On the moduli of constant mean curvature cylinders of finite type in the 3-sphere
We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.
Alexandrov  proved that there are no compact embedded surfaces with constant mean curvature (cmc) in Euclidean 3-space other than round spheres. However, while there are no compact minimal surfaces in , there is an abundance of such in the 3-sphere . For instance 2-spheres in the 3-sphere are minimal precisely when they are great 2-spheres, and Lawson proved that compact embedded minimal surfaces in exist for every genus [42, 43]. Lawson further showed  that any embedded minimal torus in is unknotted, and conjectured that up to isometry the Clifford torus is the only embedded minimal torus in . Hsiang and Lawson  proved that the only embedded minimal torus of revolution is the Clifford torus. Further results suggest that an embedded minimal torus indeed has additional symmetries: Montiel and Ros  showed that the only minimal torus immersed into by the first eigenfunctions is the Clifford torus, and Ros  proved that the normal surface of an embedded minimal torus in is also embedded. Various methods for obtaining minimal surfaces in have been employed to study specific classes, as in Karcher, Pinkall and Sterling , and more recently by Kapouleas and Yang .
Wente’s discovery  of cmc tori provided the first compact examples other than spheres in Euclidean 3-space. The studies of Abresch [1, 2], Wente  and Walter  on special classes of cmc tori in concluded in the classification by Pinkall and Sterling , and their algebro-geometric description by Bobenko [5, 6]. In fact, Bobenko gave explicit formulas for cmc tori in and hyperbolic 3-space in terms of theta–functions, and provided a unified description of cmc tori in the 3-spaceforms in terms of algebraic curves and spectral data. Independently, Hitchin  classified harmonic 2-tori in the 3-sphere, and thus also as a special case the harmonic Gauss maps of cmc tori. These ideas culminated in the description of harmonic tori in symmetric spaces by Burstall, Ferus, Pedit and Pinkall , and the generalized Weierstraß representation by Dorfmeister, Pedit and Wu .
Associated to a cmc torus in the 3-sphere is a hyperelliptic Riemann surface, the so called spectral curve. The structure equation for cmc tori is the -Gordon equation. Hitchin , and Pinkall and Sterling  independently proved that all doubly periodic solutions of the -Gordon equation correspond to spectral curves of finite genus. The genus of the spectral curve is called spectral genus. Ercolani, Knörrer and Trubowitz  proved that for every even spectral genus there exists a hyperelliptic curve which corresponds to an immersed cmc torus in . The remaining cases of odd genera was settled by Jaggy . Adapting these results, Carberry  showed that minimal tori in exist for every spectral genus. Note that while a cmc torus in has at least spectral genus 2, there is no such restriction for cmc tori in . In particular, cmc tori of revolution in have spectral genus . Pinkall and Sterling  conjectured that the only embedded cmc tori in are tori of revolution.
In parallel the global theory of embedded cmc (especially minimal) surfaces in space forms was developed using geometric PDE methods. Meeks  proved that a properly embedded end of a cmc surface is cylindrically bounded, which was used by Korevaar, Kusner and Solomon  to prove that the only embedded cmc cylinders in are surfaces of revolution - either a standard round cylinder (spectral genus ), or a Delaunay unduloid (spectral genus ). There are analogous cmc surfaces of revolution in , some of which close up into tori, see Figure 1 for some simple examples.
Kapouleas [30, 31, 32] proved the existence of compact cmc surfaces in for any genus greater than 1, as well as many new classes of non-compact cmc surfaces in , but not much is known about the moduli of cmc surfaces in general. Progress on understanding the moduli of cmc immersions of punctured spheres has recently been made in the case of three punctures by Grosse-Brauckmann, Kusner and Sullivan  and by Schmitt et. al. . Kusner, Mazzeo and Pollack  show that the moduli space of cmc surfaces is an analytic variety. The local linearization of the moduli space is described by Jacobi fields which correspond to a normal variation of the surface which preserve the constant mean curvature property. Recently Korevaar, Kusner and Ratzkin  studied Jacobi fields on a class of cmc surfaces with the additional property of being Alexandrov immersed. An Alexandrov immersed surface in is a complete noncompact properly immersed surface that is the boundary of a 3-manifold with two additional features: The mean curvature normal of points into , and extends to a proper immersion of into . When the target is the 3-sphere, we replace properness by completeness, and as Lawson  we consider in analogy a smooth immersion that we call a one-sided Alexandrov embedding if is the boundary of a connected 3-manifold and the following two conditions hold: The mean curvature of with respect to the inward normal is non-negative. Secondly, the manifold is complete with respect to the metric induced by . We prove that the property of one-sided Alexandrov embeddedness is stable under continuous deformation, which allows us to study continuous families of one-sided Alexandrov embedded surfaces.
In this paper we consider cmc cylinders which have constant Hopf differential, and whose metric is a periodic solution of the -Gordon equation of finite type. Such cmc cylinders are said to be of finite type. We describe such finite type cmc cylinders in Section 3 and 4 by spectral data, and show that the spectral data can be deformed in such a way that the corresponding family of cmc surfaces are all topologically cylinders. It turns out that the corresponding moduli space of spectral data of genus is -dimensional. Furthermore, we can control the spectral genus under the deformation, and by successively coalescing branch points of the spectral curve, we continuously deform the spectral curve in Lemma 4.5 into a curve of genus zero. In Section 5 we show that one-sided Alexandrov embedded surfaces with constant mean curvature have collars with depths uniformly bounded from below. For this purpose we use a ’maximum principle at infinity’ which was communicated to us by Harold Rosenberg . This allows us to show in Theorem 6.8 that a large class of continuous deformations of cmc cylinders of finite type preserve the one-sided Alexandrov embeddedness. In Lemma 7.3 we continuously deform any one-sided finite type cmc cylinder in into a one-sided Alexandrov embedded flat cylinder in with spectral genus zero. These are classified in Theorem 7.1. Finally this classification is extended to all possible deformations of these flat cmc cylinders in in Theorem 7.6. Since an embedded cmc torus in the 3-sphere is covered by a one-sided Alexandrov embedded cylinder, our result confirms the conjecture by Pinkall and Sterling, and since the only embedded minimal torus of revolution is the Clifford torus, also affirms Lawson’s conjecture.
Acknowledgments. We thank Fran Burstall, Karsten Grosse-Brauckmann, Ian McIntosh, Rob Kusner, Franz Pedit and Ulrich Pinkall for useful discussions. This work was mostly carried out whilst M Kilian was a research assistant at the University of Mannheim, and he would like to thank the Institute of Mathematics there for providing excellent research conditions.
We had several beneficial conversations with Laurent Hauswirth, and we are especially grateful to Antonio Ros and Harold Rosenberg who helped us close a gap in a first draft of this paper, and Univerité Paris 7 for its hospitality during these discussions.
1. Conformal cmc immersions into
This preliminary section recalls the relationship between cmc immersed surfaces in and solutions of the -Gordon equation, before considering the special case of cmc cylinders in , the notion of monodromy and the period problem.
1.1. The -Gordon equation
We identify the 3-sphere with . The Lie algebra of the matrix Lie group is , equipped with the commutator . For smooth –forms on with values in , we define the –valued –form
for vector fields on . Let be left multiplication in . Then by left translation, the tangent bundle is and is the (left) Maurer–Cartan form. It satisfies the Maurer-Cartan-equations
Complexifying the tangent bundle and decomposing into and tangent spaces, and writing , we may split into the part , the part and write . We set the –operator on to .
We denote by the bilinear extension of the Ad–invariant inner product of to . The double cover of the isometry group is via the action .
Now let be an immersion and . Then is conformal if and only if the -part of is isotropic
If is a conformal immersion then there exists a smooth function , called the conformal factor of such that
The mean curvature function of (see e.g. ) is given by
Recall the following observation of Uhlenbeck , based on an earlier result by Pohlmeyer , and suppose that in the following is a conformal immersion with non-zero constant mean curvature and conformal factor . Then (1.4) and combined give , or alternatively
Inserting respectively into (1.5) gives and . Then an easy computation shows that
satisfies the Maurer-Cartan-equations
The Maurer-Cartan-equations are an integrability condition, so we can integrate and obtain a corresponding extended frame with and . Since takes values in , we conclude that takes values in when . Now define for , the following map by the Sym-Bobenko-formula
Then for we obtain so the conformality of follows from the conformality of , since
Here we have written , and is the conformal factor of . Furthermore
In summary, by starting with one non-minimal conformal cmc immersion , we have just seen how to obtain a whole -family of solutions of the Maurer-Cartan-equations, and from the corresponding extended frame we then obtained another conformal cmc immersion . Since the mean curvature (1.9) and the conformal factor of in (1.8) only depend on the angle between , we in fact get a whole -family of isometric conformal cmc immersions, called an associated family, which is obtained by simultaneously rotating while keeping the angle between them fixed.
We next recall the following version of Theorem 14.1 in Bobenko , which provides a correspondence between solutions of the -Gordon equation and associated families of cmc surfaces in the 3-sphere.
 Let be a smooth function and define
Then if and only if is a solution of the -Gordon equation
For any solution of the -Gordon equation and corresponding extended frame , and , the map defined by the Sym-Bobenko-formula (1.6) is a conformal immersion with constant mean curvature (1.9), conformal factor , and constant Hopf differential with .
Decomposing into and parts, we compute
Now is equivalent to , which holds if and only if solves the sinh-Gordon equation (1.11).
If is a solution of the sinh-Gordon equation, then we may integrate to obtain a map . Let , and consider the map defined by the Sym-Bobenko-formula (1.6). Conformality (1.2) is a consequence of the fact that the complexified tangent vector
is isotropic with respect to the bilinear extension of the Killing form. The mean curvature can be computed using formula (1.4). The conformal factor is obtained from
From (1.9) we have , which proves the formula for the conformal factor.
Define the normal with . Then and
Consequently, , which proves the formula for the Hopf differential, and concludes the proof. ∎
There is an analogous but more general theorem (see e.g Bobenko ) than the one above, which asserts that if functions satisfy the Gauss-Codazzi equations, then one obtains a -family of solutions of the Maurer-Cartan-equations, thus an extended frame and consequently an associated family via the Sym-Bobenko-formula.
1.2. Monodromy and periodicity condition
The cmc condition implies that the Hopf differential is a holomorphic quadratic differential . On the cylinder there is an infinite dimensional space of holomorphic quadratic differentials, large classes of which can be realized as Hopf differentials of cmc cylinders . On a cmc torus the Hopf differential is constant (and non-zero). Since we are ultimately interested in tori, we restrict our attention to cmc cylinders considered via Theorem 1.1 which have constant non-zero Hopf differentials on the universal covering of . Note that for given solution of the sinh-Gordon equation an extended frame is holomorphic on and has essential singularities at .
Let be an extended frame for a cmc immersion such that (1.6) holds for two distinct unimodular numbers . Let be a translation, and assume that has period , so that . Then we define the monodromy of with respect to as
Periodicity in terms of the monodromy is then , so if and only if
If is the trace of then if and only if .
2. Finite type solutions of the -Gordon equation
In this section we introduce the solutions of the -Gordon equation which are called finite type solutions. Finite type solutions of the -Gordon equation are in one-to-one correspondence with maps called polynomial Killing fields. These polynomial Killing fields take values in certain -matrix polynomials, and solve a non-linear partial differential equation, but they are uniquely determined by one of their values. We shall call these values initial values of polynomial Killing fields or just initial values. The Symes method calculates the solutions in terms of the initial values with the help of a loop group splitting. The eigenvalues of these matrix polynomials define a real hyperelliptic algebraic curve, which is called spectral curve. One spectral curve corresponds to a whole family of finite type solutions of the -Gordon equation. We call the sets of finite type solutions (or their initial values), which belong to the same spectral curve, isospectral sets. The eigenspaces of the matrix polynomials define a holomorphic line bundle on the spectral curves called eigenbundle. These holomorphic line bundles completely determine the corresponding initial value and the corresponding solution of the -Gordon equation. Consequently, the isospectral sets can be identified with one connected component of the real part of the Picard group. In case the spectral curve has singularities, then the isospectral set can be identified with the real part of the compactification of a generalized Jacobian. These compactifications have stratifications, whose strata are the orbits under the action of the generalized Jacobian. In our case the spectral curves are hyperelliptic and we shall describe the corresponding stratifications of the isospectral sets.
2.1. Polynomial Killing fields
For some aspects of the theory untwisted loops are advantageous, and avoiding the additional covering map simplifies for example the description of Bianchi-Bäcklund transformations by the simple factors [66, 37]. For the description of polynomial Killing fields on the other hand, the twisted loop algebras as in [9, 11, 12, 17, 46] are better suited, but we remain consistent and continue working in our ’untwisted’ setting.
Let and , and consider for the finite dimensional vector space
Clearly is a real -dimensional vector space and has up to isomorphism a unique norm . These Laurent polynomials define smooth mappings from into . Note that belongs to the loop Lie algebra of the loop Lie group . For the resulting solution of the -Gordon equation to be of finite type, we need in addition the conditions . These conditions ensure that and the lower left entry of do not vanish, and therefore that is semisimple. This is the same as semisimplicity of the leading order term in the twisted setting. We thus define
for some with . Due to Pressly and Segal , the Iwasawa decomposition is a diffeomorphism between the loop group of into point wise products of elements of with elements of the loop group of holomorphic maps from to , which take at values in the subgroup of of upper-triangular matrices with positive real diagonal entries. For every there exists a unique , such that takes values in the Lie algebra of of the right hand factor in the Iwasawa decomposition (2.1).
A polynomial Killing field is a map which solves
For each initial value , there exists a unique polynomial Killing field given by
For with and the corresponding -valued 1-form is the as in (1.10) for that particular solution of the -Gordon equation corresponding to the extended frame of (2.1). For general initial values the leading term of the corresponding polynomial Killing field does not depend on the surface parameter . The corresponding differs from (1.10) by multiplication of and with constant unimodular complex numbers. Given a polynomial Killing field , we set the initial value in (2.1). Thus , or the initial value , gives rise to an extended frame, and thus to an associated family.
A solution of the -Gordon equation is called a finite type solution if and only if it corresponds to a polynomial Killing field with .
2.2. Roots of polynomial Killing fields
If an initial value has a root at some , then the corresponding polynomial Killing field has a root at the same for all . In this case we may reduce the order of and without changing the corresponding extended frame (2.1). The following polynomials transform under as
If the polynomial Killing field with initial value has a simple root at , then does not vanish at and is the polynomial Killing field with initial value . Furthermore, obviously and commute, and we next show that both polynomial Killing fields and give rise to the same extended frame (2.1).
If a polynomial Killing field with initial value has zeroes in , then there is a polynomial , such that the following two conditions hold:
is the polynomial Killing field with initial value , which gives rise to the same associated family as .
has no zeroes in .
An appropriate Möbius transformation (3.1) transforms any root into a negative root. For such negative roots the corresponding initial values and are related by multiplication with a polynomial with respect to with positive coefficients. In the Iwasawa decomposition (2.1) this factor is absorbed in . Hence the corresponding extended frames coincide, which proves (i). Repeating this procedure for every root ensures (ii). ∎
Hence amongst all polynomial Killing fields that give rise to a particular cmc surface of finite type there is one of smallest possible degree (without adding further poles), and we say that such a polynomial Killing field has minimal degree. A polynomial Killing field has minimal degree if and only if it has neither roots nor poles in . We summarize two results by Burstall and Pedit [11, 12]. The first part is a variant of Theorem 4.3 in , the second part follows immediately from results in .
(i) A cmc immersion is of finite type if and only if there exists a polynomial Killing field with initial value such that the map obtained from (2.1) is an extended frame of .
(ii) In particular there exists a unique polynomial Killing field of minimal degree that gives rise to . Thus we have a smooth 1-1 correspondence between the set of cmc immersions of finite type and the set of polynomial Killing fields without zeroes.
Point (i) is a reformulation of Theorem 4.3 in . (ii) We briefly outline how to prove the existence and uniqueness of a minimal element.
If the initial value gives rise to , then the corresponding polynomial Killing field can be modified according to Proposition 2.2 so that is of minimal degree, and still giving rise to . Hence there exists a polynomial of least degree giving rise to .
For the uniqueness, assume we have two initial values of least degree both giving rise to . Putting Proposition 3.3 and Corollary 3.8 in  together gives: Two finite type initial values give rise to the same associated family if and only if they commute and have equal residues. Since the residues coincide and both are of minimal degree, we conclude that . The unique minimal polynomial Killing field is thus .
Since the Iwasawa factorization is a diffeomorphism, and all other operations involved in obtaining an extended frame from are smooth, the resulting cmc surface depends smoothly on the entries of . ∎
2.3. Spectral curves I
Due to (2.3) the characteristic equation
of a polynomial Killing field with initial value does not depend on and agrees with the characteristic equation of the initial value . If then we may write for a polynomial of degree at most which satisfies the reality condition
Consequently the hyperelliptic curve has three involutions
If has pairwise distinct roots, then is a spectral curve of genus of a not necessarily periodic solution of the -Gordon equation. The genus is called the spectral genus.
Let be a polynomial of degree satisfying (2.6). Then the isospectral set
is compact. Furthermore, if the roots of are pair wise distinct, then .
For the compactness it suffices to show that all Laurent coefficients of a are bounded, since is a closed subset of the -dimensional vector space . For the product is skew hermitian on . The negative determinant of traceless skew hermitian matrices is the square of a norm. Hence for all the Laurent polynomial of with respect to is bounded on . Thus the Laurent coefficients are bounded.
If has pairwise distinct roots, then has no roots since at all roots of , the determinant has a root of order two. If is a root of , then vanishes and is nilpotent. For a nonzero nilpotent -matrix there exists a -matrix such that . Hence for every root of , there exists a -matrix , such that
has no pole at . The corresponding derivative of is equal to . Furthermore, is polynomial with respect to of degree . Two appropriate linear combinations with the analogous tangent element at the root of change the roots and and fixes all other roots of and respects the reality condition of . These two linear combinations belong to the tangent space of . Hence the derivatives of all the coefficients of as functions on are non-zero at all . By the implicit function theorem this set is therefore a -dimensional submanifold. The corresponding eigenspaces of depend holomorphically on the solutions of (2.5) and define a holomorphic line bundle on the spectral curve. These eigenbundles have degree , they are non-special in the sense that they have no holomorphic sections vanishing at one of the points at or , and finally they obey some reality condition. Vice versa, all holomorphic line bundles obeying these three conditions correspond to one (see McIntosh [46, Section 1.4]). Hitchin has shown in , that the third condition implies the second condition. Therefore can be identified with the real part of one connected component of the Picard group of the spectral curve, which is a -dimensional torus. ∎
If has multiple roots, then the real part of the Jacobian of the corresponding hyperelliptic curve still acts on , but not transitively. More precisely, in case of non-unimodular multiple roots of the set has a stratification, whose strata are the orbits of the action of the real part of the generalized Jacobian of the singular hyperelliptic curve defined by . The elements of different strata have different orders of zeroes at the multiple roots of . In Proposition 2.2 we have seen that all are products of of lower degree with polynomials of the form (2.4).
Every finite type solution of the -Gordon equation corresponds to a unique polynomial Killing field without zeroes and initial value . The curve defined by has a unique compactification to a projective curve without singularities at and . If has multiple roots, then we say that the solution contains bubbletons. The arithmetic genus of this hyperelliptic curve is equal to .
We briefly motivate Definition 2.5 above, and refer the reader to [8, 12, 37, 46, 64] for further details. If an initial value gives rise to a cmc cylinder with extended frame and monodromy , and is a point at which , we define a simple factor
Then for any the dressed extended frame, obtained from the -Iwasawa factorization  of , is an extended frame of a cmc cylinder with a bubbleton. On the initial value level this dressing action corresponds to which obviously has singularities at . To eliminate these, consider . If , then , so the polynomial of a bubbleton has double zeroes.
If has multiple roots, then is open and dense in . If has no unimodular zeroes then it is a -dimensional submanifold. If has unimodular zeroes, then let denote the quotient of by all real zeroes. Then is the image of the multiplication with an appropriate rational function from to .
Similar arguments as in the proof of Lemma 2.4 carry over to this situation. ∎
Suppose is polynomial of degree satisfying the reality condition (2.6). Assume has precisely pairwise distinct non-unimodular roots and pairs of unimodular roots of order 2. Then .
Since is the square of a norm on all skew-hermitian matrices, all have a zero at the unimodular double roots of . Let be the corresponding decomposition of into an with pairwise distinct roots and the corresponding factors (2.4). Due to Proposition 2.2 the one-to-one correspondence between polynomial Killing fields with roots and polynomial Killing fields without roots induces an isomorphism . The assertion now follows from Lemma 2.4. ∎
2.5. Spectral curves II
We also utilize the description of finite type cmc surfaces in via spectral curves due to Hitchin , and relate this to our previous definition of spectral curves due to Bobenko . While Hitchin defines the spectral curve as the characteristic equation for the holonomy of a loop of flat connections, Bobenko defines the spectral curve as the characteristic equation of a polynomial Killing field. We shall use both of these descriptions, and briefly recall their equivalence: Due to (1.12), the monodromy is a holomorphic map with essential singularities at . By construction the monodromy takes values in for . The monodromy depends on the choice of base point, but its conjugacy class and hence eigenvalues do not. With the characteristic equation reads
The set of solutions of (2.8) yields another definition of the spectral curve of periodic (not necessarily finite type) solutions of the -Gordon equation. Moreover, the eigenspace of depends holomorphically on and defines the eigenbundle on the spectral curve. Let us compare this with the previous definition of a spectral curve of periodic finite type solutions of the -Gordon equations. Let be a polynomial Killing field with initial value , with period so that for all . Then also the corresponding is -periodic. Let and be the monodromy with respect to . Then for we have and thus
All eigenvalues of holomorphic matrix valued functions depending on and commuting point wise with or define the sheaf of holomorphic functions of the spectral curve. Hence the eigenvalues of and are different functions on the same Riemann surface. Furthermore, on this common spectral curve the eigenspaces of and coincide point-wise. Consequently the holomorphic eigenbundles of and coincide.
A finite type solution of the -Gordon equation is periodic if and only if
There exists a meromorphic differential on the spectral curve with second order poles without residues at the two points and .
This differential is the logarithmic derivative of a function on the spectral curve which transforms under the involutions (2.7) as , and .
Conversely, a periodic solutions of the -Gordon equation is of finite type if and only if the monodromy (1.12) fails at only finitely many points to be semisimple.
Due to Krichever , the translations by act on the eigenbundle by the tensor product with a one-dimensional subgroup of the Picard group. In Sections 1.4-1.7 McIntosh  describes this Krichever construction for finite type solutions of the -Gordon equation. The line bundle corresponding to is trivial if and only if there exists a non-vanishing holomorphic function on the compactified spectral curve with essential singularities at and , whose logarithm has a first order pole at and with singular part equal to and . This implies the characterization of periodic finite type solutions.
At all simple roots of the monodromy (1.12) cannot be semisimple. Furthermore, at a double root of the monodromy fails to be semisimple, if and only if it is dressed by a simple factor and contains a corresponding bubbleton. An asymptotic analysis shows that there can exists at most finitely many roots of of order larger than two. ∎
Pinkall and Sterling , and independently Hitchin  proved that doubly periodic solutions of the sinh-Gordon are of finite type. Thus all metrics of cmc tori are of finite type. We enlarge this class by relaxing one period, and make the following
The cmc cylinders with constant Hopf differential and whose metric is a periodic solution of finite type of the -Gordon will be called cmc cylinders of finite type.
We compute some examples of initial values, polynomial Killing fields and extended frames for spheres, and spectral genus surfaces. Formulas for all finite type surfaces in terms of theta-functions are given by Bobenko .
We start with a discussion of spheres. Since the Hopf differential vanishes identically, spheres constitute a degenerate case since their conformal factor is a solution to the Liouville equation rather than the -Gordon equation, a fact also reflected in the initial value which does not satisfy the semi-simplicity condition. Consider