with strongly irreducible Heegaard splittings and M has a bumpy metric, in the sense of White ([53]).
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Theorem 1.8. ([37]) Suppose that M is closed and has a bumpy Riemannian metric and S is a strongly irreducible Heegaard splitting. Then either S is isotopic to a minimal surface of index one, or S is isotopic to the boundary of a regular neighbourhood N (S 0 ) of a onesided splitting surface S 0 , with a single handle added by taking the boundary of a regular neighbourhood of an arc ` which is an I fibre in the I bundle structure on N (S 0 ). In the latter case, S 0 is geometrically incompressible and is isotopic to a stable minimal surface in M . We can view S as collapsing onto S 0 with multiplicity 2. Sketch By an extension of the methods of [35], [47], one can take a limit of a sequence of surfaces Sn , all in the isotopy class of the Heegaard surface S, approaching the minimax bound in area and show that a smooth minimal surface is the result. Moreover this surface can be formed topologically by handle pinching and collapsing sheets onto multiply covered components. A handle pinch ( or disk surgery) is obtained by taking an embedded disk D with D ∩S = ∂D a non-contractible loop on S. Let N (D) be a small product neighbourhood of D. The annulus A = N (D) ∩ S is then replaced by the two parallel disks ∂N (D) \ int(A). Handle pinching is associated with the curvature on the sequence of surfaces blowing up at a point or a finite number of points. Now the assumption that M is strongly irreducible means that only pinching occurs on one side of the Heegaard surface S, or else a single handle pinch takes place, followed by collapsing onto a non-orientable surface S 0 with multiplicity 2. In the latter case, it is easy to check that S 0 is geometrically incompressible and is a one-sided Heegaard splitting as claimed. The fact that S 0 can be isotoped to a stable minimal surface follows by [31]. In the former case, we need the additional information that the limit surface has components which have total index one. A bumpy metric is required for this step. (Otherwise we can only show that the minimax limit surface has nullity. A bumpy metric implies that minimal surfaces have no Jacobi fields, so the index must be non-zero). Now it can be observed that the components of the limit surface all have multiplicity one ( possibly throwing away some nesting spheres) and all bound disjoint balls or handlebodies which are core for one of the original handlebodies , say H+ for S. By this we mean that the complement of the interiors of these handlebodies in H+ is a compression body. Moreover, by Casson Gordon ([6]), if all these small open handlebodies are removed from M , we obtain a new manifold M 0 with incompressible minimal boundary of index one. The unstable component can be pushed into the interior of M 0 and shrunk to least area by Meeks, Simon, Yau ([31]). Define M 00 by remove the open collar between the stable surface just constructed and the boundary. Next, apply a new minimax procedure in M 00 using the compression body decomposition defined by S. ( Note that we do not quite have a
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compression body decomposition defined by S in M 00 , since the unstable component might be a sphere so a ball is removed from the compression body. This does not affect the argument). We can iterate this procedure - since the manifold is strictly decreasing in size at each step, we obtain a nested sequence of compact sets which has a limit intersection. The boundary of this limit compact set is a smooth minimal surface and the limiting process contradicts the assumption that the metric is bumpy ([53]). We conclude that after a finite number of steps the procedure terminates, ie no handles are pinched except possibly to split off some spheres, and a minimal surface isotopic to S has been obtained. Corollary 1.9. If M has an arbitrary Riemannian metric then the conclusion of the previous Theorem holds, except that the minimal surface has index at most one, as it may have non-zero nullity. Proof It suffices to show that if a sequence ρn of bumpy metrics is chosen converging to the original metric ρ, then the resulting sequence of minimal surfaces ( in the case of no collapsing to a non-orientable surface with multiplicity 2) converges smoothly to a minimal surface in the metric ρ. We can apply the technique of [53] since in either case, there are a priori area bounds on the sequence of minimal surfaces by the fact that the minimax areas for the metrics ρn converge to the minimax area for ρ. Since the minimal surfaces cannot converge to a multiply covered component ( by strong irreducibility, such pinching is ruled out, except in the special case of non-orientable surfaces covered twice), the smooth convergence follows by [53]. Note that the limiting minimal surface cannot have index more than one, but can also have nullity. To complete this section, we discuss the notion of telescoping of an irreducible but not strongly irreducible splitting S as in [46] ( cf also [42]). The idea is that given such a splitting, there is a collection of disjoint incompressible surfaces S1 , .., Sk so that if M is split open along these surfaces, then there are strongly irreducible compression body decompositions Sk+1 , .., Sm for the components. The original splitting surface S is then formed by tubing together all these surfaces in a standard way. Now we can apply the previous Theorem to this situation in a natural way. In particular, we can first isotop all the incompressible surfaces S1 , .., Sk to be least area ([31]) and to be either disjoint or coincide. Moreover then the compression body decompositions Sk+1 , .., Sm can be isotoped to be either minimal surfaces of index one, disjoint from the incompressible minimal surfaces ( which act as barriers by the maximum principle), or collapse onto geometrically incompressible non-orientable surfaces with multiplicity 2, with a single handle being pinched out in the process. These latter surfaces can also be isotoped to be minimal in the complement of the incompressible minimal surfaces.
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Example 1. An interesting example to illustrate these ideas is the Heegaard splittings of a hyperbolic surface bundle M over a circle. By [50] a pseudo-Anosov gluing φ of a surface S of genus at least 2 to itself can be used as the monodromy for such a bundle. Let S be isotoped to be least area by [31] and let d be the length of a shortest non-contractible closed ˜ of M , with monodromy φn for some curve isotopic onto S. Next, take a cyclic cover M large n. We can choose n so that the distance D around the bundle between the two sides ˜ has a strongly irreducible splitting S˜ of of S is as big as we like. Now assume that M ˜ genus g. By corollary 1.9, S is isotopic to a minimal surface or collapses onto a minimal one-sided surface with multiplicity 2. Now using the coarea formula plus slicing, we see that the area of S˜ or the minimal one-sided surface can be made arbitrarily large. For if a slice by an S-fibre contains a non-contractible curve, its length is at least d. So if we have a long interval of such fibres then the resulting area is large. On the other hand, if a long interval of slices has only contractible curves and no handles, we see that this part of S˜ lifts to the ˜ and by monotonicity, has large area. Finally if there are many universal covering of M handles in such a long interval of slices, then the genus of S˜ is large. So we conclude by ˜ must be Gauss-Bonnet in all cases that the genus of any strongly irreducible splitting of M arbitrarily large. Notice we have actually proved more by telescoping, namely that any irreducible splitting must telescope into incompressible surfaces and strongly irreducible splittings, where any component of genus which is not large must be a copy of S. So we conclude (by an easy ˜ which does not have large genus must be argument) that the only irreducible splitting of M the standard splitting, ie two parallel copies of S joined by two tubes running around the bundle in the two product regions S × I, so that the tubes have cores isotopic to x × I and y × I, for two points x, y on S In the third section on hyperbolic 3-manifolds, we will extend these results to the case of complete hyperbolic metrics of finite volume on manifolds with cusps. In particular we will show that strongly irreducible compression body decompositions can still be isotoped to index one closed minimal surfaces or collapse to non-orientable surfaces, i.e do not move out into the cusps to become non-compact. We will also extend the above example to manifolds with infinite first homology, so that a shortest loop of infinite order in homology is long relative to the injectivity radius of the manifold. 2. 3-manifolds with non-negative curvature and Seifert fibred spaces We discuss first the geometries S3 , R3 , S2 × R. These have sectional curvatures ≥ 0. So it is convenient to extend the discussion to the more general classes of metrics where the Ricci curvature or scalar curvature is non-negative. Two fundamental results about the first case were given in [24] and [9].
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Theorem 2.1. [24] Suppose that M is a closed orientable irreducible 3-manifold with a Riemannian metric of positive Ricci curvature and let S be a closed orientable minimal surface. If M is split open along S into two components P, Q then the induced maps π1 (S) → π1 (P ) and π1 (S) → π1 (Q) are onto. Consequently P, Q are both handlebodies. Theorem 2.2. [9] Suppose that M is a closed orientable irreducible 3-manifold with a Riemannian metric of positive Ricci curvature and let S be an embedded closed minimal surface. Then there is an area bound for S in terms of the genus and a lower bound for Ricci curvature. As a consequence, the minimal surfaces of bounded genus form a compact set. We now discuss a variation on the proof of [24] which gives a little more information. In particular, this raises interesting issues about minimal 3-manifolds in 4-manifolds of positive Ricci curvature. Lawson uses a second variation argument of Frankel in the following way. Cut M open along S into two domains X, Y ( by Myer’s Theorem we know that M has finite fundamental ˜ Y˜ of group so any orientable surface must separate M ). Then the universal coverings X, X, Y are shown to have connected boundaries, ie the induced maps π1 (S) → π1 (X) and π1 (S) → π1 (Y ) are shown to be onto. For if there were two separate components S1 , S2 ˜ then one can connect them by a shortest geodesic arc ` which is orthogonal to of say ∂ X both surfaces. But Frankel’s formula says that such an arc must be unstable, ie have second variation negative in some direction. Hence this gives a contradiction. Lawson then goes on to use Dehn’s lemma and the Loop Theorem of Papakyriakopoulos to show that X, Y are both handlebodies and so S is a Heegaard surface for M . By taking double covers, it follows then that any non-orientable minimal surface defines a one-sided Heegaard splitting. We sketch why the conclusion of Lawson can be obtained without using the 3-manifold techniques, but rather by showing directly that X, Y have one-dimensional spines, ie retract onto graphs. To do this, we show that there is a height function ( ie Morse function) on each of X, Y so that S is a level surface and there are no local minima or index one critical points. So critical points are of index 2 or 3 and the dual handle decomposition has only 0 and 1 handles, so gives a handlebody structure on X, Y . Consider the distance function d of points in X to S = ∂X. The Frankel-Lawson argument shows that this function has no local minima on the focal set of S. So we use the standard argument ([32]) of approximating d by a Morse function f . It is easy to check that this can be done so that f has no critical points of index 0 or 1, since d has no local minima and an index 1 critical point would correspond to a local minimum on the focal set. So this completes the sketch.
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Note however this method works in all dimensions. In dimensions higher than 4, the conclusion about fundamental groups enables one to eliminate 1-handles by surgery theory. However this cannot be done smoothly in dimension 4 and only topologically in the presence of suitable fundamental group to apply Freedman’s techniques. So we have proved; Theorem 2.3. Suppose that W is a compact orientable 4-manifold with a metric of positive Ricci curvature and M is a closed orientable embedded minimal 3-manifold. Then the complementary domains X, Y for M in W both have 2-dimensional spines, or dual handle presentations with only 2-,3- and 4-handles attached to a thickened copy of M . Remarks 1. Notice we have shown that the focal set of an orientable minimal surface in an orientable 3-manifold with positive Ricci curvature retracts onto a graph. Similarly in the corresponding case for minimal 3-manifolds in 4-manifolds, the focal set retracts onto a 2-complex. It would be interesting to find other properties of such focal sets. 2. A well known source of possible examples is given by the Mazur construction ([27]). Start with a 0-handle (4-ball) and attach first n 1-handles and then n 2-handles which cancel algebraically but not geometrically. Let W 4 be the resulting contractible 4-manifold, its boundary ∂W = M 3 is a homology 3-sphere. Then the 5-manifold W × I has the same collection of handles which now cancel geometrically, giving the 5-ball. One concludes that the boundary S 4 = 2W 4 . Note that the spines of the two identical complementary domains of M 3 in S 4 are 2-complexes. In Mazur’s original construction, these spines are dunce hats, obtained by gluing together the three sides of a triangle with orientations a, a, a−1 . Notice that there is a simple foliation of B 5 by copies of W 4 ; the natural question is can one get an unstable minimal embedding of W 4 ⊂ B 5 so that W 4 meets S 4 orthogonally in a minimal copy of M 3 . 3. The examples of White ([52]) show that finding embedded minimal hypersurfaces, with control on the topology, in dimension greater than 3, is very difficult. We note an interesting example in [26] Example 2. Lawson gives an action of SO(3) on S 4 with regular orbits which have stabilisers Z2 × Z2 , so are copies of the quaternionic space M 3 = S 3 /Γ, where Γ is the group of 8 units in the quaternions with integer entries. There are two exceptional orbits with SO(2) as stabiliser which are minimally embedded copies of RP 2 . So we get the focal sets of the orbit of maximum volume, which is a minimal embedding of M 3 , are both copies of the projective plane. The two complementary regions of this minimal hypersurface are disk bundles over the projective plane. We next discuss the case of non-negative scalar curvature. The fundamental result was proved in [49].
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Theorem 2.4. Suppose M is a closed Riemannian 3-manifold with a metric of non-negative scalar curvature. If M has an immersed π1 -injective orientable surface S then S must be a torus and M has a flat metric. The only stable orientable minimal surfaces immersed in M are spheres and tori. Using this, it is fairly easy to get a good topological picture of embedded minimal surfaces ([43], [28])). Theorem 2.5. Suppose M is a closed Riemannian 3-manifold with a metric of non-negative scalar curvature. If M is not flat and S is an embedded orientable minimal surface then there is a collection of embedded 2-spheres in the complement of S so that S together with some of these spheres bounds a punctured handlebody on each side ( a punctured Heegaard splitting). The spheres split M into pieces, chosen so that none are punctured 3-balls. In the non-flat case, non-orientable minimal surfaces together with spheres give punctured one-sided Heegaard splittings. In the flat case, either an orientable minimal surface gives a Heegaard splitting or it is a flat torus. Similarly a non-orientable minimal surface gives a one-sided Heegaard splitting or a flat Klein bottle. Remarks 1. In the non-flat case, one can use Dehn’s lemma and the Loop Theorem to show that an orientable minimal surface S must compress on both sides to spheres, since otherwise there would be disjoint stable minimal surfaces which are not spheres. For non-orientable minimal surfaces the argument is similar. 2. For the flat case, the same argument shows that either a surface can be completely compressed (defining a Heegaard splitting which may be one-sided) or there is a disjoint flat torus. (There are no minimal spheres in flat manifolds, by Gauss-Bonnet.) In the latter case, by translation and the maximum principle, one sees the original minimal surface is a flat torus or Klein bottle. This result was first proved by Meeks [29], using a second variation argument in the style of Lawson. He showed that for an orientable closed minimal surface in an orientable closed flat 3-manifold, either the fundamental group maps onto the complementary regions or the surface is a flat torus. In this case, second variation of a shortest geodesic arc may be zero or positive. So he used parallel translation of a piece of the minimal surface along the geodesic to give a contradiction, by the maximum principle. To complete this section, we discuss minimal surfaces in Seifert fibred spaces. Note that there are 8 geometries of Thurston (see [45] for an excellent discussion of these metrics). ^R), H2 × R, Nil, Solv, H3 . Closed manifolds modelled These are S3 , S2 × R, R3 , PSL(2, on these geometries have Seifert fibred space structures, except for the most interesting hyperbolic case and Solv. By passing to a 2-fold cover, we will always assume for simplicity that a closed orientable Seifert fibred spaces has an S 1 -action, where there are a finite
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number of orbits with finite non-trivial stabilisers ( called exceptional orbits) and all other orbits are called regular. To understand the topological possibilities for embedded minimal surfaces, it turns out to be only necessary to assume that the metric is chosen so that the S 1 -action is by isometries. The first key result was proved in [16] Theorem 2.6. Suppose that an orientable Seifert fibred space has a metric which is invariant under the S 1 -action. Then any orientable incompressible surface is either horizontal or vertical, ie is either transverse everywhere to the S 1 -orbits or is a union of regular orbits. Vertical surfaces are tori or Klein bottles. The general case for minimal surfaces which are not necessarily incompressible was given in [38] Theorem 2.7. If M is a closed orientable Seifert fibred space with an S 1 -invariant metric and S is a closed embedded minimal surface, then there are two compression bodies bounded by S and a collection of disjoint vertical embedded tori, if S is orientable. If S is non-orientable, then the boundary of a small regular neighbourhood of S together with some vertical embedded tori bound one compression body. The other components of the complement of these tori are sub-Seifert fibred spaces of M . Remark So the situation is rather similar to telescoping, described in the previous section. Note that many interesting geometric examples of closed embedded minimal surfaces in spherical spaces forms ( with constant sectional curvature +1) or in flat manifolds or Seifert fibred spaces can be found in e.g [25], [23], [29], [30], [17], [38], [39]. We include here a brief account of minimal surfaces in Solv geometry, in response to a question of P. Shalen. In [45], p. 470, this geometry is described as a split extension of R acting on R2 . There is a horizontal foliation by planes, which can easily be seen to be minimal. In fact, with coordinates (x, y, z) where the planes are z = c, multiplication is given by (x, y, z)(x0 , y 0 , z 0 ) = (x + e−z x0 , y + ez y 0 , z + z 0 ) and the metric is ds2 = e2z dx2 + e−2z dy 2 + dz 2 . Now it is easy to see that the horizontal planes are all minimal, by showing their mean curvatures are zero. We can use the isometry φ fixing the origin taking (x, y, z) → (y, x, −z). This maps the normal (0, 0, 1) to the plane P given by z = 0 to the opposite normal (0, 0, −1). Clearly the principal directions on P at the origin must be invariant or interchanged by φ. In the latter case, since the signs of the principal curvatures are reversed,
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we see that their sum must be zero and the plane has zero mean curvature at the origin. In the former case, both principal curvatures would be zero and the plane would be flat. This is actually not the case, but we dont need this fact. Since Solv acts isometrically on itself by left multiplication. we conclude that P and all other planes z = c are minimal in the Solv metric. Now given any embedded closed orientable minimal surface S which is not a torus in a closed orientable 3-manifold M with the Solv metric, it follows by the maximum principle that any tangencies with the induced foliation of M by minimal tori must be saddles. Now it is easy to use standard 3-manifold techniques to deduce the following Theorem. Theorem 2.8. Let M be closed orientable with the Solv geometry. If S is a closed orientable embedded minimal surface in M then either S is part of the foliation of M by minimal tori, or S is a Heegaard surface for M .
3. 3-manifolds with complete hyperbolic metrics of finite volume By the results of Thurston ([50]), we know that the most common type of geometric structure is a hyperbolic metric. Moreover Thurston’s hyperbolisation Theorem shows that all orientable closed irreducible atoroidal Haken 3-manifolds have hyperbolic metrics. Also for compact, orientable, irreducible, atoroidal, non-Seifert fibred manifolds with incompressible tori boundary components, there is a complete hyperbolic metric of finite volume on the interior of the manifold. (Here atoroidal means that any incompressible torus is boundary parallel). By Mostow rigidity, this hyperbolic metric is unique. Furthermore, by recent results ([13], [14]) any irreducible manifold homotopy equivalent to such a hyperbolic manifold is homeomorphic to it. So the hyperbolic metric is a topological invariant of the manifold in this strong sense. Many interesting invariants of the hyperbolic structure have been extensively studied, such as volume, Chern-Simons invariants, geodesic length spectrum etc. However, minimal surfaces may give a lot of new information about the hyperbolic geometry of 3-manifolds. In particular, area, second fundamental form, principal curvatures etc are natural invariants, although difficult to calculate explicitly. The aim in this section is to suggest some preliminary ideas in this direction, coming from elementary bounds on minimal surfaces, e.g using monotonicity and curvature. We refer to [1], [51], [54], [48] for useful background information. A first key observation is that for a closed orientable minimal surface immersed or embedded in a hyperbolic 3-manifold, by the Gauss equation, the induced metric has Gaussian curvature κ ≤ −1. So by Gauss-Bonnet, there is an area bound in terms of the genus g, A ≤ 4π(g−1). In particular, there are no minimal 2-spheres, projective planes, Klein bottles
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or tori. This type of estimate works also in 3-manifolds with all sectional curvatures strictly negative, which occur when doing Gromov-Thurston Dehn filling ([15], [5], [33], [18]). A consequence is compactness of the space of minimal surfaces of bounded genus in closed hyperbolic or negatively curved 3-manifolds ([1], [54]). So any infinite sequence of such surfaces has a convergent subsequence. At a finite number of points, the curvature can blow up and handles can pinch; away from these points the convergence is smooth. Such pinching only happens when the limit surface occurs with multiplicity. We note that compactness also works on non-compact hyperbolic 3-manifolds with complete metrics of finite volume, so long as the minimal surfaces can be bounded away from the cusps. Note that a cusp is defined by taking a region R = {(x, y, z) : z ≥ z0 > 0} in the upper half space model of H3 and dividing out by a parabolic group Γ = Z × Z of isometries of the form (x, y, z) → (x + c, y + c0 , z) generating a lattice in the horizontal planes ( horospheres). Such horospheres project to horotori in M . Now if an embedded minimal surface S of genus at most g meets the horotorus coming from z = z0 in parallel essential curves C1 , C2 , ... of length at most L, then it is easy to bound the ‘depth’ ie the minimum value of z for the horotori intersected by S. For we can find a foliation of R by minimal strips bounded by lines in the homotopy class of the lifts of C1 , C2 , ... and use the maximum principle, since the distance between lifts of C1 , C2 , ... is bounded, depending only on the homotopy class of the curves and not on the surface S, since S is embedded. Therefore to complete the discussion, we must first show that a sequence of embedded minimal surfaces of bounded genus cannot meet the horotorus z = z0 in longer and longer essential curves and must then eliminate inessential curves. The first fact is easy and is proved even for immersed surfaces in [18]. The coarea formula and the area bound above give the result. It is also straightforward to eliminate long tubes, ie regions between horotori z1 ≤ z ≤ z0 where z0 − z1 is large and the surface S meets every horotoral level in homotopically trivial curves. For such a tube lifts to the universal cover and the monotonicity formula gives that the area of the tube grows exponentially with the size of z0 − z1 . ( A similar argument was used in [33]). By our genus bound, there is a bounded number of critical points of intersection of the surface and the horotori ( there are no local maxima relative to the z coordinate by the maximum principle). So all cases are now done. Notice that Freedman twisting of immersed essential surfaces about a cusp ([12]) indicate that this result cannot be extended to immersed minimal surfaces. One can construct such surfaces which are separable, ie lift to embeddings in finite sheeted covers, so can be isotoped to be closed and minimal. However it is easy to see that the surfaces will penetrate deeper and deeper into the cusp and in the limit will give a non-compact complete minimal surface. ([7]). We will construct some examples in the next section. We summarise;
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Theorem 3.1. Let M 3 be a complete hyperbolic 3-manifold of finite volume. Then the collection of closed embedded minimal surfaces of bounded genus forms a compact space. We note that the minimax construction for strongly irreducible Heegaard splittings, extends to the case of compression body decompositions of complete hyperbolic 3-manifolds of finite volume. By essentially the same argument as above, given a critical sequence of sweepouts with minimax surfaces converging in the varifold metric to a smooth minimal surface S ( possibly non-orientable with multiplicity 2), we can show that S is closed. For the regularity argument in [36], [47], is local. So we need only show that this critical sequence only penetrates a bounded distance into any cusps. But the argument follows the same lines as above to conclude this. The minimax construction plus telescoping now enables us to conclude finiteness of the collection of irreducible and strongly irreducible Heegaard splittings of bounded genus of either closed or complete finite volume hyperbolic 3-manifolds. (cf [22] for a different solution of this problem - the Waldhausen conjecture). For given any infinite sequence of such splittings, they can be isotoped to be closed minimal surfaces - possibly non-orientable surfaces with multiplicity 2 - ( the pieces of the splitting in the case of telescoping) and so converge. Now in the case of pinching occurring, the limit surface must have multiplicity. So either the surfaces converge to some non-orientable components with multiplicity 2 or to a surface with multiplicity 1 with no handles pinching, using strong irreducibility ( of the pieces in case of telescoping). Theorem 3.2. ([40]) Let M be a complete finite volume hyperbolic 3-manifold. Then there are finitely many irreducible Heegaard splittings of bounded genus up to isotopy. Our next idea is to relate the Heegaard genus of a complete finite volume hyperbolic 3-manifold M to its geometry. The Heegaard genus is the smallest genus of all Heegaard splittings of M . In particular, the genus should reflect the smallest cross-sectional area under a sweepout. Definition 3.3. A sweepout of a 3-manifold consists of a pseudo-isotopy St , where 0 ≤ t ≤ 1 and S0 , S1 are graphs, St is embedded for 0 < t < 1, ( For many purposes it is useful to allow handles of St to pinch and regrow during the sweepout at finitely many values of t cf [42]). Let us first consider the case of a closed hyperbolic 3-manifold M . Denote the injectivity radius (half the length of the shortest closed geodesic) by ρ. We give a simple lower bound for the Heegaard genus of M . By monotonicity, the area of a minimal surface S passing through the centre of a ball of radius ρ is at least 2π(coshρ − 1), which is the area of a totally geodesic disk of this radius. On the other hand, we know by Gauss Bonnet as at the beginning of this section, that the area A of S satisfies A ≤ 4π(g − 1). Putting these
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together gives g≥
coshρ + 1 . 2
Now this estimate can be improved by eliminating any very short closed geodesics from consideration. So we investigate the thick/thin decomposition of M (cf [50]). Let C1 , C2 , ..., Ck be a collection of disjoint short closed geodesic simple loops in M . Take tubular neighbourhoods of these loops and expand out at the same rate until the boundaries bump into each other. Let N1 , N2 , ..., Nk denote these maximal embedded neighbourhoods. Keep S S expanding and flatten out into a 2-complex X. So the complement M \ (N1 N2 ... Nk is a neighbourhood of X which we denote by N (X). The solid tori Ni are foliated by tori with inward pointing mean curvature vectors, by taking tori at constant distance from Ci . Let ρ0 denote the injectivity radius of loops based at N (X), ie we take half the length of the shortest geodesic loop based at some point of N (X) as ρ0 . Now the same estimate as 0 +1 before applies, namely g ≥ coshρ . We need only observe that any minimal surface S must 2 meet X. For otherwise S would like in one of the open solid tori complementary domains of N (X). But these solid tori are foliated by tori with inward pointing mean curvature vectors, so by the maximum principle, it is follows that there are no closed minimal surfaces in such regions. Note the same argument works well for non-compact complete finite volume hyperbolic 3-manifolds with cusps. For we can remove any short closed geodesics and expand out both the cusps and neighbourhoods of the short geodesics simultanously to again find a neighbourhood N (X) of a 2-complex X and compute ρ0 . In both cases we will refer to ρ0 as the injectivity radius of the thick part of M . Theorem 3.4. Let M be a closed or complete finite volume hyperbolic 3-manifold. Let ρ0 be the injectivity radius of the thick part of M . Then the minimal Heegaard genus g of M satisfies g≥
coshρ0 + 1 . 2
Remarks 1. Note that if the smallest genus Heegaard splitting is irreducible but not strongly irreducible, ie telescoping occurs, then this estimate can be improved. The same is true if the minimal surface is obtained by collapsing onto a non-orientable surface with multiplicity 2. 2. The same estimate also works for bounding the smallest genus of embedded or immersed closed π1 -injective surfaces in M . For by [49], such surfaces can be homotoped to least area immersions. It is easy to show as above that such least area maps remain away from the cusps and so give closed minimal immersions. So the monotonicity bounds apply.
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3. An interesting question about Heegaard genus g is whether the rank r of π1 (M ), ie the smallest number of generators, can be arbitrarily smaller than g, ie whether g − r can be arbitrarily large. ( It is obvious that g ≥ r.) However it is hard to find a good upper bound for r. Example 3. We can now extend example 1. Let M be a closed hyperbolic 3-manifold with an element α of infinite order in H1 (M, Z) or M complete finite volume hyperbolic with cusps and H1 (M, Z) has an element α of infinite order which is non-cuspidal. Assume that the length of a shortest loop in the homology class of α is L and ρ0 is the injectivity radius of the thick part of M . Then the smallest genus g 0 of a strongly irreducible Heegaard splitting for M satisfies 4π(g 0 − 1) ≥ 2αρ0 − and so g 0 ≥ αρ 0 2π. The reason is that we can map M to a circle S 1 by the argument of Stallings, sending the class α to a generator of the homology of S 1 . Now perturb the minimal surface S corresponding to a strongly irreducible splitting ( or the corresponding non-orientable surface with multiplicity 2) into general position relative to this map. If the level sets on S are always essential, the estimate follows by the coarea formula. If there are only inessential curves at some levels, this cannot occur except for a very short interval, using monotonicity (see previously - we add in a small correction term to take account of this case.) So this give the result. We conclude that if L is large, then any strongly irreducible splitting must have large genus. By taking cyclic covers, lifting α to a finite multiple, we can make L as large as we like. Our next observation is that some information can be obtained on the Heegaard genus of finite sheeted coverings of M with a complete hyperbolic metric of finite volume. It is well-known that π1 (M ) is residually finite. So any non-zero element β of π1 (M ) can be ‘unwound’ so that some non-trivial multiple lifts to a finite sheeted covering. In this way, we can choose a sequence of finite sheeted coverings to ensure that the injectivity radius ( of the whole manifold or of the thick part) increases as much as desired. In this way, by Theorem 3.4, it follows that the Heegaard genus of this covering space also increases to any desired lower bound. On the other hand, suppose that S is a strongly irreducible splitting of M . Let D1 , D2 , .., Dk and D10 , D20 , .., Ds0 be two complete systems of meridian disks for the compression bodies on either side of S, ie if the compression bodies are cut open along these disks, the result is products of closed surfaces and intervals or 3-balls. Now if the total intersection number of these disks is N and we take a d fold covering space, then we get kd and sd disks respectively which meet in dN points. Now if d2 ks > dN , ie d > ks N , then clearly some of the lifted disks in one compression body do not meet some of the disks on the other side, ie the lifted splitting is no longer strongly irreducible. Hence either it is irreducible and telescopes or
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has some trivial handles and destabilises. In either case we can find new minimal surfaces in the lifted manifold, by using Corollary 1.9. ( Collapsing onto a non-orientable minimal surface with multiplicity 2 is included as a possibility. We will see however that there can be a problem with telescoping - see the next paragraph.) Projecting back to M , this leads to a sequence of immersed minimal surfaces which are not coverings of each other. To complete this discussion, we need to consider the case where no new strongly irreducible Heegaard splittings come from this covering construction. because telescoping may not produce any interesting surfaces. In this case, some finite sheeted covering of M is a ˜ over a circle. For then, telescoping only gives two copies of the fibring surface bundle M surface and no strongly irreducible splitting surfaces of the pieces. Moreover any finite ˜ is again a bundle so we may be stuck with no further new minimal sheeted cover of M surfaces. It is easy to see this is the only case where telescoping fails to come up with new strongly irreducible compression body decompositions between incompressible surfaces. ¯ of M ˜ where the Now an easy way of proceeding is to pass to a finite sheeted covering M monodromy map is trivial on H1 (F, Z2 ), where F is the fibre of the bundle structure. Since ˜ induces a finite order permutation on this homology, by taking a the monodromy for M ¯ . Now the rank of H1 (M ¯ , Z2 ) is at least the genus of suitable cyclic cover, we can find M F . So we get a collection of incompressible orientable or non-orientable embedded surfaces ¯ , depending on whether the dual homology classes come from classes in H1 (M ¯ , Z) or in M not. These can all be isotoped to least area maps by [49]. To get more, we can pass to a higher covering, eg by taking the kernel of the map from π1 (F ) → H1 (F, Z2 ) and forming ¯ preserves this kernel. the induced covering space, since the monodromy of the bundle M If we find new non-orientable surfaces in the higher covering space, these cannot cover ¯ lift to orientable surfaces in previously found non-orientable surfaces since all the ones in M the new covering space. If incompressible orientable surfaces are found which are not fibres of bundles structures, then telescoping gives new minimal surfaces as previously. Finally if new fibrings are found then again these cannot cover previously found fibrings and we have proved; Theorem 3.5. Let M be a 3-manifold with a complete hyperbolic metric of finite volume. Then M admits infinitely many distinct minimal immersions of closed surfaces. Remarks In [55], the question is asked whether all closed Riemannian 3-manifolds admit infinitely many embedded closed minimal surfaces. It would be good to at least answer this question for the case of hyperbolic 3-manifolds. To complete this section, we mention two recent applications of minimal surfaces to questions about hyperbolic 3-manifolds. In [18], a question of P. Shalen about boundary
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slopes of immersed proper π1 -injective and π1 -∂-injective surfaces in 3-manifolds is answered. In particular it is shown that for a complete hyperbolic 3-manifold with finite volume and one cusp, there is a uniform bound on the number of boundary slopes of such surfaces of genus at most g, which is quadratic in g. The main idea is to use area bounds, estimating the amount of area inside a maximal cusp as a function of the length of the boundary slope. (The boundary slope of a proper surface is the homology class of the boundary curves on the boundary torus - we are assuming here that the surface has a single boundary slope, even though it is only immersed. This is natural when considering closed immersed incompressible surfaces arising from Dehn filling on knots). A similar result is also obtained for hyperbolic 3-manifolds with several cusps. In [33], the irreducible Heegaard splittings of the collection of Dehn fillings on a complete hyperbolic 3-manifold M with finite volume and several cusps is studied. (Recall that Dehn filling means, remove an open horotoroidal neighbourhood of each cusp and glue in a solid torus). In [50], it is shown that except for a finite number of exceptions for each cusp, almost all Dehn fillings give a closed hyperbolic 3-manifold N . The main result of [33] is that except for finitely many exceptions at each cusp, there are finitely many surfaces embedded in M so that any Heegaard splitting of genus at most g of N is isotopic to one of these surfaces. Moreover each of these surfaces is either a Heegaard splitting of M or there is a unique curve on the surface isotopic into one of the cusps. Again the key idea is to use area bounds - Dehn filling has the property that if the meridian curve for the glued in solid torus is very long on a maximal horotorus, then the area of the meridian disk becomes very large. So any irreducible Heegaard surface cannot cross the solid torus in a meridian disk without having large genus. ( This works well in case of telescoping as well as strong irreducibility). We conclude any splitting surface of bounded genus must lie outside the solid torus, ie is in M as claimed. This result shows that small genus Heegaard splittings of nearly all the hyperbolic 3manifolds obtained by Dehn filling on several cusps of M come from the same surfaces in M . Thurston and Jorgensen ( cf [50]) have shown that complete hyperbolic 3-manifolds of finite volume form a well-ordered set using volume as the order. Limit points correspond precisely to the collection of all Dehn fillings on a cusped manifold, which converge (in the Gromov-Hausdorff metric) to the cusped manifold. 4. Surfaces with small principal curvatures and the topology of minimal surfaces in hyperbolic 3-manifolds In this section, we first look at immersed closed surfaces with small principal curvature in complete hyperbolic 3-manifolds of finite volume. In [51] and [10], some properties of such surfaces were discussed. Here small principal curvature means either ≤ 1 or < 1 depending on circumstances. A key difference between the two cases occurs for 3-manifolds
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with cusps. It is not difficult to show that an immersed closed surface S with principal curvatures < 1 cannot have any essential loops homotopic into the cusps. ( These are often called accidental parabolics). However we will give many examples of such surfaces with principal curvatures ≤ 1 with accidental parabolics. Note that Freedman twisting ([12]) is associated with accidental parabolics. The main result about surfaces of small principal curvature is that they are π1 - injective. Such surfaces are of great interest in 3-manifold topology - a fundamental problem - the Waldhausen-Thurston conjecture states that every aspherical 3-manifold has such surfaces. In fact, one approach to Thurston’s geometrisation conjecture is to try to show the existence of π1 -injective surfaces which are separable, ie lift to embeddings in finite sheeted covering spaces ( cf for example [7]). It is not hard to prove that if a π1 -injective surface immersed in a hyperbolic 3-manifold with cusps has no accidental parabolics, then the surface remains π1 -injective under most Dehn fillings (cf [4] for a proof of Thurston). So this gives a good way of constructing such examples. Moreover, Thurston has conjectured that all complete finite volume hyperbolic 3-manifolds have immersed surfaces with principal curvature < 1. Note also that under sufficiently large hyperbolic Dehn filling, a surface with principal curvature bounded below 1, will have its metric perturbed only a small amount, so the surface will remain in this class. Theorem 4.1. ([51], [10]) Suppose that M is a complete hyperbolic 3-manifold with finite volume. Let S be an immersed closed surface with all principal curvatures at most one. Then S is π1 -injective and lifts to an embedding in the covering space MS corresponding to the subgroup π1 (S) of π1 (M ). The Gauss or normal map gives a foliation of MS by copies of S at constant distance along normal geodesics. Finally if S has accidental parabolics then S has points with principal curvature exactly 1.
Proof There are several ways of proving this; we use the foliation of the universal cover by horospheres, since this will be useful for the new method of constructing examples in [3]. Assume first that the principal curvatures of S are < 1. Choose a lift S˜ of S to the universal covering space H3 and a point x on the sphere at infinity, which is not in the limit set of ˜ To find such point is easy - just choose a horosphere touching S˜ at some point. Then S. it is easy to see that the centre of this horosphere is such a point x. Now the foliation of H3 by horospheres centred at x will have tangencies on S which can be minima but never maxima nor saddles, since the principal curvatures of S are less than those of the horospheres. We conclude that S˜ must be a plane, with only one local minimum relative to this foliation. Moreover it also follows that S˜ is embedded. For the intersection curves with the horospheres are always embedded loops and there are no self intersections.
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If we allow principal curvatures up to 1, accidental parabolics can arise. Note that if there is an essential loop C on S homotopic into a cusp, and if C is shrunk to a geodesic loop on S then this loop lifts to a line in a horosphere in H3 . Let Z denote the subgroup of π1 (S) ⊂ π1 (M ) generated by C. Let MZ denote the corresponding covering of M and let SZ be the covering of S. Then SZ is an annulus with centre C and MZ is the quotient of H3 by the action of Z. We can foliate MZ by the horoannuli with centre at the fixed point of the action of Z on the sphere at infinity. It is easy to see that there must be a tangency of SZ and one of these horoannuli along the geodesic representative of C lifted to SZ . If there are no accidental parabolics and S is not a horotorus, then there must be points where at least one of the principal curvatures is < 1. So we can choose a horosphere touching S˜ at such a point x and use the horospheres with centre at x as the foliation. Although now the foliation can touch S˜ in regions such as disks or graphs, one can still show there is a single such (simply connected) region which corresponds to a minimum and no other critical points. Consequently S˜ is an embedded plane. As in the previous paragraph, if S has accidental parabolics, it follows that S˜ and one of the horospheres touch along a line with ends at the centre of the horosphere. At any rate, the same conclusion works that S˜ is an embedded plane. Finally the observation about the foliation of MS follows easily - there are clearly no focal conjugate points, by looking at Jacobi fields along a geodesic ray. If two normal geodesics of the same length end at a common point y, then by taking metric spheres centred at y we can touch S˜ at some point in a saddle or local maximum, contradicting the principal curvature ≤ 1 assumption. Ben Andrews has described a beautiful heat flow method for deforming any surface with all principal curvatures < 1 to a canonical minimal surface with the same property. In fact, the minimal surface has the amazing property that it has the smallest maximum principal curvature in its homotopy class. So the flow proceeds by ‘flattening’ the surfaces as well as decreasing their area. Note it is easy to see that any minimal surface S with principal curvatures ≤ 1 is unique in its homotopy class. For lifting to the covering space MS , the foliation by surfaces at a constant distance from a lift of S all have inward pointing mean curvature. So any other minimal surface S 0 homotopic to S would lift to MS and be touched by one of these surfaces, contradicting the maximum principle. To complete this discussion, we sketch the main idea of [3] ( cf [2] for background). Thurston ([50]) described a beautiful triangulation of the figure-8 knot space, ie the result M of removing this knot from S 3 . He took two regular ideal tetrahedra, with all dihedral
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angles π3 between their faces and glued together the faces to form M . In particular, this exhibits the complete finite volume hyperbolic metric on M in a simple way. Now Haken’s theory of normal surfaces can be viewed as a form of discrete minimal surface theory (cf [42]). A normal surface consists of elementary disks properly embedded in each tetrahedron. There are two types of disks, triangles cutting off vertices and quadrilaterals separating opposite pairs of edges in a tetrahedron. Our idea is to find an analogue of small principal curvature for such immersed normal surfaces. We follow the outline in Theorem 4.1 by using foliations by horospheres to bound the bending of immersed normal surfaces. Theorem 4.2. ([3]) Let S be a closed surface which is tiled by triangles and quadrilaterals satisfying the following conditions; -every vertex of the tiling is of degree 6. - no two quadrilaterals share a common edge. - every vertex has an even number of quadrilaterals. Then after possibly passing to a small finite sheeted covering of S we get a surface S˜ which immerses as a normal surface in the figure-8 knot space M and is π1 -injective. Moreover this surface can be deformed to a smooth surface with principal curvatures ≤ 1 and has accidental parabolics if and only if there is an essential annulus of triangles in the tiling. If there are no accidental parabolics, the surface can be deformed to have principal curvatures < 1. Sketch proof The main ideas of the proof are as follows. Firstly one can locally immerse such a tiling into M . However to match up around essential loops, the surface may not glue up as one may have different normal arc types at a given face of the triangulation around such a loop. However this is corrected by passing to a covering space. Next, there is a convenient way to place all the elementary disks, ie triangles and quadrilaterals. All the vertices are placed at the midpoints of the edges of the triangulation, ie the points of symmetry ( all edges of the triangulation are infinitely long geodesics). The edges of the elementary disks are geodesic arcs in the faces between the vertices. Finally, by studying a neighbourhood of an elementary disk (i.e all elementary disks with a vertex or edge in common with a given disk), it can be shown that given the three conditions of the Theorem, that such a neighbourhood can be placed ‘outside’ of a horosphere. For example, given a triangle, its 3 vertices lie on a unique horosphere. All the other vertices of such a neighbourhood then lie outside this horosphere. For a quadrilateral, one takes any 3 vertices and checks that the fourth lies outside automatically as do the other vertices of the neighbourhood. In this way, we can see that a foliation by horospheres can only touch
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the surface at local minima. These local minima must include elementary triangular disks. So the conclusions of Theorem 4.1 apply. Note that the 8 triangles at the corners of the two tetrahedra form a horotorus in M . So any annulus of triangles in the tiling gives an accidental parabolic. It is easy to check that the absence of such an annulus means that there are no accidental parabolics. Finally we can round off the edges and vertices of the surface to form a smooth surface which still has principal curvatures ≤ 1. More work is required in the case of no parabolics to deform the surface to have all principal curvatures < 1; the main idea is to slightly bend regions of triangles to lie on surfaces of constant principal curvature. < 1 Remarks The same method works with other link complements with regular cell decompositions, such as the Whitehead link and Borromean rings, which are divided into regular ideal octahedra with all dihedral angles π2 . In theory the method should work for general ideal cell decompositions and triangulations, but it is much more difficult to characterise the tilings of surfaces and to locate the vertices so conveniently. Possibly the class of simple 2-bridge knot and link complements could be analysed. Our final topic is the topology of embedded minimal surfaces in hyperbolic 3-manifolds. In particular, we consider two related questions. By Corollary 1.9, we know that any strongly irreducible Heegaard splitting can either be isotoped to be minimal, or else collapses onto a non-orientable minimal surface with multiplicity 2 and the resulting surface defines a one-sided Heegaard splitting. So the first question, asked by S.T. Yau is whether there can be a disjoint minimal surface inside a handlebody bounded by a minimal surface. The second question is can such a surface be knotted, ie can we find embedded closed orientable minimal surfaces which are neither incompressible nor bound handlebodies. We sketch a construction giving both types of examples. The key idea is to form a barrier by using a big tube about a short closed geodesic, ie the thin part of the hyperbolic manifold. Theorem 4.3. ([44]) Hyperbolic 3-manifolds which are complete of finite volume or closed, can be constructed with arbitrarily many disjoint closed embedded minimal surfaces, which are neither incompressible nor bound handlebodies nor compression bodies ( in the case of cusps). Moreover such surfaces can be found inside handlebodies bounded by minimal surfaces. Sketch proof The basic idea is to choose two simple closed curves C, C 0 on a strongly irreducible Heegaard surface S so that C, C 0 are ‘disk busting’ in the handlebodies, H+ , H− respectively, bounded by S. By this we mean that C meets every meridian disk for H+ and similarly
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for C 0 and H− . Then push C into X and C 0 into Y . We also require that M \ (C hyperbolic i.e is irreducible and atoroidal.
S
C 0 ) is
Next do large longitudinal Dehn surgery along both C and C 0 . It is easy to see that the handlebodies H+ , H− remain handlebodies, only the gluing between their boundaries along S is changed. Moreover one can obtain that the new Heegaard splitting is still strongly irreducible if desired, following an argument of Casson-Gordon ([34]). This is useful if we want to find stable and unstable minimal surfaces which are parallel. (Note longitudinal surgery means that the new meridian disk for the glued in solid torus meets a longitude for the original solid torus once.) Now we will form a barrier surface using S and the big tubes about C, C 0 . The intuitive idea is that in the limit of Dehn surgeries, C, C 0 become cusps and so S is now an incompressible surface. In this case we can find a stable minimal surface isotopic to S. Note that the property that C, C 0 are disk busting comes in here, since we want no compressing disks for S remaining when these curves are removed. Remove smaller open tubes about the core circles C, C 0 , of say half the radius of the ˜ with two boundary tori T, T 0 . We can find disjoint big tubes, to form a new manifold M ¯ S¯0 compact properly embedded incompressible and boundary incompressible surfaces S, with two boundary essential boundary curves on T, T 0 respectively. These are formed by pushing an annulus on S out into the tubes about C, C 0 respectively. (The disk busting assumption is needed here). Now as in [18], we can taper off the metric to make the two boundary tori totally geodesic ˜ . We can then isotope S, ¯ S¯0 to be stable minimal surfaces which have geodesic boundary in M ˜ orthogonally. Now a simple area estimate shows that inside a collar curves and meet ∂ M ¯ S¯0 meet each horotorus in two curves which are close to of the boundary, the surfaces S, ¯ S¯0 being geodesics. So we can choose least area annuli A, A0 between two such curves for S, ¯ S¯0 by removing two annuli and replacing them by A, A0 , respectively so that if we cut off S, then new closed surfaces S1 , S2 isotopic to S are formed. These are piecewise least area and form barriers. To justify this, we must check that the angles between the annuli and the surfaces are less than π. This can be done by knowing that the horotori are met in curves close to geodesics. In particular, at this stage explicit estimates can be obtained on the size of Dehn surgery required to perform the construction, depending only on the genus of S. So we can isotope S to a stable minimal surface in the region between these surfaces. Now this argument can be applied to find new types of minimal surfaces, as well as many parallel surfaces inside handlebodies. Choose any surface U inside one of the handlebodies, say H+ , to bound a smaller handlebody X, so that there is a compression body Y between U and S. Now perform exactly the same procedure, using X, Y in place of H+ , H− . We see that after longitudinal Dehn surgery of curves pushed off V , that S still defines a strongly
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irreducible compression body splitting for M \ intX. So U can be isotoped to be a stable minimal surface inside the handlebody bounded by the minimal surface isotopic to S. Finally, we can choose a separating compressible surface V which does not bound a handlebody on either side. Again loops C, C 0 can be selected to be disk busting for any compressing disks on each side of V . The same construction will give a new 3-manifold for which V still does not bound handlebodies on either side ( longitudinal Dehn surgery does not affect the homeomorphism type of the two sides of V , only their gluing along V ). Here V can be isotoped to a stable minimal surface by the previous barrier argument. It is not difficult now to find many disjoint embedded minimal surfaces of different types, using this procedure a number of times. References [1] M. Anderson, Curvature estimates for minimal surfaces in 3-manifolds, Ann. Scient. Ecole Norm. Sup. 18 (1985), 89-105. [2] I. Aitchison, S. Matsumoto and J.H. Rubinstein, Surfaces in the figure-8 knot: complement, Journal of Knot theory and its ramifications, 7 (1998), 1005-1025. [3] I. Aitchison, S. Matsumoto and J.H. Rubinstein, Immersed incompressible surfaces of small curvature in the figure-8 knot complement, in preparation. [4] I. Aitchison and J.H. Rubinstein, Incompressible surfaces and the topology of 3-manifolds, Journal of the Aust Math Soc. Series A, 55 (1993), 1-22. [5] S. Bleiler and C. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), 809-833. [6] A. Casson and C. Gordon, Reducing Heegaard splittings of 3-manifolds, Topology and its applications, 27 (1987), 275-283 [7] D. Cooper and D. Long, Virtually Haken Dehn filling, J. of Diff, Geom. 52 (1999), 173-187. [8] T. Colding and W. Minnicozzi, Minimal surfaces, Courant Lecture Notes in Mathematics, Vol 4, 1999 [9] H. Choi and R. Schoen, The space of embeddings of a minimal surface into a three dimensional manifold of positive Ricci curvature, Invent. Math. 85 (1985), 387-394. [10] C. Epstein, The hyperbolic Gauss map and quasiconformal refections, Journal fur Mathematik, 372 (1986), 96-135. [11] M. Freedman, J. Hass and P. Scott, Least area incompressible surfaces in 3-manifolds, Invent Math. 7 (1983), 609-642. [12] B. Freedman and M.H. Freedman, Kneser-Haken finiteness for bounded 3-manifolds, locally free groups and cyclic covers, Topology37 (1998), 133-147. [13] D. Gabai, On the geometric and topological rigidity of hyperbolic 3-manifolds, Journal of the Amer. Math. Soc. 10 (1997), 37-74. [14] D. Gabai, W Meyerhoff and N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, to appear in Annals of Math. [15] M. Gromov and W. Thurston, Pinching constants for hyperbolic 3-manifolds, Invent. Math. 89 (1987), 1-12. [16] J. Hass, Minimal surfaces in Seifert fibred spaces, Topology Appl. 18 (1984), 145-151. [17] J. Hass, J. Pitts and J.H. Rubinstein, Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces, Proc. Symposium Pure App. Math., 54, I Amer. Math. Soc. Providence R. I. 1993, 147-162. [18] J. Hass, J.H. Rubinstein and S. Wang, Boundary slopes of immersed surfaces in 3-manifolds, J. Diff. Geom. 52 (1999), 303-325. [19] J. Hass and P. Scott, The existence of least area surfaces in 3-manifolds, Trans. Amer. Math. Soc. 310 (1988), 87-114.
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