The structure of singly-periodic minimal surfaces - Springer Link

Invent. math. 99, 455~481 (1990)

Inventiones mathematicae 9 Springer-Verlag 1990

The structure of singly-periodic minimal surfaces* Michael Callahan, D a v i d Hoffman and William H. Meeks II! Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003, USA

I. Introduction 1.1. The structure theorems The only properly embedded minimal surface with more than one topological end and a continuous g r o u p of symmetries is the catenoid. Considering periodic minimal surfaces with m o r e than one end (a surface in N 3 is periodic if it is invariant under an infinite discrete g r o u p of isometrics that act freely on 1R3), the only classical examples are those discovered by Riemann. The R i e m a n n examples ~ possess a quite special set of properties: 1. They have an infinite n u m b e r of flat annular ends 1; 2. They are invariant under a nontrivial translation T; 3. The surfaces ~ / T h a v e genus 1 and 2 ends; 4. The surfaces ~ / T have total curvature equal to - 8 zr. Motivated by the R i e m a n n examples, we established the existence of an infinite family of properly embedded periodic minimal surfaces, d//k, each with an infinite n u m b e r of ends [1]. These surfaces, which are discussed below (see Sect. 1.2 on Rigidity), also have an infinite n u m b e r of flat annular ends, and are invariant under a translation T. The quotient of these surfaces by T is a surface of genus 2 k + 1 with two ends and total curvature - 8 r e ( k + 1), k > l . We have been able to "twist" each of these new surfaces; that is, we construct surfaces with the same properties that are invariant under screw motions T + R, where R is a rotation a r o u n d an axis in the direction of T (see Sect. 1.3 below). T o g e t h e r with the catenoid a n d the examples of Riemann, these new surfaces comprise all the k n o w n properly embedded minimal surfaces with more than one topological end whose s y m m e t r y g r o u p contains an infinite discrete subgroup. These examples share m a n y of the properties of the R i e m a n n examples. A major goal of this paper is to show that this is not accidental. * The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8611574 and DMS-8802858 i By an annular end of a surface we mean an end that has a representative homeomorphic to a punctured disk. Often such a representative is referred to as the end itself. An annular end is flat provided it is asymptotic to a plane in ~3

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Theorem 1. Suppose M is a properly embedded minimal surface, with more than one end, whose symmetry group is infinite. Then either M is the catenoid or: 1. M has an infinite number qf ends ; 2. M is invariant under a screw motion 5P ; 3. all annular ends of M are fiat ends; 4. the total curvature of M = M / Y is finite if and only if )f4 has finite topology, in which case C (ffI) = 2 ~z(X (if/I) - r (M)), where r(lfI) is the number of ends of [~I. We emphasize that it follows from Statement 4 that, for the surfaces we are considering, finite topology of the quotient surface M/5 e is equivalent to finite total curvature. See also Sect. 1.4 for related results. Theorem 1 is proved in Sect. 3 below. Using Theorem 1, it can be shown that if M has one end which is annular, it has infinitely many (see Corollary 1 of Sect. 3) and they are asymptotic to an infinite family of parallel planes. We believe that Property 3 of Theorem 1 can be strengthened. Conjecture 1. Under the hypotheses of Theorem 1, either M is a catenoid or M/Sg has a finite number of ends, each with quadratic area growth. One is tempted to conjecture even more: namely, that M/Se is a Riemann surface of finite topology. However, the authors believe that is not the case because of the following hueristic construction. Take one of the surfaces of Theorem 3 below, say J/g1, together with a horizontal translate of itself. Each flat end intersects its translate transversely in a finite number of divergent arcs. We believe that the union of these two surfaces can be approximated by an embedded periodic minimal surface whose quotient by the vertical translation group T has two ends of infinite genus. The new surface looks like the union of the two surfaces, with each transverse intersection curve replaced by a ribbon of tunnels. We note that in Theorem I and Conjecture 1, the assumption that the symmetry group of M is infinite may be replaced by the weaker assumption that the isometry group of M (in the induced metric) is infinite. This is a consequence of the Isometry Theorem of [2] : Every intrinsic isometry of a properly embedded minimal surface M with more than one end is induced by a symmetry of M. Theorem 1 is a consequence of a somewhat more general result, which in turn depends on an analysis of the asymptotic behavior of a properly embedded minimal surface with more than one topological end. We show in Sect. 3 that to every end of such a surface one can associate a limit tangent plane and this limit tangent plane is unique (i.e. does not change from end to end). This is proved in Sect. 3, Lemma 1 and Theorem 6. In this way, these surfaces behave as if they had finite total curvature. The existence of a unique limit tangent plane for a properly embedded minimal surface is a key element in the proof of

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Theorem 2 (The structure theorem). Suppose M is a properly embedded minimal surface in I~ 3 with infinite symmetry group and more than one end. Then either M is a catenoid or else M possesses the following properties: 1. Sym (M) contains an infinite cyclic subgroup T of finite index, generated by a screw motion T. 2. There exists a plane P, parallel to the limit tangent plane of M, whose intersection with M consists of a finite number of simple closed curves. 3. I f the screw motion T has nontrivial rotational part, the limit tangent plane of M is orthogonal to the axis of T. Using this structure theorem, we can establish the following important topological result: A doubly-periodic, properly embedded, minimal surface in IR 3 has one end. (Corollary 2, Sect. 3.) The proof of Theorem 2 requires a technical tool important enough to state separately.

Proposition 1 (Canonical representation of annular ends). Each annular end of a complete, nonsimply-connected, oriented minimal surface has a unique representative whose boundary is a closed geodesic. These representatives have pairwisedisjoint interiors. If the boundaries of two such annular ends touch, they coincide and M is an annulus. We will refer to such an end as a canonical end. Its existence has a direct consequence in the case that M is complete embedded minimal surface of finite total curvature.

Proposition2. Let

M u I R 3 be a complete oriented minimal surface of genus k with r ends, all separately embedded, and finite total curvature. Suppose M is neither the plane nor the catenoid. Then M may be decomposed in a unique manner as the union of r canonical ends, each of which has total curvature - 2 n , and a surface of genus k whose boundary consists of r closed geodesics and whose total curvature is - 2 n(2 k + r - 2).

These two propositions are proved in Sect. 2. The second proposition may be proved directly in the case of finite total curvature, and we have been informed that this was done by Celso Costa, (unpublished, oral communication).

1.2. Rigidity of the surfaces J/Ik The theoretical results described above were motivated by a desire to understand the qualitative behavior of the surfaces recently discovered by the authors in [1]. These surfaces are described by the following theorem.

Theorem 3 [1]. For every positive integer k there exists a properly embedded minimal surface JIk with the following properties: 1. d/lk has an infinite number of annular ends. 2. dgk is invariant under the group of translations T generated by T: ~-+~ + (0, O, 2). 3. J/tk/T has genus 2 k + 1 and two ends. 4. The symmetry group of d/lk/T has order 8(k+ 1).

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5. Reflection in the plane {x 3 = n + 1/2}, n6~, is a symmetry of .Xg~. 6. ,//gk/T has ]~nite total curvature -- 4 ~ (2 k + 2). 7. All the ends of ~ k are fiat; they are asymptotic to the planes x3 = n, n ~ . 8. ~t'k~ {X3 =n}, n~Z, consists of k + 1 equally spaced straight lines meeting at (0, O, n). 9. ~/~kC~{X3 = C}, CCZ is a simple closed curve. 10. The subgroup R of the symmetry group of J//[k consisting of rotations about the xa-axis has order k + 1 and is generated by rotation by 2~z/(k + 1). 11. J/tk is symmetric under reflection through the k + 1 vertical planes containing the xa-axis and bisecting the lines of property 8. 12. The full symmetry group of ~gk is generated by T, R, one of the reflections in 5, rotation about one of the lines in 8, and reflection through one of the planes in 11. See [1] for pictures of the surface Jgl and ~g2. In Sect. 5, we use the structure theorems to prove that Properties 1-5 of Theorem 3 imply Properties 6-12. In particular, the surface J[k/T must have finite total curvature. There is strong computational evidence that the surfaces Jgk are unique. For each k, there is a one-parameter family of immersed minimal surface which must contain any surface satisfying the conditions of Theorem 3. A surface in this family will be embedded and singly-periodic, and will satisfy Theorem 3, if a period vanishes. This period is a smooth function of the parameter describing the family, and in [1], we show that it possesses a zero and is asymptotic to a linear function. Computations indicate that it is also monotonic and, hence, that its zero is unique. Nonetheless, there is the remote possibility that this is not the case and that there is more than one JLk satisfying the conditions of Theorem 3. 1.3. Construction of new examples invariant under a screw motion In Sect. 4, we construct analogs of J/Lk that are invariant under nondegenerate screw motions. Theorem 4. For every positive integer k and angle O, 00}. We define M + = M n/~. In Statement 1, A c M + so, in particular M + # : 0 and M + is noncompact. In Statement 2, the assumption that M + 4 0 is equivalent (see the discussion before the statement of Lemma 2) to the assumption that M does not lie on one side of the catenoid barrier A. Proof of Statement I. Suppose that Sym(M) contains a screw motion whose translational part is not parallel to P. Without loss of generality we may asssume that the translational part has a positive x3-component. The rotational part of the screw motion must take P into a plane parallel to P. Since A is the graph of a function with logarithmic growth, a sufficiently large power of this symmetry t a k e s / ) into R. Label this symmetry f Since c~M + c D , f ( O M + ) c R . Hence f ( M + ) c M +. In fact, f ( M + ) c M + - A. Let M~- denote the subset of M + between the planes P and fk(p), k ~ Z +. Note that the sets t?M + and fk(t?M+) are isometric compact sets, and the distance between these sets grows without bound as k--.oe. If OMk+ = t?M + u f k(OM +) for all k ~ Z +, we would have a sequence of compact minimal surfaces whose boundaries are contained in the union of two disjoint balls of fixed size, with the distance between balls growing without bound. This is an impossibility. Therefore, t?M~ ~ t?M + w fk(t?M +) for sufficiently large k > 0. But since o m ~ = P w fk(P), we now know that fk(t~ m +) ~ t?m ~ n fk(p). Let V be the constant unit-length vector field on R3 whose value at each point is the vertical vector (0, 0, 1). The vector field V is the gradient of the

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M+

k(p)

Fig. 2. The barrier d u D. In the proof of Statement 1, A c M +.

x3-coordinate function, whose restriction to M is harmonic. If n is the outwardpointing normal to t?M~, it follows from the divergence theorem and the fact that the intrinsic gradient of the restriction of x3 to M is the tangential projection of V, that ~ V. n = 0. t?M~

But since t3M + is isometric to fk(t?M+), V is invariant under f, and (from the previous p a r a g r a p h ) f ~ ( t ? M + ) ~ = ~ M ] c~fk(P), it follows that this integral is not zero. This contradiction proves that if M has a catenoid end E, no screw motion symmetry of M can have a translational component that is not parallel to P, and therefore to the limit tangent plane of E. Before completing the proof of Statement 1, we observe that the same argument applied to a catenoid barrier in Statement 2 will show that Observation 1. I f E is a catenoid barrier for M and M + 4=O, then no symmetry of M can have a translational component that is not parallel to the (xl, x2)-plane P. To complete the proof of Statement 1, we must show that a screw motion symmetry of M cannot have a translational component that is purely horizontal and nonzero. Suppose such a symmetry, say s, exists. Observe that sk(A)+A, for all k > 1. But since the rotational part of s must take P into a plane parallel to itself, s Z ( A ) n A ~ O . This contradicts the fact that A ~ M and M is embedded. []

Proof of Statement 2. Suppose E is a catenoid barrier for M. We assert that E can be chosen to be the end of an actual catenoid. The m a x i m u m principle

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The component of R - A * touching D Fig. 3. The three catenoids comprising A*, drawn here in the case where v is a translation

at infinity for minimal surfaces, Theorem 1 in [15], states that two disjoint, properly immersed, minimal surfaces in N~3 with compact boundary are a positive distance apart, and so E is a positive distance from M. Therefore, the end E' of the catenoid to which E is asymptotic can also be chosen to be a positive distance from M. This proves the assertion and henceforth in the proof of the lemma we will assume that the catenoid barrier referred to in Statement 2 is the end of an actual catenoid. F r o m Observation 1, we know that a screw motion symmetry of M must have a translational component that is parallel to P, i.e. horizontal. Since P is parallel to the limit tangent plane of M at infinity, the screw motion symmetry must have its axis parallel to a line in P and the rotational part must be the identity or rotation by n around the axis. It follows that the composition of a screw motion symmetry of M with itself is a translation in I13. Hence, if Sym (M) contains a screw motion, then it also contains a translation that leaves P invariant. Suppose T is such a translational symmetry of M. Let M ' = M c~ {x 3 ~_~0} and note that M' is z-invariant. Because A c~ M ' = 0 it follows that zk(A)c~M'=O for all k ~ Z . In particular A * = z - X ( A ) w A wz(A) is disjoint from M + c M'. Thus M + is contained in the component of R - A * whose boundary contains D. (See Fig. 3). But observe that this component lies between two vertical planes, and hence it lies in a half-slab or a half-wedge of R3. But M + is a noncompact, properly immersed minimal surface with compact boundary. According to Theorem 3 of [8], its convex hull is equal to one of the following: IR3; a halfspace; a slab; a plane. This contradiction shows that z cannot have a translational component, and completes the proof of Statement 2. []

Remark 3. With a bit more effort, one can strengthen Statement 1 of L e m m a 2 to say that a connected, properly embedded minimal surface with a catenoid end and an infinite symmetry group must be the catenoid. This in fact follows from Theorem 2 (see R e m a r k 5). Theorem 2 (The structure theorem). Suppose M is a properly embedded minimal surface in 1lt3 with infinite symmetry group and more than one end. Then either M is a catenoid or else M possesses the following properties:

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1. Sym (M) contains an infinite, cyclic subgroup T of finite index, generated by a screw motion T. 2. There exists a plane P, parallel to the limit tangent plane of M, whose intersection with M consists of a finite number of simple closed curves. 3. If the screw motion T has nontrivial rotational part, the limit tangent plane of M is orthogonal to the axis of T.

Remark 4. It should be pointed out that if T is a pure translation, the plane P in Property 2 of the theorem is not necessarily orthogonal to the direction of T. F o r example, the axis of T in Riemann's example is not orthogonal to the plane P. Proof First we show that if M is not the catenoid, it has discrete symmetry group. Suppose that Sym ( M ) c Sym (IR3) is not discrete. There are two cases: Sym (M) contains either an Sl-subgroup or an R-subgroup. If it contains an Sl-subgroup, M is a surface of revolution and, since M has more than one end, M is a catenoid. We assert that the second case cannot occur. Since M is not simply-connected, it must contain, by Proposition 3, an embedded, closed least-length geodesic that is unique in its h o m o t o p y class. Under a smooth action by isometries, this geodesic is invariant. Therefore, any smooth subgroup of symmetries must consist of rotations and M is the catenoid. Assume now that Sym (M) is discrete. Since every discrete subgroup of Sym (1/3) contains a screw motion, we m a y assert that there is a screw motion in Sym (M). We will find a plane P satisfying Property 2 of the theorem. Then we use the plane P to prove statements 1 and 3 of the theorem. Since M has more than one end, there exists a simple closed curve that separates the surface into two noncompact components. As shown in the proof of L e m m a 1, we may construct a least-area orientable surface 2; with the curve as boundary, which lies in the closure N of a component N of 1 1 3 M in which t?2; is not homologous to zero, and 2; has finite total curvature. We will now show that 2; necessarily has at least one flat end. If 2 ; c M , then by Statement 1 of L e m m a 2, every end of 2; must be flat, since Sym (M) must contain screw motions. Otherwise, 2; c~ M = ~72;. Each catenoid end Ei of 2: may be represented by a graph A i over the exterior of a convex disk Di in a plane P~, with all the P~ parallel to the limit tangent plane of M. If all the ends of 2; are of catenoid type and (U~D~)nM=0, then (?2; bounds the piecewise-smooth compact surface (2;-U~A~)UiDi which lies in N. But by assumption, 02: is not homologous to zero in N. Hence we may assume that at least one of the disks, say DI, intersects M. Since Mc~DI#:O, it is clear that M is not contained on one side of the catenoid barrier E. This contradicts Statement 2 of L e m m a 2. Thus 2; must have at least one flat end. We can now assert the existence of a noncompact minimal annulus with compact boundary that is asymptotic to a plane and lies in a component of I R 3 - M . This is trivial if 2; does not lie on M; we simply take the flat end of 2;. In the other case, M has a flat end A and we can divide the surface into three pieces: the end A ; a compact annulus A, which is a small neighborhood of ~?A in M - I n t ( A ) ; and the rest of the surface M'. We can choose A and A such that their union is a graph. Theorem 2.2 in 1-2] states that a bounded

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minimal graph on the exterior of a disk in •2 is a positive distance from any properly-immersed surface with compact boundary that does not intersect it. Hence, a small translation of A orthogonal to its asymptotic plane must have positive distance from the surface M and thus must lie in one of the components of ]R3--M. Let P' be the asymptotic plane of the minimal annulus just constructed. Again, by Theorem 2.2 of [2], the end of this plane is a positive distance from M. Therefore M r P ' is compact. Clearly, we can choose a plane P parallel and close to P' which intersects M transversely in a compact set. This proves statement 2 of the theorem. We now construct the subgroup T c Sym (M) satisfying statement 1. Assume that P is the (xl, x2)-plane. Let Sy~"m(M) consist of the orientation preserving elements of Sy~'~(M) whose linear part fixes the vector (0, 0, 1). It consists of the following elements of S ~ ( M ) : translations; screw motions with vertical axis; and rotations with vertical axis. By Theorem 6, every element of Sym (M) either preserves or reverses the vertical direction; hence Sym (M) has index 1, 2 or 4 in Sym (M). Consider the orbit of P n M under Sym (M). Since the end of P is a positive distance from M (again, Theorem 2.2 in [2]), and Sym (M) acts on the heights by translation, there exists an element T of Sym (M) taking P c~ M to an orbit of smallest height. (If there were no such element, then the heights of the orbits would have to accumulate at the height of PoeM, which is impossible.) Since T~Sym (M), it must be a screw motion. Since the height of P is zero, the heights of the T-orbit of P form an infinite cyclic group G under addition. Note that the natural map f : Sy~'m( M ) ~ , which gives the vertical translation part of an element in S ~ (M), has image G by our choice of T. Since every element in the kernel of f must leave the compact set P n M invariant, the kernel of f is a compact subgroup of the Lie group Sym (M). Since M is not invariant under a continuous group of symmetries, the kernel of f must be finite, which proves Property 1 in the theorem. Furthermore, if T ~ S y m ( M ) is a screw motion with nontrivial rotational part, it must have vertical axis. This proves statement 3. []

Remark 5. In the proof of Theorem 2, it was necessary to produce a fiat end disjoint from M. This was done using the results of Lemma 2 and the construction of the finite total curvature surfaces S in Lemma 1. We note that as a consequence of Theorem 2, a connected, properly embedded, minimal surface with more than one end and an infinite isometry group does not have a catenoid barrier unless it is a catenoid.

Proof of Theorem 1. We now prove Theorem 1, which is stated in the Introduction. Let M be a properly embedded minimal surface with more than one end whose symmetry group is infinite. Suppose that M is not the catenoid. Property 1 of Theorem 2 implies that M is invariant under a screw motion S. Furthermore, Theorem 2 states that there exists a plane P, with M c~ P compact, whose S-orbit is an infinite collection of parallel planes. Consider the closed slab L c ] R 3 whose

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boundary consists of the planes P and S(P). Clearly L n M is not compact. Hence the S-orbit of L n M gives rise to an infinite number of distinct ends of M. This proves statement 1 and 2 of Theorem 1. Let g denote the subgroup of Sym (M) generated by S. Clearly, g acts freely on the collection of canonical annular ends of M. The Annular End Theorem [7-] states that M can have, at most, two pairwise-disjoint annular ends of infinite total curvature. Since the S-orbit of a canonical annular end E c M consists of an infinite collection of pairwise-disjoint ends with the same total curvature, every annular end of M must have finite total curvature. These annular ends converge to flat planes or catenoids. By Lemma 2, Statement 1, all the annular ends of M are flat. It remains to prove that C(~I)=2rc()~(Mi)-r(l~4)) where M = M / S . Huber [10] proved that a complete Riemannian surface S of nonpositive curvature and finite total curvature must be conformally diffeomorphic to a closed surface punctured in a finite number of points; in particular, finite total curvature implies finite topology. If M has finite topology, then the surface ML = M c~L has finite topology, where L is the slab between P and S(P). Since ML has a finite number of ends, we may perturb P, if necessary, to insure that P is not asymptotic to an end of ML. Clearly, the annular ends E 1. . . . . E, of ML give rise to representatives of the annular ends of A4. In this way we see that the associated canonical annular ends/~1, ..-,/~, of ML give rise to the n "canonical ends" of M, where each of the canonical ends of M has finite total curvature -2~z. Suppose El, E2 ..... E, are the canonical ends of A4. Then Gauss-Bonnet shows C(M)= ~ KdA=

~

~/

M-U~/~i

KdA+

~ KdA:2~(Z(MI)--r(~4)). UiE~

This completes the proof of Theorem 1.

[]

Corollary 1. Suppose M is a properly embedded minimal surface with infinite symmetry group and more than one end. Assume M is not a catenoid. I f M has an annular end, then it has an infinite number of annular ends. Proof By Theorem 2, M is invariant under a screw motion S. If E is a canonical annular end of M, then, since S(E):gE (because ~E is compact), the S-orbit of E consists of infinitely many canonical ends of M. [] Corollary 2. A doubly-periodic, properly embedded minimal surface in ~ 3 has one end. Proof Since the symmetry group of a doubly-periodic minimal surface does not have a cyclic subgroup of finite index, Theorem 2 implies that a connected doubly-periodic minimal surface has one end. []

4. The existence of minimal surfaces with screw motion symmetry The following lemma constructs the basic building block of the surfaces of Theorem 4.

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Fig. 4

Lemma 3. Let L k be a set of k+ 1 lines contained in the (xl, x2)-plane that meet at equal angles at the origin, k >=l. Let Lk, o denote the image of L k under the composition of a rotation around the xa-axis by 0 and a vertical translation by (0, 0, 1). Then LR U Lk, 0 is the boundary of a properly immersed minimal annulus Ak, o satisfying: 1. Ak, 0--{(0, 0, 0), (0, 0, 1)} is an embedded ,surface in the slab {0-0. Since N n { 0 < x 3 < 1} contains no points with vertical normals, it is a collection of minimal annuli. If n > 0, either N is not connected or it has genus greater that 2k + 1, both of which are contradictions. Hence n = 0, as required. Arguments similar to the one in the previous paragraph prove the following two properties. Property 11. M k is symmetric under reflection through the k + 1 vertical planes containing the x3-axis and bisecting the lines of Property 8. Property 12. The full symmetry group of Mk is generated by, T, R, one of the reflections in Property 5, rotation about one of the lines in 8, and reflection through one of the planes in Property 11. The remaining property is not so easy to check. Property 9. M ~ {x3 = c}, cr

is a simple closed curve.

We observed above that N n { 0 < x 3 < 1} consists of a collection of annuli. Denote the number of such annuli by n, and let A be one of them. Suppose ~ is a ray, emanating from the origin, that is contained in the boundary of A at height 0. Near the origin the annulus A is a graph over the (x l, xa)-plane, so it is clear from Property 11 that A is invariant under reflection in a vertical plane V of symmetry of M containing the x3-axis and bisecting consecutive rays in {x3=0}. Furthermore, reflection in this plane of symmetry takes ~ into another ray, ~ ' c ~3A. Since M is invariant under reflection in the plane {x3=89 and this plane intersects A, it must be a symmetry plane of A. It follows that the reflected image of Lo = ~w ~' is another boundary curve of A. If n > 1, then it is clear that the curve Lo and its reflected image, L~ c {x3 =1}, are precisely the boundary of A. (Otherwise Properties 10 and 11 would show that OA contained all the rays in {x3 =0} and {x 3 = 1}.) Assuming that n > 1, we will derive a contradiction. We now prove that A is contained in the convex hull of OA=Lo wLa. Supn pose V+ and V_ are obtained from V by rotations around the x3-axis by k + 1 --TZ

and k ~ '

respectively. Since V+ and V_ are symmetry planes of M but not

of A, they must be disjoint from A. Let C be a compact catenoid with boundary in the planes {x 3 =0}, {x 3 = 1}, choosen so that C is contained in the component of IR3-(V+ w V_) disjoint from A. It follows from the maximum principle that any smooth family of catenoids produced from C by horizontal translation must first contact A on t?A. Since we can "roll" C along OA, A must lie in the envelope created by these catenoids together with the planes {x3 =0} and {x3 = 1}. By choosing the radius of these catenoids sufficiently large, we can show that every point in the slab {0 0 . We now wish to prove that it is true for all t > 0 . Suppose

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it fails. Let to be the largest value for which the condition holds for all t 0 . In particular, we note that P, can never meet A orthogonally, or else A~= At by the boundary maximum principle. We can conclude that the projection from At to Pt is a submersion. Since this is true for all t > 0 and the planes P~are parallel, it follows that projection of A onto Po is a submersion. We now want to show that the projection of A consists of the entire strip 0 < x3 < 1 in the (xz, x3)-plane. Toward that end, we observe that the preceding argument will work with the family of planes Pt replaced by any similar family of parallel vertical planes. The only requirement is that the plane at time t = 0 contain the x3-axis and intersect the (xl, Xz)-plane in a line outside of the wedge Lo. With this in mind, suppose that the projection of A is not onto the strip. Then there must be a divergent curve on A whose projection terminates in a point in the interior of the strip and the tangent plane to A becomes orthogonal to Po along the curve as it diverges. However, as observed, the projection of A onto a vertical plane produced by a small rotation of Po about the x3-axis is a submersion. But somewhere along the curve the tangent plane to A is orthogonal to this rotated plane. This contradiction proves that the projection of A is onto the strip. We will now show that A is a graph over the strip. Suppose q is a point in the strip over which there are two points, say Pl and P2, on A. Ifx~ coordinates of the points over q are hi and h 2, h l > h 2 then Pl reflects into P2 through the plane Pt,, where t,=(h2-hO/2. Thus A~,c~.4t, contains P2, but pzq~Ac~Pt,. This contradiction shows that projection of A onto the strip is one-to-one, so A is a graph. We observe that A must be asymptotic to a flat strip as it diverges. This can be seen by taking a very large circle inside the wedge Lo and a vertical translate of it to the plane {x3 = 1}. These circles form the boundary of a catenoid that is disjoint from A, yet very close to a cylinder. This catenoid forms a barrier for A, forcing it to lie closer and closer to the boundary of the convex hull of L 0 w L~, as it diverges. Thus any minimal graph over the strip is asymptotic to a linear function of x2. An application of the maximum principle will show that such a graph is unique. It remains to prove existence. There are two ways to proceed. First way: Truncate Low L~ at the level of the plane Pt, producing a quadrilateral boundary It that projects one-to-one onto a rectangle in Po. The quadrilateral l~ bounds a unique minimal graph. As t-~ oe, these graphs are close to linear at their extremities and will converge to a minimal graph over the entire strip. Second way: If the angle formed by L0 is equal to 2 ' observe that Scherk's singly-periodic surface, properly scaled, is built up of minimal graphs bounded by Low LI. By modifying the Weierstrass representation for this surface in the

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manner of Karcher [11], one can explicitly produce a solution to this boundary value problem for angles between 0 and rr. [] Remark 7. Karcher and Pitts [12] have constructed properly embedded minimal

surfaces that are asymptotic to two coaxial helicoids. They have one topological end and screw motion symmetry. If one twists L1 by an angle 0, in the proceeding construction, the new boundary, say L0 w L1,0, can be easily shown, in a manner analogous to the argument in the proof of the Lemma 4, to bound a unique minimal graph. Extending this graph by rotation about its line boundaries produces the Karcher-Pitts examples.

References 1. Callahan, M., Hoffman, D., Meeks III, W.H.: Embedded minimal surface with an infinite number of ends. Invent. Math. 96, 459-505 (1989) 2. Choi, T., Meeks III, W.H., White, B.: A rigidity theorem for properly embedded minimal surfaces in ~3. j. Differ. Geom. 3. Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in 3-manifolds. Invent. Math. 82, 121-132 (1985) 4. Freedman, M., Hass, J., Scott, P.: Closed geodesics on surfaces. Bull. Lond. Math. Soc. 14, 385 391 (1982) 5. Hadamard, J.-J.: Les surfaces k, courbures oppos6es et leurs lignes g+od6siques. J. Math. Pures Appl. 4, 27-73 (1898) 6. Hardt, R., Simon, L.: Boundary regularity and embedded minimal solutions for the oriented Plateau problem. Ann. Math. 110, 439-486 (1979) 7. Hoffman, D., Meeks III, W.H.: The asymptotic behavior of properly embedded minimal surfaces of finite topology. J. Am. Math. Soc. 2, No. 4, 667~i81 (1989) 8. Hoffman, D., Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. (to appear) 9. Hoffman, D., Meeks III, W.H.: A variational approach to the existence of complete embedded minimal surfaces. Duke J. Math. 57, 877-894 (1988) 10. Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv. 32, 181-206 (1957) 11. Karcher, H.: Embedded minimal surfaces derived from Scherk's examples. Manuscr. Math. 62, 83-114 (1988) 12. Karcher, H., Pitts, J.: (personal communications) 13. Meeks III, W.H., Rosenberg, H.: The geometry of periodic minimal surfaces (Preprint) 14. Meeks III, W.H., Rosenberg, H.: The global theory of doubly periodic minimal surfaces. Invent. Math. 97, 351-379 (1989) 15. Meeks IIl, W.H., Rosenberg, H.: The maximum principle at infinite for minimal surfaces in flat three-manifolds. Comm. Math. Helv. (to appear) 16. Meeks III, W.H., Yau, S.T.: The topological uniqueness theorem of complete minimal surfaces of finite topological type (Preprint) 17. Meeks III, W.H.: Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179, 151-168 (1982) 18. Morse, M.: Collected Papers. World Scientific, Singapore, 1987 19. Osserman, R.: Minimal surfaces in the large. Comment. Math. Helv. 35, 65-76 (1961) 20. Osserman, R.: Global properties of minimal surfaces in E 3 and E". Ann. Math. 80, 340-364 (1964)

The structure of singly-periodic minimal surfaces

481

21. Schoen, R.: Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Differ. Geom. 18, 791-809 (1983) 22. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Center for Mathematical Analysis, vol. 3, Canberra, Australia, 1983. Australian National University Oblatum 12-IX-1988 & 7-VI-1989

Note added in proof New singly-periodic, properly embedded minimal surfaces with an infinite number of flat ends have been constructed (D. Hoffman and M. Wohlgemuth, in preparation). These surfaces are invariant under a translation T, and the quotient by Thas genus 4k + 1.