MINIMAL SURFACES IN THE 3-SPHERE BY DESINGULARIZING INTERSECTING CLIFFORD TORI NIKOLAOS KAPOULEAS AND DAVID WIYGUL 7 Abstract. For each integer k ≥ 2, we apply gluing methods to construct sequences of minimal 1 surfaces embedded in the round 3-sphere. We produce two types of sequences, all desingularizing 0 collections of intersecting Clifford tori. Sequences of the first type converge to a collection of k 2 Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, n after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower a of order k. Sequences of the second type desingularize a collection of the same k Clifford tori J supplementedbyanadditionalCliffordtorusequidistantfromtheoriginaltwocirclesofintersection, 0 so that the latter torus orthogonally intersects each of the former k tori along a pair of disjoint 2 orthogonal circles, near which the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type resemble surfaces constructed by Choe and ] G Soret[2]bydifferentmethodswherethenumberofhandlesdesingularizingeachcircleisthesame. There is a plethora of new examples which are more complicated and on which the number of D handles for the two circles differs. Examples of the second type are new as well. . h t a m 1. Introduction [ The general framework. 1 After totally geodesic 2-spheres, Clifford tori represent the next simplest minimal embeddings v of closed surfaces in the round unit 3-sphere S3. In fact Marques and Neves [19], in their proof 8 of the Willmore conjecture, have identified Clifford tori as the unique area minimizers among all 5 embedded closed minimal surfaces of genus at least one, and Brendle [1] has shown they are the 6 5 only embedded minimal tori, affirming a conjecture of Lawson. The first examples of higher-genus 0 minimalsurfacesinS3 wereproducedbyLawsonhimself[18],andfurtherexampleswerefoundlater . 1 by Karcher, Pinkall, and Sterling [17]. Both constructions proceed by solving Plateau’s problem 0 for suitably chosen boundary and extending the solution to a closed surface by reflection. 7 In this article we carry out certain constructions by using gluing techniques by singular per- 1 : turbation methods. One begins with a collection of known embedded minimal surfaces. These v ingredients are then glued together to produce a new embedded surface, called the initial surface, i X having small but nonvanishing mean curvature introduced in gluing. The construction is successful r when the initial surfaces are close to a singular limit. The construction is then completed by per- a turbing the surface to minimality without sacrificing embeddedness. Of course the size of the mean curvature and the feasibility of perturbing the surface so as to eliminate it both depend crucially on the design of the initial surface. Gluing methods have been applied extensively and with great success in gauge theories by Don- aldson, Taubes, and others. In many geometric problems similar to the one studied in this article obstructions appear to solving the linearized equation. An extensive methodology has been devel- oped to deal with this difficulty in a large class of geometric problems, starting with [25], [5] and with further refinements in [6]. We refer to [8] for a general discussion of this gluing methodology and[9]foradetailedgeneraldiscussionofdoublinganddesingularizationconstructionsforminimal surfaces. In this article, however, we limit ourselves to constructions of unusually high symmetry (except in section 8) and this way we avoid these difficulties entirely. 1 Thefirstdesingularizationconstructionbygluingmethodsforminimalsurfaceswithintersection curves which are not straight lines was carried out in [7] and serves as a prototype (see for example [12,13,20]) for desingularizations of rotationally invariant surfaces with transverse intersections without triple or higher points. (An independent construction by Traizet [26] has straight lines as intersections.) For one earlier application of the gluing methodology in the context of minimal surfaces in S3 see [14], where a “doubling” construction of the Clifford torus is carried out; this work has been extended in [27] for “stackings” of the Clifford torus and in [10] for doublings of the equatorial two-sphere. The present construction also glues tori, but by desingularization rather than doubling. The idea of a desingularization construction for intersecting minimal surfaces in a Riemannian three-manifold is to start with a collection of minimal surfaces intersecting along some curves and to produce a single embedded minimal surface by desingularizing the curves of intersection. Assuming transverse intersection, this is accomplished, at the level of the initial surface, through the replacement of a tubular neighborhood of each component curve of the intersection set by a surface which on small scales approximates a minimal surface in Euclidean space desingularizing the intersection of a collection of planes along a single line. In prior desingularization constructions the appropriate models for these desingularizing surfaces were furnished by the classical Scherk towers of [24], which desingularize the intersection of two planes. One novelty of the present article is our use of the more general Karcher-Scherk towers, introduced in [15], which come in families desingularizingany numberofintersectingplanesandso accommodatecurvesof intersectionwhose complements, in small neighborhoods of the curves, contain more than four components. Note that althoughhavingmorethantwominimalsurfacesintersectalongacurveisnotagenericsituation,it canhappeninrotationallyinvariantcasesasforexampleinthecaseofcoaxialcatenoids. Extending the results of [7] to such situations for example is an interesting but difficult problem because one would have to use the full family of Karcher-Scherk towers as studied in [22]. A motivation for our construction is the observation [15, Section 2.7] that Lawson’s surfaces may be regarded as desingularizations of a collection of great 2-spheres intersecting with maximal symmetry along a common equator. In this article we pursue analogous constructions with tori instead of spheres as proposed in [9, Section 4, page 300]. Pitts and Rubinstein have described one class of surfaces (item 10 on Table 1 of [23]), similar to some of our surfaces, to be obtained by min-max methods. Recently Choe and Soret [2] have produced examples by solving Plateau’s problem for a suitably selected boundary, in the spirit of [18]. Their examples resemble the simpler examplesweconstruct. (Toprovetheyarethesameonewouldhavetoprovethatthesolutionofthe Plateau problem is unique; see remark 4.19). Our construction has been developed independently and is more general in ways we describe below, and our strategy is quite different, based as we already mentioned on gluing techniques by singular perturbation methods. On the other hand our methods work only for high-genus surfaces. To outline, we construct two infinite families of embedded minimal surfaces in S3. The first familyconsistsofdesingularizationsofaconfigurationW (see3.8)ofk ≥ 2Cliffordtoriintersecting k symmetricallyalongapairofdisjointgreatcirclesC andC whichlieontwoorthogonaltwo-planes 1 2 in R4. The second family consists of desingularizations of a configuration W(cid:48) (see 3.8) which is the k previous one augmented by the Clifford torus which is equidistant from C and C . In both cases 1 2 the construction is based on choosing “scaffoldings”, that is unions of great circles contained in the given configuration, and which we demand to be contained in the minimal surfaces we construct. Moreover, reflections with respect to the great circles contained in the scaffoldings are required to be symmetries of our constructions. We denote the scaffoldings we choose by C ⊂ W or k,m k C(cid:48) ⊂ W(cid:48) (see 3.23). k,m k To construct the initial surfaces we replace tubular neighborhoods of the intersection circles withsurfacesmodeledonappropriatelyscaledandtruncatedmaximallysymmetricKarcher-Scherk 2 towers. Towers with 2k ends are used along C and C , while classical Scherk towers with 4 ends 1 2 are used along other circles of intersection (present only in W(cid:48)). The replacements are made k so that each initial surface is closed and embedded, contains the applicable scaffolding, and is invariant under reflection through every scaffold circle (see Definition 4.13). These initial surfaces are perturbed then to minimality in a way which respects the reflections, so the surfaces produced are closed embedded minimal and still contain the scaffolding (see the Main Theorem 7.1). Note that Lawson’s approach also makes use of a scaffolding. Our approach, however, gives much more freedominthenumberofhandlesweincludeinthefundamentaldomain, whileinLawson’smethod thefundamentaldomainisadiscsothatPlateau’sproblemcanbesolved. Thismakesnodifference when considering the original construction of Lawson in [18]: we expect that the construction with more handles in the fundamental domain will still produce a Lawson surface even though it does not a priori impose all the symmetries of the surface. When there are more than two circles of intersection involved, however, we can choose different numbers of handles on each of them and this gives a plethora of new surfaces as in the present constructions (see the Main Theorem 7.1). It seems also rather daunting to try to construct even the simplest of our surfaces desingularizing W(cid:48) by Lawson’s method. k The present constructions motivate two important new directions for further study. First, what othersimilardesingularizationconstructionscanbecarriedoutincaseswherethereareobstructions due to less symmetry? One has to deal then with the obstructions in the usual way by introducing smoothfamiliesofinitialsurfaceswiththeparameterscorrespondingtotheobstructionsasinearlier work (see [7–9,11]). We discuss this question in Section 8 and we provide some partial answers. Second, asremarkedintheendof[9, Section4.2], therearevariousnaturalquestionsaboutrigidity and uniqueness for the surfaces presently constructed which are similar to those asked [9, questions 4.3,4.4and4.5]abouttheLawsonsurfaces. Inparticularwearecurrentlyworkingtoprovethatthe surfaces desingularizing W can not be smoothly deformed to surfaces desingularizing k Clifford k tori still intersecting along C and C but with different angles (that is they cannot “flap their 1 2 wings”). More precisely we hope to prove that even with reduced symmetries imposed so “flapping the wings” is allowed, there are no new Jacobi fields on our surfaces. Outline of the approach. As we have already mentioned the main difficulty of this construction as compared to earlier results is proving that under the symmetries imposed there is no kernel on the Karcher-Scherk towers. AsfortheclassicalsinglyperiodicScherksurfaces(k = 2)[7],thisisachievedbysubdividing the surface suitably and applying the Montiel-Ros approach [21]. Our approach is also somewhat different than usual in some other aspects and we employ the high symmetry we have available. Organization of the presentation. In Section 2 we study in sufficient detail the maximally symmetric Karcher-Scherk towers using the Enneper-Weirstrass representation and following [15]. In Section 3 we study the geometry of the Clifford tori and the initial configurations we will be using later, their symmetries, and the symmetries and scaffoldings we will impose in our constructions later. In Section 4 we discuss in detail the construction of the initial surfaces and we study their geometry. In Section 5 we provide estimates for the geometric quantities on the initial surfaces. In Section 6 we study the linearized equation and estimate its solutions on the initial surfaces. We finally combine these results to establish the main theorem in Section 7. Finally in Section 8 we discuss further results using more technology. Notation and conventions. Given an open set Ω of a submanifold immersed in an ambient manifold endowed with metric g, an exponent α ∈ [0,1), and a tensor field T on Ω, possibly taking 3 values in the normal bundle, we define the pointwise H¨older seminorm |T(x)−τ T(y)| yx g (1.1) [T] (x) = sup , α d(x,y)α y∈Bx where B denotes the open geodesic ball, with respect to g, with center x and radius the minimum x of 1 and the injectivity radius at x; |·| denotes the pointwise norm induced by g; τ denotes g yx parallel transport, also induced by g, from y to x along the unique geodesic in B joining y and x; x and d(x,y) denotes the distance between x and y. Given further a continuous positive function f : Ω → R and a nonnegative integer (cid:96), assuming that T is a section of the bundle E over Ω all of whose order-(cid:96) partial derivatives (with respect to any coordinate system) exist and are continuous, we set (cid:13) (cid:13) (cid:88)(cid:96) (cid:12)(cid:12)DjT(x)(cid:12)(cid:12) (cid:2)D(cid:96)T(cid:3) (x) (1.2) (cid:13)T : C(cid:96),α(E,g,f)(cid:13) = (cid:107)T(cid:107) = sup + α , (cid:13) (cid:13) (cid:96),α,f f(x) f(x) x∈Ω j=0 where D denotes the Levi-Civita connection determined by g. In case E is the trivial bundle Ω×R, instead of C(cid:96),α(E,g,f) we write C(cid:96),α(Ω,g,f). When f ≡ 1, we write just C(cid:96),α(Ω,g), and when α = 0, we write just C(cid:96)(Ω,g). Additionally, if G is a group acting a on a set B and if A is a subset of B, then we refer to the subgroup (1.3) StabG(A) := {g ∈ G | gA ⊆ A} as the stabilizer of A in G. When A is a subset of Euclidean 3-space (or the round 3-sphere), we will set (1.4) G (A) := Stab A, sym Isom where Isom = O(3) (or O(4)). For a subset C of Euclidean space (or the round 3-sphere) we define (1.5) G (C) refl to be the group generated by reflections with respect to the lines (or great circles) contained in C. Here reflection through a great circle can be defined as the restriction to the 3-sphere of reflection in R4 through the 2-plane containing the circle. If G is a group of isometries of a Riemannian manifold with a two-sided immersed submanifold Σ, then we call g ∈ StabG(Σ) even if g preserves the sides of Σ and odd if it exchanges them. In this two-sided case, sections of the normal bundle of Σ may be represented by functions, on which StabGΣ then acts according to (gu)(x) = (−1)gu(cid:0)g−1x(cid:1), where (−1)g is defined to be 1 for g even and −1 for g odd. We append a subscript G to the spaces of functions defined above to designate the subspace which is StabG(Σ)-equivariant in this sense so that for instance (cid:110) (cid:12) (cid:111) (1.6) CGk,β(Σ,g,f) = u ∈ Ck,β(Σ,g,f) (cid:12)(cid:12) u(cid:0)g−1(x)(cid:1) = (−1)gu(x) ∀x ∈ Σ ∀g ∈ StabG(Σ) . Finally, we make extensive use of cutoff functions, and for this reason we fix a smooth function Ψ : R → [0,1] with the following properties: (i) Ψ is nondecreasing, (ii) Ψ ≡ 1 on [1,∞] and Ψ ≡ 0 on (−∞,−1], and (iii) Ψ− 1 is an odd function. 2 Given then a,b ∈ R with a (cid:54)= b, we define a smooth function ψ[a,b] : R → [0,1] by (1.7) ψ[a,b] = Ψ◦L , a,b where L : R → R is the linear function defined by the requirements L(a) = −3 and L(b) = 3. a,b Clearly then ψ[a,b] has the following properties: 4 (i) ψ[a,b] is weakly monotone, (ii) ψ[a,b] = 1 on a neighborhood of b and ψ[a,b] = 0 on a neighborhood of a, and (iii) ψ[a,b]+ψ[b,a] = 1 on R. Acknowledgments. The authors would like to thank Richard Schoen for his continuous support andinterestintheresultsofthisarticle. N.K.waspartiallysupportedbyNSFgrantsDMS-1105371 and DMS-1405537. 2. The Karcher-Scherk towers In [15] Karcher introduced generalizations of the classical singly periodic Scherk surfaces, includ- ing the maximally symmetric Karcher-Scherk towers of order k ≥ 2, which we will denote by S . k S is a singly periodic, complete minimal surface embedded in Euclidean 3-space, asymptotic to k k planes intersecting at equal angles along a line. This line, which we call the axis of S , is parallel k to the direction of periodicity. The classical singly periodic Scherk tower [24] asymptotic to two orthogonal planes is recovered by taking k = 2. Although in this article we will only use S in our k constructions, it is worth mentioning that there is a continuous family of singly periodic minimal surfaces with Scherk-type ends which has been studied by P´erez and Traizet in [22] and in which family S is the most symmetric member. k We proceed to outline the construction of Karcher [15]. Note that S , which we will define in k detail later, differs by a scaling and rotation from the surface described now. Karcher considered the Enneper-Weierstrass data of 1 dζ (2.1) Gauss map ζk−1 and height differential ζk +ζ−k ζ on the closed unit disc in C punctured at the 2kth roots of −1. The embedding defined by the data mapstheorigintoasaddlepoint,thek linesegmentsjoiningoppositerootsof−1tohorizontallines of symmetry intersecting at equal angles, the 2k radii terminating at roots of unity to alternately ascending and descending curves of finite length lying in k vertical planes of symmetry, and the 2k circumferential arcs between consecutive roots of −1 to curves of infinite length lying alternately in one of two horizontal planes of symmetry. The union of this region with its reflection through either horizontal plane of symmetry is a fundamental domain for the tower under periodic vertical translation. The images of small neighborhoods of the roots of −1 are asymptotic to vertical half-planes, which bisect the vertical planes of symmetry. For future reference the following proposition fills in the details of the above outline and sum- marizes the basic geometric properties of S , including its symmetries and asymptotic behavior. k To state the lemma we make a few preliminary definitions. First we define the sets H(cid:98), a union of horizontal planes, and V(cid:98)k, a union of vertical planes, by (cid:91) (cid:91) (2.2) H(cid:98) := {z = nπ} and V(cid:98)k := {y = xtanjπ/k}, n∈Z j∈Z whose intersection is the union of horizontal lines (2.3) Ck := H(cid:98) ∩V(cid:98)k. We enumerate the connected components of the complement of H(cid:98) ∪V(cid:98)k by (cid:26) (cid:18) (cid:19) (cid:27) j j +1 (2.4) B := (rcosθ,rsinθ,z) : r ≥ 0, θ ∈ π, π , z ∈ (lπ,(l+1)π) k,l,j k k 5 (cid:16) (cid:17) for each l,j ∈ Z, and we also define a partition of R3\ H(cid:98) ∪V(cid:98)k into disjoint sets Ak and A(cid:48)k given by (cid:91) (cid:91) (2.5) A := B and A(cid:48) := B . k k,l,j k k,l,j l+j∈2Z l+j∈2Z+1 To describe the symmetries of Sk we write R(cid:98)P for reflection in R3 through the plane P, T(cid:98)a(cid:96) for translation in R3 by a units along the directed line (cid:96), and R(cid:98)φ for rotation in R3 through angle φ (cid:96) about the directed line (cid:96) (according to the usual orientation conventions). A horizontal bar over a subset of R3 denotes its topological closure in R3, and angled brackets enclosing a list of elements (or sets of elements) of O(3) indicate the subgroup generated by all the elements listed or included in the sets mentioned. Proposition 2.6. For every integer k ≥ 2 there is a complete embedded minimal surface S ⊂ R3 k such that (i) Sk ∩H(cid:98) = Ck, Sk\Ck ⊂ Ak, and every straight line on Sk is contained in Ck; (ii) for any connected component B of A the intersection S ∩B is an open disc with ∂(S ∩B) = k k k C ∩B (the union of four horizontal rays); k (cid:68) (cid:69) (iii) for each non negative integer m the quotient surface Sk/ T(cid:98)2z-maxπis has 2k ends and genus (k−1)(m−1); (iv) G (C ) (cid:40) G (S ) = G (A ) = G (A(cid:48)) (cid:40) G (C ) (recall 1.4 and 1.5) and G (S ) refl k sym k sym k sym k sym k sym k acts transitively on the set of connected components of S \C , the set of connected components k k of A , and the set of connected components of A(cid:48); k k (cid:68) (cid:69) (v) Grefl(Ck) = R(cid:98)2z-πa/xkis, R(cid:98)πx-axis, T(cid:98)2z-πaxis ; (cid:68) (cid:69) (cid:68) (cid:69) (vi) Gsym(Sk) = R(cid:98)2z-πa/xkis, R(cid:98)πx-axis, T(cid:98)2z-πaxis, R(cid:98)θ=π/2k, R(cid:98)z=π/2 = Grefl(Ck), R(cid:98)θ=π/2k, R(cid:98)z=π/2 ; (vii) Gsym(Ck) = Gsym(Sk) ∪ R(cid:98)πz-/akxisGsym(Sk) = Gsym(Sk) ∪ T(cid:98)πz-axisGsym(Sk); and (cid:112) (viii) there exists R > 0 so that S \{ x2+y2 < R } has 2k connected components, all isometric k k k by the symmetries, exactly one of which lies in the region π {(rcosθ,rsinθ,z) : r > 0, |θ| < , z ∈ R}, 2k and the intersection of this component with {x ≥ R } is the graph k {(x,W (x,z),z) : x ≥ R , z ∈ R} k k over the xz-plane of a function W : [R ,∞)×R → R, which decays exponentially in the sense k k that ∀j,(cid:96) ≥ 0 we have (cid:12) (cid:12) (2.7) (cid:12)∂j∂(cid:96)W (x,z)(cid:12) ≤ C(k,j +(cid:96))e−x. (cid:12) x z k (cid:12) Proof. The usual Inner-Weierstrass recipe (see [16] for example or any standard reference for the classicaltheoryofminimalsurfaces)definesfromthedata2.1aminimalsurfaceinR3 parametrized by the closed unit disc in C punctured at the 2kTh roots of −1. Requiring the parametrization to 6 take the origin in C to the origin in R3, it takes the form (cid:90) w 1−ζ2k−2 1 (cid:88)2k x(w) = Re dζ = (−Reω )ln|w−ω |, 1+ζ2k k j j 0 j=1 (cid:90) w 1+ζ2k−2 1 (cid:88)2k (2.8) y(w) = Re i dζ = (Imω )ln|w−ω |, 1+ζ2k k j j 0 j=1 z(w) = Re(cid:90) w 2ζk−1 dζ = 1 (cid:88)2k (−1)j−1arg ωj −w, 1+ζ2k k ω 0 j j=1 where ωj = eiπ1+2k2j is the jTh 2kTh root of −1 and arg is the imaginary part of the branch of the logarithmic function cut along the ray from 0 to −∞ and taking the value 0 at 1. The symmetries can be read from the data by the following standard argument. Looking at the expression for the metric (2.9) D(cid:48)s2 = (cid:18)|ζ|k−1+ 1 (cid:19)2(cid:12)(cid:12)(cid:12) 1 dζ(cid:12)(cid:12)(cid:12)2 |ζ|k−1 (cid:12)ζk +ζ−k ζ (cid:12) in terms of the Enneper-Weierstrass data 2.1, one identifies as intrinsic geodesics both circumfer- ential arcs on the unit circle and diametric segments joining opposite 2kth roots of ±1. Looking next at the expression for the second fundamental form (cid:20)(cid:18) (cid:19)(cid:18) (cid:19)(cid:21) dζ 1 dζ (2.10) A (V,V) = 2(k−1)Re V V tow ζ ζk +ζ−k ζ in terms of the Enneper-Weierstrass data, it becomes apparent that the diametric segments joining opposite 2kth roots of −1 are extrinsic geodesics as well, which are therefore mapped to Euclidean lines, while the diametric segments joining opposite 2kth roots of unity along with the circumfer- ential arcs are lines of curvature, which, being also intrinsic geodesics, are necessarily mapped to planar curves. Consultation of 2.8 confirms that (a) the straight lines lie along the intersection of the single horizontal plane z = 0 with the vertical planes of the form θ = π/2k +nπ/k for n ∈ Z, (b) the images of the circumferential arcs lie, alternatingly, in the two horizontal planes z = ±π/2k, and (c) the images of the remaining lines of curvature lie, consecutively, in the vertical planes of the form θ = nπ/k for n ∈ Z. Below we will check that the parametrization 2.8 is in fact an embedding of the punctured unit disc. In fact we will show that the image of the punctured sector (cid:110) π (cid:111) (2.11) D := reiθ : 0 ≤ r ≤ 1, 0 ≤ θ ≤ \{ω } 1 2k is embedded and intersects the planes z = 0, z = π/2k, θ = 0, and θ = −π/2k only along the curves just mentioned. The reflection principle for harmonic functions then implies that this image may be extended to a complete embedded minimal surface Sˇ , invariant under reflection through any k line in 1C and through any plane of the form θ = π/2k+jπ/k or z = π/2k+jπ/k for j ∈ Z. We k k define Sk := kR(cid:98)πz-/a2xkisSˇk. Items (ii) and (iii) and the first two claims of item (i) follow immediately, as do the contain- ments G (C ) (cid:40) G (A ) ⊆ G (S ), considering that item (v) and the equalities G (A ) = refl k sym k sym k sym k (cid:68) (cid:69) Gsym(A(cid:48)k) = R(cid:98)2z-πa/xkis, R(cid:98)πx-axis, T(cid:98)2z-πaxis, R(cid:98)θ=π/2k, R(cid:98)z=π/2 are clear from the definitions (2.3 and 2.5) of C , A , and A(cid:48) alone. To see the containment G (S ) ⊆ G (C ) note that any symmetry of k k k sym k sym k the tower must permute the asymptotic planes, so must preserve their intersection, so must take 7 any line in C to a line orthogonally intersecting the z-axis; we are assuming (and confirm with the k maximum-principle argument below) that S intersects the z-axis only where C does, and from k k the expression (2.10) for the second fundamental form we know we have already accounted for all lines on Sk through such points. The equalities Gsym(Ck) = Gsym(Ak) ∪ R(cid:98)πz-/akxisGsym(Ak) = Gsym(Ak) ∪ T(cid:98)πz-axisGsym(Ak) are also immediate consequences of the definitions. In particular reflection through the z = 0 plane preserves C , but it does not preserve S (because, for example, k k the normal to S is not constantly vertical along the lines S ∩{z = 0}), so G (S ) (cid:54)= G (C ). k k sym k sym k Now, since G (A ) has index 2 in G (C ) and G (A ) ⊆ G (S ) (cid:40) G (C ), in fact sym k sym k sym k sym k sym k G (A ) = G (S ). sym k sym k Thus we have checked items (ii)-(vii), as well as the first two claims of (i), and the transitivity claim in (iv) is now obvious. To verify the remaining claim in (i), note that any straight line in S , k by virtue of the latter’s minimality, is a line of reflectional symmetry. On the other hand reflection through a straight line in R3 preserves A only if the line is the z-axis (and then only for k even) k π/2 π/2k or if it is contained in Ck or T(cid:98)z-axisR(cid:98)z-axisCk. It is easy to see, however, that of these lines only those contained in C lie on the surface. (For example 2.1 and 2.8 reveal that at easily identified points k where any of the other lines do intersect the surface the normal to the surface there is parallel to the line.) In addition to checking (viii), it remains to show that the image of the region D is embedded and intersects the planes z = 0, z = π/2k, θ = 0, and θ = −π/2k as described above. That the tower is embedded may be established by recognizing the conjugate surface of the region between two consecutive horizontal planes of symmetry as a graph—specifically the solution to the Jenkins- Serrin problem [4] on a regular 2k-gon—and then appealing to a theorem of Krust. See [16] for details on this approach. Alternatively we show embeddedness more directly as follows and in the process identify the intersection of the image of D with these four planes. Recall that D is the punctured sector {reiθ : 0 ≤ r ≤ 1, 0 ≤ θ ≤ π }\{ω }. From 2.8 it is clear 2k 1 that dz is positive along the radial segment from the origin to 1 and vanishes along both the radial segment from the origin to ω and the circular arc from 1 to ω . A sufficiently small circular arc, 1 1 centered at ω , which originates on the segment from 0 to ω and terminates on the circumferential j 1 arc from ω to 1, can be seen from 2.8 to have height monotonically increasing from 0 to π . The 1 2k maximum principle then implies that the image of D is contained in the slab {0 ≤ z ≤ π } and 2k intersectsz = 0onlyalongthe(straight,horizontal)imageoftheradialsegmenttoω andz = π/2k 1 only along the (horizontal) image of the circumferential arc. Similarly, from 2.8 one may readily check monotonicity of dx, dy, and (Reω )dy+(Imω )dx on 1 1 the boundary curves of D in order to establish that the boundary has image contained in the wedge {(rcosθ,rsinθ,z) : r ≥ 0, − π ≤ θ ≤ 0, z ∈ R}. Moreover, 2.8 reveals that in D 2k (2.12) lim x(w) = ∞, w→ω1 (2.13) lim y(w) = −∞, and w→ω1 (2.14) lim [(Reω )y(w)+(Imω )x(w)] = 0, 1 1 w→ω1 so that another application of the maximum principle (to the harmonic coordinate functions x, y, and (Reω )y+(Imω )x, the last extended to the closure of D) establishes that the image of D is 1 1 contained in {(rcosθ,rsinθ,z) : r ≥ 0, − π ≤ θ ≤ 0, 0 ≤ z ≤ π } and intersects θ = 0 only along 2k 2k the image of the radial segment to 1 and θ = −π/2k only along the image of the radial segment to ω . 1 BecauseofthesymmetriesitthereforesufficestoshowthattheEnneper-Weierstrassparametriza- tion restricts to D as an embedding. To this end observe first that each level curve of y in D is connected. Indeed one can verify that dy vanishes nowhere on D and has norm (relative to |dz|2) 8 tending to infinity at ω , so ∇y defines a smooth vector field on the closure of D. The corre- 1 |∇y|2 sponding flow for time t, when it exists, maps points (other than the fixed point ω ) with y = y 1 0 to points with y = y +t. Examining the field at the boundary, one can check that the backward 0 flow of a point in D exits D only after it reaches the segment joining 0 and ω , while the forward 1 flow leaves D only through the real boundary segment. Thus, if the flow for time t ≤ 0 of a point x on the real segment from 0 to 1 lies in D, then the flow for time t of any real point to the right of x will also lie in D. Now given w ,w ∈ D with y(w ) = y(w ) = y < 0, the flow for time −y 1 2 1 2 0 0 takes each point to a point on the real segment of D (since by the maximum principle and earlier monotonicity arguments this segment is the entirety of the y = 0 curve in D). By the previous considerations the flow for time y exists for every point on the real segment joining these points, 0 so, the flow for time y being a continuous function of the initial point, we get a level y = y path 0 0 joining w and w . 1 2 Now suppose there exist points w ,w ∈ D such that x(w ) = x(w ) and y(w ) = y(w ). Then 1 2 1 2 1 2 thereisapathcontainedinasinglelevelcurveofyjoiningw andw ,and,ifw andw aredistinct, 1 2 1 2 by the mean value theorem there exists a third point w between w and w on this path at which 3 1 2 dx vanishes along the path. Thus at w the gradients ∇x and ∇y must be parallel. One finds from 3 2.8 that these gradients are parallel only on the circular boundary of D, where they are tangential to the boundary, and therefore they are endpoints of level curves, so that we may assume w is 3 not such a point. Thus we conclude that w = w , showing not only that the Enneper-Weierstrass 1 2 parametrization is an embedding but also that its image of the unit disc is actually a graph over a region in the z = 0 plane. Now we prove (viii). It is clear from 2.8 that for R sufficiently large the set of w in the unit disc with x2(w)+y2(w) > R2 has 2k components, each containing exactly one of the 2kth roots of −1. We define (2.15) s(w) = Reω x(w)−Imω y(w) and 1 1 (2.16) t(w) = Imω x(w)+Reω y(w), 1 1 so that t(w) is the signed distance of the image of w from the plane θ = −π/2k and (2.17) (x(w),y(w),z(w)) = s(w)(Reω ,−Imω ,0)+z(w)(0,0,1)+t(w)(Imω ,Reω ,0), 1 1 1 1 the three terms on the right being pairwise orthogonal. We will show that for R sufficiently large k there is a correspondingly small neighborhood Ω of ω in the closed unit disc such that the map 1 1 f : C\(cid:83)2k {cω | c ∈ [1,∞)} → C defined by j=1 j (2.18) f(w) := (s(w),z(w)) restricts to a diffeomorphism from Ω onto the half-strip [R ,∞)×(cid:2)− π , π (cid:3). 1 k 2k 2k Working from 2.8 we find (cid:89)k (cid:12)(cid:12) w−ωj (cid:12)(cid:12)Imω1ωj (2.19) kt(w) = ln (cid:12) (cid:12) , (cid:12)w−ω2ω (cid:12) j=2 1 j (cid:18) w (cid:19) (cid:89)2k (cid:18) w (cid:19)(−1)j−1 kz(w) = arg 1− +arg 1− ω ω 1 j (2.20) j=2 (cid:18) (cid:19) w = arg 1− +ψ (w), z ω 1 9 and k (cid:88) ks(w) = −ln|w−ω |+ln|w+ω |− Reω ω ln|w−ω ||w−ω2ω | 1 1 1 j j 1 j (2.21) j=2 = −ln|w−ω |+c+ψ (w), 1 s where c = lim (ks(w)+ln|w−ω |) and ψ and ψ are defined by the equalities where they are j s z w→ω1 introduced. Then lim ψ (w) = lim ψ (w) = 0, and for each nonnegative integer (cid:96) there exists a s z w→ω1 w→ω1 constant C(k,(cid:96)) > 0 such that (cid:96) (2.22) sup (cid:88)(cid:16)(cid:12)(cid:12)∂j∂(cid:96)−jψ (w)(cid:12)(cid:12)+(cid:12)(cid:12)∂j∂(cid:96)−jψ (w)(cid:12)(cid:12)(cid:17) < C(k,(cid:96)). (cid:12) w w s (cid:12) (cid:12) w w z (cid:12) w∈Ω1j=0 Defining the map g : C → C by (cid:16) (cid:17) (2.23) g(w) := ω 1−e−kw+c , 1 we see that for R sufficiently large the composite f ◦g(cid:12)(cid:12)[R,∞)×(−π,π) is well-defined (identifying C with R2 as usual) and 1 (2.24) f(g(w)) = w+ (ψ +iψ )(g(w)), s z k so, since lim g(w) = ω and lim ∂jg(w) = 0foreachintegerj > 0, bytakingRlargeenough 1 w Rew→∞ Rew→∞ (cid:12) wecanensurethatf◦g(cid:12)[R,∞)×(−π,π) isasmallperturbationoftheidentityandsoadiffeomorphism with image containing the half-strip [R ,∞)×(cid:2)− π , π (cid:3) for some R > R. k 2k 2k k Thus f itself restricts to a diffeomorphism from some region Ω onto this half-strip, as asserted 1 above, which shows that the image of Ω under the Enneper-Weierstrass parametrization is the 1 graph of t◦f−1 over the half-strip {(rcos−π/2k,rsin−π/2k,z) : r ≥ R ,|z| ≤ π/2k}. Since t is k smooth on Ω , t(ω ) = 0, and f−1 = g◦(f ◦g)−1, we finally obtain the estimates 1 1 (cid:12) (cid:12) (2.25) (cid:12)∂j∂(cid:96)−j(t◦f−1)(s+iz)(cid:12) ≤ C(k,(cid:96))e−ks. (cid:12) s z (cid:12) (cid:3) The union C of all horizontal lines on S can be regarded as a scaffolding for the tower, but we k k emphasize that a tower is not uniquely determined by a choice of scaffold: Remark 2.26. (i) The surface T(cid:98)πz-axisSk = R(cid:98)πz-/akxisSk satisfies all conditions in the lemma provided the roles of Ak and A(cid:48)k are reversed in the statement, so in particular its intersection with H(cid:98) is Ck. (ii) For m a positive integer the surface m−1Sk also has Ck as its intersection with H(cid:98), but the (cid:68) (cid:69) quotient surface (cid:0)m−1Sk(cid:1)/ T(cid:98)2z-πaxis has genus (k−1)(m−1) (cid:54)= 0 instead of 0 (recall 2.6.iii). 3. The initial configurations The Clifford tori. In this subsection we discuss the Clifford tori and their geometry. We first introduce some helpful notation. Given a great circle C in S3 we will write C⊥ for the furthest great circle from it. (Note that the points of C⊥ are at distance π/2 in S3 from C and any point of S3\C⊥ is at distance < π/2 from C). As viewed from R4, the planes containing C and C⊥ are orthogonal complements. On the other hand, C and C⊥ may be regarded as parallel in that the function on S3 measuring distance from one of the circles is constant on the other. (This relation 10