Constant mean curvature surfaces in 3-dimensional Thurston geometries

Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 Constant mean curvature surfaces in 3-dimensional Thurston geometries Isabel Fern´andez and Pablo Mira Abstract. This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3,H3,S3,H2×R,S2×R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented. Keywords. Constant mean curvature surfaces, homogeneous spaces, Thurston geometries, harmonic maps, minimal surfaces, entire graphs. 1. Introduction Constant mean curvature (CMC) surfaces appear as critical points of a natural geometric variational problem: to minimize surface area with or without a volume constraint (the unconstrained case corresponds to zero mean curvature, i.e. to minimal surfaces). A fundamental problem of this discipline is the geometric study and classification of CMC surfaces under global hypotheses like compactness, completeness, properness or embeddedness. The study of this problem for CMC surfaces in the model spaces R3,S3and H3has produced a very rich theory, in which geometric arguments interact with complex analysis, harmonic maps, integrable systems, maximum principles, elliptic PDEs, geometric measure theory and so on. One of the most remarkable achievements of this field in the last decade has been the extension of this classical theory to the case of CMC surfaces in simply connected homogeneous 3-dimensional ambient spaces. Apart from R3,S3and H3, these spaces are the remaining five Thurston 3-dimensional geometries (i.e. H2×R,S2×R, the Heisenberg group Nil3, the universal covering of PSL2(R) and the Lie group Sol3), together with 3dimensional Berger spheres and some other Lie groups with left-invariant metrics (see Section 2). It must be said here that there is an important number of contributions regarding CMC surfaces in general Riemannian 3-manifolds (not even homogeneous), many of which deal for instance with isoperimetric questions or with geometric consequences derived from the stability operator associated to the second variation of the surface. The achievement in the case of homogeneous ambient 3-spaces has been the construction of a very rich global theory of CMC surfaces, analogous to the case of R3,S3and H3, with an emphasis on the geometric classification (up to ambient isometries) of properly immersed or properly 2Isabel Fern´andez and Pablo Mira embedded CMC surfaces. The fact that the ambient space is homogeneous, i.e. it has the same local geometry at all points, makes this problem extremely natural. Our aim here is to present a survey on some fundamental aspects of the global theory of CMC surfaces in homogeneous 3-manifolds. We do not plan, however, to give a systematic account of all important results of this already broad theory, but to discuss some specific problems at the core of it. Hence, there will be many important results omitted, and we apologize in advance for that. In order to explain the problems we shall be dealing with, let us distinguish between compact and non-compact CMC surfaces in these spaces. In the case of compact CMC surfaces, three fundamental problems are the Alexandrov problem (i.e. to classify compact embedded CMC surfaces), the Hopf problem (i.e. to classify CMC spheres), and the isoperimetric problem (recall that isoperimetric regions on a Riemannian 3-manifold are bounded by compact embedded CMC surfaces, but the converse is not always true). By classical results, round spheres constitute the solution to each of these three problems in the case of CMC surfaces in R3. One of our main objectives will be to explain what is known (and what is not known) for these problems in the broader context of CMC surfaces in homogeneous 3-manifolds. In the case of non-compact CMC surfaces, one of the basic problems is to study the properly embedded CMC surfaces of finite topology. A classical result in that direction is given by Bernstein’s theorem: planes are the only entire minimal graphs in R3. As in all Thurston 3-dimensional geometries there is a natural notion of entire graph, it is an important problem of the discipline to solve the Bernstein problem for CMC graphs, i.e. to classify all entire CMC graphs in these 3-dimensional ambient spaces. This will be our other main objective. The theory of CMC surfaces in Thurston 3-dimensional geometries started to develop as a consistent unified theory after some pioneer works by Harold Rosenberg, jointly with William H. Meeks [MeRo1, MeRo2, Ros] for the case of minimal surfaces in product spaces, and jointly with Uwe Abresch [AbRo1, AbRo2] for the case of CMC surfaces in homogeneous spaces with a 4-dimensional isometry group. On one hand, Meeks and Rosenberg established many results on complete minimal surfaces in M2×R, what has guided a large number of subsequent works in the field. A recent major contribution in this sense is the Collin-Rosenberg theorem [CoRo] on the existence of harmonic diffeomorphims from Conto the hyperbolic plane H2, obtained by constructing an entire minimal graph of parabolic conformal type in H2×R. On the other hand, Abresch and Rosenberg discovered a holomorphic quadratic differential for CMC surfaces in these homogeneous spaces with 4-dimensional isometry group (the E3(κ, τ) spaces), and solved the Hopf problem for them. The general integrability theory of CMC surfaces in the homogeneous E3(κ, τ) spaces was then established by B. Daniel [Dan1]. The discovery by the authors of a harmonic Gauss map into H2 for H= 1/2 surfaces in H2×Rturned into a series of papers by Daniel, Fern´andez, Hauswirth, Mira, Rosenberg, Spruck [FeMi1, Dan2, FeMi2, HRS, DaHa] in which the Bernstein problem for CMC graphs of critical mean curvature (including minimal graphs in Heisenberg space Nil3, see Section 6) was solved. Very recently, the Hopf and Alexandrov problems for CMC surfaces have been solved by Daniel-Mira and Meeks [DaMi, Mee] in the remaining Thurston 3-dimensional geometry: the Lie group Sol3, whose isometry group is only 3-dimensional. We have organized this exposition as follows. In Section 2 we will introduce the 3-dimensional homogeneous ambient spaces. In Section 3 we will present the basic integrability equations by Daniel for CMC surfaces in the homogeneous spaces E3(κ, τ), Constant mean curvature surfaces in 3-dimensional Thurston geometries 3 together with the holomorphic Abresch-Rosenberg differential, and with some basic definitions on stability of CMC surfaces. In Section 4 we will discuss the Hopf, Alexandrov and isoperimetric problems in the homogeneous spaces E3(κ, τ). Section 5 will be devoted to solving the Hopf and Alexandrov problems in the eighth Thurston geometry, i.e. the Lie group Sol3. In Section 6 we will present the solution to the Bernstein problem for entire graphs of critical CMC in the homogeneous E3(κ, τ) spaces. Finally, in Section 7 we shall expose the Collin-Rosenberg theorem on parabolic entire minimal graphs in H2×R, together with some developments on the theory of complete minimal surfaces of finite total curvature in H2×R. Most sections finish with a selection of important open problems. See [Mee, DHM] for more open problems in the theory. A more detailed introduction to the global theory of CMC surfaces in homogeneous 3-spaces can be found in the Lecture Notes by Daniel, Hauswirth and Mira [DHM]. The authors are grateful to H. Rosenberg, B. Daniel and J.A. G´alvez for useful observations about this manuscript. 2. Homogeneous 3-spaces and Thurston geometries Homogeneous spaces are the natural generalization of space forms. By definition, a manifold is said to be homogeneous if the isometry group acts transitively on the manifold. Roughly speaking, the manifold looks the same at all the points, even though, standing at one point, the manifold can look different in different directions. In the simply connected case, the classification of the 3-dimensional homogeneous spaces is well-known. It turns out that any simply connected homogeneous 3-space must have isometry group of dimension 6, 4 or 3. The complete list of these spaces is the following (see subsections below for more details): •The spaces with 6-dimensional isometry group are the space forms: the Euclidean space R3, the hyperbolic space H3(κ), and the standard sphere S3(κ). For simplicity we will assume that κ=±1 and write H3=H3(−1) and S3=S3(1). •The spaces with 4-dimensional isometry group are fibrations over the 2-dimensional space forms. They are the product spaces H2×Rand S2×R, the Berger spheres, the Heisenberg space Nil3and the universal covering of the Lie group PSL(2,R). •The spaces with 3-dimensional isometry group are a certain class of Lie groups; among them we specially quote the space Sol3. These spaces are closely related with Thurston’s Geometrization Conjecture. This recently proved conjecture states that any compact orientable 3-manifold can be cut by disjoint embedded 2-spheres or tori into pieces, each one of them, after gluing 2-balls or solid tori along its boundary components, admits a geometric structure. A 3-manifold without boundary is said to admit a geometric structure if it can be endowed with a complete locally homogeneous metric. In this case, by considering its universal covering we obtain a complete simply-connected locally homogeneous space and hence, by a result of Singer, homogeneous. Thus, a 3-manifold admitting a geometric structure can be realized as the quotient of a homogeneous simply connected 3-space under the action of a subgroup of a Lie group acting transitively by isometries. The list of the maximal geometric structures that give compact quotients consists of eight of the previously described spaces: the three space forms, the two product spaces, Nil3, the universal covering of PSL(2,R) and Sol3(Berger spheres must be excluded from this list because they are 4Isabel Fern´andez and Pablo Mira not maximal, their isometry group are contained in the one of the standard sphere S3). We refer to [Sco, Bon] for more details. 2.1. Homogeneous spaces with 4-dimensional isometry group. Denote by M2(κ) the 2-dimensional space form of constant curvature κ(for example, M2(κ) = R2,H2,S2for κ= 0,−1,1 respectively). As commented above, any simply connected homogeneous 3-space with 4-dimensional isometry group admits a fibration over M2(κ), for some κ∈R. Moreover, these spaces can be parameterized in terms of the base curvature κand the bundle curvature τ, that satisfy κ−4τ26= 0. We will use the notation E3(κ, τ) for these homogeneous spaces. 1. When τ= 0, we have the product spaces M2(κ)×R, i.e. up to scaling, the spaces S2×Rwhen κ > 0, and H2×Rwhen κ < 0. 2. When τ6= 0 and κ > 0, the corresponding spaces are the Berger spheres, a family of 2-parameter (1-parameter after a homothetical change of coordinates) metrics on the sphere, obtained by deforming the standard metric in such a way that the Hopf fibration is still a Riemannian fibration. They can also be seen as the Lie group SU(2) endowed with a 1-parameter family of left-invariant metrics. 3. When τ6= 0 and κ= 0, E3(κ, τ) is the Heisenberg group Nil3, the nilpotent Lie group     1a b 0 1 c 0 0 1 ;a, b, c ∈R  , endowed with a 1-parameter family of left-invariant metrics, all of them isometrically equivalent after a homothetical change of coordinates. 4. When τ6= 0 and κ < 0, we obtain the universal covering of the Lie group PSL(2,R), endowed with a 2-parameter (again 1-parameter after homotheties) family of leftinvariant metrics. There exists a common setting for all these spaces. Indeed, label D(ρ) = {(x1, x2)∈ R2;x2 1+x2 2< ρ2}. Then, if κ= 0 (resp. κ < 0), the space E3(κ, τ) can be viewed as R3 (resp. D2/√−κ×R) endowed with the metric ds2=λ2(dx2 1+dx2 2) + τλ(x2dx1−x1dx2) + dx32, λ =1 1 + κ 4(x2 1+x2 2).(1) Also, for κ > 0, (R3, ds2) corresponds to the universal cover of E3(κ, τ) minus one fiber. In all cases, up to a homothetical change of coordinates we can suppose without loss of generality that κ−4τ2=±1. The corresponding Riemannian fibration π:E3(κ, τ)→ M2(κ) is given here by the projection on the first two coordinates. The unitary vector field ξ=∂ ∂x3 is a Killing field tangent to the fibers of π, and will be referred to as the vertical field of the space E3(κ, τ). It satisfies the equation b ∇Xξ=τX ×ξ for all vector fields Xin E3(κ, τ). Here b ∇is the Levi-Civita connection, ×the cross product and τthe bundle curvature (this is basically the definition of τ). Constant mean curvature surfaces in 3-dimensional Thurston geometries 5 A remarkable difference between the spaces E3(κ, τ) is that their isometry group has four connected components in the case τ= 0, and only two when τ6= 0. This follows from the fact that any isometry in the product spaces can either preserve or reverse the orientation of the base and the fibers independently, while in the case τ6= 0 it can only either preserve or reverse both orientations. In particular, reflections only exist in product spaces. Also, when τ6= 0 the spaces E3(κ, τ) are Lie groups, and if we set σ:= κ 2τ, an orthonormal frame of left-invariant vector fields (called the canonical frame) is given by E1=λ−1cos(σx3)∂ ∂x1 + sin(σx3)∂ ∂x2+τ(x1sin(σx3)−x2cos(σx3)) ∂ ∂x3 , E2=λ−1−sin(σx3)∂ ∂x1 + cos(σx3)∂ ∂x2+τ(x1cos(σx3) + x2sin(σx3)) ∂ ∂x3 , E3=ξ=∂ ∂x3 . 2.2. Homogeneous spaces with 3-dimensional isometry group. Of all homogeneous spaces with 3-dimensional isometry group, Sol3is specially important, since it is the only Thurston geometry among them. We will now describe some aspects of this space. A useful representation of Sol3is the space R3with the metric ds2=e2x3dx2 1+e−2x3dx2 2+dx2 3, that is left-invariant for the structure of Lie group given by (x1, x2, x3)·(y1, y2, y3) = (x1+e−x3y1, x2+ex3y2, x3+y3). The following vector fields form an orthonormal left-invariant frame E1=e−x3∂ ∂x1 , E2=ex3∂ ∂x2 , E3=∂ ∂x3 . The isometries in Sol3are generated by the three 1-parameter groups of translations (x1, x2, x3)7→ (x1+c, x2, x3),(x1, x2, x3)7→ (x1, x2+c, x3), (x1, x2, x3)7→ (e−cx1, ecx2, x3+c), and by the orientation reversing isometries fixing the origin (x1, x2, x3)7→ (−x1, x2, x3),(x1, x2, x3)7→ (x2,−x1,−x3). A remarkable fact is the existence of two canonical foliations, namely F1={x1= constant},F2={x2= constant}, whose leaves are totally geodesic surfaces isometric to the hyperbolic plane H2. Reflections across any of these leaves are orientation reversing isometries of Sol3. 6Isabel Fern´andez and Pablo Mira 3. CMC surfaces: basic equations In this section we present three important tools for our study. One is the set of integrability equations of CMC surfaces in E3(κ, τ) by Daniel [Dan1]. Another one the Abresch-Rosenberg differential, a holomorphic quadratic differential geometrically defined on any CMC surface in E3(κ, τ). The third one is a local isometric correspondence for CMC surfaces in E3(κ, τ) via which one can pass from one homogeneous space into another when studying CMC surfaces [Dan1]. Some notions about the stability operator of CMC surfaces are also given. 3.1. Integrability equations in E3(κ, τ). It is well known that the GaussCodazzi equations are the integrability conditions of surface theory in R3,S3and H3. In other homogeneous spaces, the situation is more complicated. Let ψ: Σ →E3(κ, τ) be an isometric immersion with unit normal map η, and consider on Σ the conformal structure given by its induced metric via ψ. Associated to a conformal parameter z=s+it on Σ, we will consider the usual operators ∂z= (∂s−i∂t)/2 and ∂¯z= (∂s+i∂t)/2. Also denote by ξthe vertical Killing field of E3(κ, τ). Definition 3.1. We call the fundamental data of ψthe 5-tuple (λ|dz|2, u, H, p dz2, A dz) where His the mean curvature and λ= 2hψz, ψ¯zi, u =hN, ξi, p =−hψz, Nzi, A =hξ, ψzi. The function uis commonly called the angle function of the surface. Once here, a set of necessary and sufficient conditions for the integrability of CMC surfaces in E3(κ, τ) can be written in terms of these fundamental data. This is a result by B. Daniel [Dan1], although the formulation that we expose here (i.e. in terms of a conformal parameter on the surface) comes from [FeMi2]. Theorem 3.2 ([Dan1, FeMi2]). The fundamental data of an immersed surface ψ: Σ → E3(κ, τ)satisfy the following integrability conditions:                  (C.1)p¯z=λ 2(Hz+uA(κ−4τ2)). (C.2)A¯z=uλ 2(H+iτ). (C.3)uz=−(H−iτ)A−2p λ¯ A. (C.4)4|A|2 λ= 1 −u2. (2) Conversely, if Σis simply connected, these equations are also sufficient for the existence of a surface ψ: Σ →E3(κ, τ)with fundamental data (λ|dz|2, u, H, p dz2, A dz). This surface is unique up to ambient isometries preserving the orientations of base and fiber of E3(κ, τ). We see then that, in the spaces E3(κ, τ), more equations apart from the GaussCodazzi ones are needed, due to the loss of symmetries. As a matter of fact, (C.1) is the Codazzi equation, while the Gauss equation does not appear (it is deduced from the rest). These new equations evidence the special character of the vertical direction in the E3(κ, τ) spaces. Constant mean curvature surfaces in 3-dimensional Thurston geometries 7 Definition 3.3. The Abresch-Rosenberg differential of the immersion is defined as the quadratic differential on Σgiven by Qdz2=2(H+iτ)p−(κ−4τ2)A2dz2. It is then easy to see by means of (C.2) that the Codazzi equation (C.1) can be rephrased in terms of Qas Q¯z=λHz+ (κ−4τ2)H¯zA2 (H+iτ)2.(3) Consequently, one has the following theorem, which generalized the classical fact that the Hopf differential is holomorphic for CMC surfaces in R3,S3and H3. Theorem 3.4 ([AbRo1, AbRo2]). Qdz2is a holomorphic quadratic differential on any CMC surface in E3(κ, τ). This is a crucial result of the theory, since it allows the use of holomorphic functions in the geometric classification of CMC surfaces in E3(κ, τ) (see Section 4 and Section 6, for instance). An important tool in the description of CMC surfaces in R3,S3and H3is the classical Lawson correspondence. It establishes an isometric one-to-one local correspondence between CMC surfaces in different space forms that allows to pass, for instance, from minimal surfaces in R3to H= 1 surfaces in H3. The Lawson correspondence was generalized by B. Daniel to the context of homogeneous spaces. Indeed, Daniel discovered in [Dan1] an isometric local correspondence for CMC surfaces in all the homogeneous spaces E3(κ, τ), which can be described as follows in terms of the fundamental data defined above. Theorem 3.5 (Sister correspondence, [Dan1]). Let (λ|dz|2, u, H1, p1dz2, A1dz)be the fundamental data of a simply connected H1-CMC surface in E(κ1, τ1), and consider κ2, τ2, H2∈Rso that κ2−4τ2 2=κ1−4τ2 1, H2 2+τ2 2=H2 1+τ2 1. Then if we set θ∈Rgiven by H2−iτ2=eiθ(H1−iτ1), the fundamental data given by (λ|dz|2, u, H2, p2dz2=e−iθp1dz2, A2dz =e−iθA1dz) (4) give rise to a (simply connected) H2-CMC surface in E3(κ2, τ2), which is locally isometric to the original one. Two surfaces related by the above correspondence are called sister surfaces with phase θ. In particular, the corresponding Abresch-Rosenberg differentials of sister surfaces are related by Q2=e−2iθQ1. As special cases of this correspondence we obtain the associate family of minimal surfaces in M2(κ)×R, and a correspondence between minimal surfaces in Nil3and CMC 1 2surfaces in H2×R. Generically, and up to ambient isometries and dilations, the family of sister surfaces for a given choice of (H, κ, τ) is a continuous 1parameter family. There is a natural notion of graph in these spaces. Since E3(κ, τ) has a canonical fibration over M2(κ) (see Section 2), we will say that an immersed surface Σ in E3(κ, τ) 8Isabel Fern´andez and Pablo Mira is a (local) graph if the projection to the base is a (local) diffeomorphism. The CMCequation for the graph of a function u=u(x, y) is the PDE (see [Lee]) 2H δ2=∂ ∂x α ω+∂ ∂y β ω,(5) where δ= 1 + κ 4(x2+y2), ω =p1 + δ2(x2+y2), α=ux+τy δ, β =uy−τx δ. For instance, a graph u=u(x, y) in Nil3≡E3(0, τ) is minimal if and only if it satisfies the elliptic PDE (1 + β2)uxx −2αβ uxy + (1 + α2)uyy = 0,(6) where α:= ux+y/2 and β:= uy−x/2. 3.2. Stability and index of CMC surfaces. As it is well known, CMC surfaces in Riemannian 3-manifolds appear as the critical points for the area functional associated to variations of the surface with compact support and constant enclosed volume. Equivalently, an immersed surface Shas constant mean curvature Hif and only if it is a critical point for the functional Area −2HVol. The second variation formula for this functional is given by Q(f, f) = −ZS fL(f), where Lis the Jacobi operator (or stability operator) of the surface: L= ∆ + ||B||2+ Ric(η). Here ∆ is the Laplacian for the induced metric on the surface, Bis the second fundamental form, ηis the unit normal vector field, and Ric is the Ricci curvature in the ambient manifold. As a particular case, the Jacobi operator for CMC surfaces in the spaces E3(κ, τ) can be rewritten (see [Dan1]) as L= ∆ −2K+ 4H2+ 4τ2+ (κ−4τ2)(1 + u2), being Kthe Gaussian curvature of the surface and uthe angle function (see Definition 3.1). A Jacobi function is a function ffor which L(f) = 0. A CMC surface Sis said to be stable (resp. weakly stable) if Q(f, f) = −ZS fL(f)≥0 holds for any smooth function fon Swith compact support (resp. with compact support and RSf= 0). For instance, CMC graphs in E3(κ, τ) are stable, and compact CMC surfaces bounding isoperimetric regions are weakly stable (but not necessarily stable, as round spheres in R3show). An important concept related to stability is the index of a CMC surface. The index of a compact CMC surface is defined as the number of negative eigenvalues of its Jacobi operator. Thus, stable CMC surfaces (in particular CMC graphs) have index zero. Round spheres in R3have index one. We refer to [MPR] for more details about stability of CMC surfaces. Constant mean curvature surfaces in 3-dimensional Thurston geometries 9 4. Compact CMC surfaces in E3(κ, τ) In this section we explain the most important results that are known regarding the existence and uniqueness of compact CMC surfaces in the homogeneous 3-spaces E3(κ, τ). The fundamental examples are the rotational CMC spheres, and we shall be interested in their uniqueness among compact embedded CMC surfaces, and among immersed CMC surfaces. These problems are called, respectively, the Alexandrov and Hopf problems. 4.1. Rotational compact CMC surfaces. Although round spheres in the model spaces R3,S3,H3are CMC spheres, this does not hold for the rest of homogeneous spaces. However, in all the spaces E3(κ, τ) there exist rotations with respect to the vertical axis, and so there is a natural notion of rotational surface. It is hence natural to seek CMC spheres (and CMC tori) in E3(κ, τ) among the class of rotational surfaces. This can be done by ODE analysis, and the result of this can be summarized as follows: Theorem 4.1. (Structure of rotational CMC spheres in E3(κ, τ)). 1. If κ−4τ2>0, then for every H∈Rthere exists a unique rotational CMC Hsphere (up to isometries) in E3(κ, τ). These spheres are embedded if τ= 0, i.e. in S2×R, and also for most Berger spheres. However, for some Berger spheres with small bundle curvature τ(with respect to a fixed κ) there is a certain region of variation of the parameters (H, τ)where the spheres are non-embedded. This region can be explicitly described, see [Tor]. 2. If κ−4τ2<0, then •if H2⩽−κ 4, then there exists no rotational CMC Hsphere in E3(κ, τ), •if H2>−κ 4, then there exists a unique rotational CMC Hsphere (up to isometries) in E3(κ, τ). All these spheres are embedded. Let us remark that all these CMC spheres can be constructed explicitly. We shall call them canonical rotational CMC spheres. For example, the rotational CMC Hspheres in S2×R⊂R4are given by the formula ψ(u, v) = (−cos k(u),sin k(u) cos v, sin k(u) sin v, h(u)), where −1≤u≤1, H∈Rand k(u) := 2 arctan 2H √1−u2, h(u) := 4H √4H2+ 1 arcsinh u √1−u2+ 4H2. Besides these rotational CMC spheres, there also exist rotational CMC tori in E3(κ, τ) when (and only when) κ−4τ2>0 (excluding minimal surfaces in S2×R). For S2×R, they are all embedded (see Pedrosa [Ped]). For Berger spheres the situation is explained by Torralbo and Urbano in [Tor, ToUr]; one has for every Hrotational embedded CMC tori given by the Hopf lift of a circle in S2, but there also exist some other non-flat rotational CMC tori. The embeddedness problem for such tori is open in general, but for the minimal case there are embedded rotational tori other than Clifford tori. This contrasts with the case of embedded minimal tori in S3. A general study of CMC surfaces in H2×Rand S2×Rinvariant by a continuous 1-parameter subgroup of ambient isometries can be found in [SaE, SaTo]. 16 Isabel Fern´andez and Pablo Mira Definition 6.3. We will say that a harmonic map Ginto H2admits Weierstrass data {Q0, τ0}if the pullback metric induced by Gcan be written as hdG, dGi=Q0dz2+µ|dz|2+¯ Q0d¯z2, µ =τ0 4+4|Q0|2 τ0 , τ0being a positive smooth function. 6.1.1. H= 1/2 surfaces in H2×R.We will regard H2×R=E3(−1,0) in its Minkowski model, i.e. H2×R={(x0, x1, x2, x3) : x0>0,−x2 0+x2 1+x2 2=−1} ⊂ L3×R= L4. Using this model, the unit normal vector ηof an immersed surface ψ= (N, h) : Σ → H2×Rtakes values in the de Sitter 3-space, and {η, N}is an orthonormal frame for the Lorentzian normal bundle of ψin L4. Moreover, if uis the angle function of the surface (that is, the last coordinate of η) and we assume that u6= 0 (that is, that ψis nowhere vertical, or equivalently, that it is a multigraph), then we can write 1 u(η+N) = (G, 1),(12) for a certain map G: Σ →H2. Definition 6.4 ([FeMi1]). The map Ggiven by (12) will be called the hyperbolic Gauss map of an immersed (nowhere vertical) surface in H2×R. The main property of the hyperbolic Gauss map is the following [FeMi1]: Theorem 6.5 (Fern´andez-Mira). The hyperbolic Gauss map of a CMC surface with H= 1/2in H2×Ris a harmonic map into H2, and admits Weierstrass data {−Q, λu2}, where Qdz2,λ|dz|2and uare, respectively, the Abresch-Rosenberg differential, the metric, and the angle function of the surface. Conversely, if Σis simply connected, any harmonic map G: Σ →H2admitting Weierstrass data is the hyperbolic Gauss map of some H= 1/2surface in H2×R. Moreover, the space of H= 1/2surfaces in H2×Rwith the same hyperbolic Gauss map Gis generically two-dimensional, and it can be recovered from Gby a representation formula. The proof of the direct part of the above result follows from equations (2) and the very definition of G. The converse part is an integrability argument. This result is of great importance for the rest of this section, since it allows the use of harmonic maps in the description of surfaces of critical CMC. 6.1.2. Minimal surfaces in Nil3.The existence of this harmonic Gauss map for H= 1/2 surfaces in H2×Rwas extended by B. Daniel [Dan2] to the case of minimal surfaces in Nil3=E3(0,1 2). This time, the harmonic Gauss map is given by the Lie group Gauss map of the surface. Indeed, if we identify the Lie algebra of Nil3with the tangent space at a point by left multiplication, we can stereographically project the unit normal vector field to obtain a map taking values in the extended complex plane. More specifically, we will consider the model of Nil3given in Section 2 and its canonical frame of left-invariant Constant mean curvature surfaces in 3-dimensional Thurston geometries 17 fields {E1, E2, E3}. If N=PNiEiis the unit normal of X: Σ →Nil3, then the Gauss map of Xis given by g=N1+iN2 1 + N3 : Σ →C. Now, if the surface is nowhere vertical we can orient it so that u=hN, E3iis positive, and so gtakes values in the unit disc D. By identifying H2with (D, ds2 P), where ds2 Pis the Poincar´e metric, Daniel obtained in [Dan2]: Theorem 6.6 (Daniel). The Gauss map of a nowhere vertical minimal surface is harmonic into H2. Conversely, let g: Σ →H2be a harmonic map defined on a simply connected oriented Riemann surface into H2, and assume that gis nowhere antiholomorphic (i.e., gzdoes not vanish at any point). Take z0∈Σand X0∈Nil3. Then there exists a unique conformal nowhere vertical minimal immersion X: Σ → Nil3with X(z0) = X0having gas its Gauss map. Moreover, Xcan be uniquely recovered from gthrough an adequate representation formula. Furthermore, it can be checked that the Weierstrass data of gas above are {−Q, λu2}, where Qdz2,λ|dz|2and uare, respectively, the Abresch-Rosenberg differential, the metric, and the angle function of the surface defined in Section 2. As we saw in Section 3, minimal surfaces in Nil3and H= 1/2 surfaces in H2×Rare related by the sister correspondence, and sister surfaces have the same metric and angle function (in particular, the condition of being nowhere vertical is preserved). As in this case the sister surfaces have opposite Abresch-Rosenberg differentials, it turns out that their respective harmonic Gauss maps are conjugate to each other. The relation between minimal surfaces in Nil3and H= 1/2 surfaces in H2×Rcan be made more explicit by means of the theory of spacelike CMC surfaces in L3, as follows. Theorem 6.7 ([FeMi3]). Let X= (F, t) : Σ →Nil3be a simply connected nowhere vertical minimal surface with metric λ|dz|2and angle function u, and ψ= (N, h) : Σ → H2×Rits sister surface. Then f:= (F, h) : Σ → L3is a spacelike H= 1/2surface in the Minkowski 3-space with metric λu2|dz|2and Hopf differential −Qdz2, where Qdz2is the Abresch-Rosenberg differential of X. 6.1.3. CMC √−κ/2 surfaces in ^ PSL(2,R). In a forthcoming paper [DFM], the authors and B. Daniel will prove that there exists also a harmonic Gauss map for critical CMC surfaces in the remaining case, i.e. the universal covering of the group PSL(2,R), and will derive a representation formula for them. 6.2. Half-space theorems. One of the most important results in the global study of minimal surfaces in R3is the classical half-space theorem by Hoffman and Meeks [HoMe]. This theorem says that any properly immersed minimal surface in R3lying in a half-space must be a plane parallel to the one determining the half-space. The main tools used here are the maximum principle and the existence of catenoids, a 1-parameter family of minimal surfaces converging to a doubly-covered punctured plane P, and intersecting the planes parallel to Pin compact curves. The analogous version for CMC one half surfaces in H2×Rwas proved in [HRS]. In this setting, horocylinders play the role of the planes in R3. 18 Isabel Fern´andez and Pablo Mira Theorem 6.8 (Hauswirth-Rosenberg-Spruck). The only properly immersed CMC one half surfaces in H2×Rthat are contained in the mean convex side of a horocylinder C are the horocylinders parallel to C. Also, the only properly embedded CMC one half surfaces in H2×Rcontaining a horocylinder in its mean convex side are the horocylinders. Proof. The main point here is to construct a family of CMC one half surfaces in H2×R to be used in the same way as catenoids in the proof of the half-space theorem in R3. This is achieved by means of compact annuli with boundaries, contained between two horocylinders. For the case of Nil3, we must distinguish between horizontal and vertical half-spaces. The equivalent to the half-space theorem for surfaces lying in a horizontal half-space is proved by using the family of rotational annuli [AbRo2]. The corresponding vertical version has been obtained in [DaHa], by constructing first a family of horizontal catenoids, i.e. properly embedded minimal annuli (non-rotational) with a geometric behaviour good enough to apply the Hoffman-Meeks technique. Theorem 6.9 (Daniel-Hauswirth). The only properly immersed minimal surfaces in Nil3that are contained in a vertical half space are the vertical planes parallel to the one determining the half-space. Proof. Using the representation formula for minimal surfaces in Nil3(see Theorem 6.6), it is possible to construct horizontal catenoids in Nil3. These surfaces are a 1-parameter family of properly embedded minimal annuli, intersecting vertical planes {x2=c}in a non-empty closed convex curve. Moreover, the family converges to a double covering of {x2= 0}minus a point. They are obtained by integrating a family of harmonic maps that belong to a more general family used in the construction of Riemann type minimal surfaces in H2×R[Ha]. Once we have these catenoids, we finish by using the maximum principle similarly to the Euclidean case. 6.3. The classification of entire graphs. In this section we will describe the space of entire graphs of critical CMC in E3(κ, τ). Such a description follows from the works of Fern´andez-Mira [FeMi1, FeMi3], Hauswirth-Rosenberg-Spruck [HRS] and Daniel-Hauswirth [DaHa], and is contained in Theorems 6.10 and 6.11. We expose here a unified perspective to this subject. First, we have Theorem 6.10 ([DaHa, FeMi3, HRS]). The following conditions are equivalent for a surface of critical CMC in E3(κ, τ): (1) It is an entire graph. (2) It is a complete multigraph. (3) u2ds2is a complete Riemannian metric (where uis the angle function and ds2the metric of the surface). In particular, the sister correspondence preserves entire graphs of critical CMC. Let us make some comments on this theorem. First, Hauswirth, Rosenberg and Spruck proved (2) ⇒(1) for H= 1/2 surfaces in H2×R. Second, the authors proved in [FeMi3] that (3) ⇒(1) (for any surface in E3(κ, τ), not necessarily CMC), and that (1) ⇒(3) holds for minimal surfaces in Nil3. Finally, Daniel and Hauswirth showed that (2) ⇒(1) holds for minimal surfaces in Nil3. The rest of the cases can be easily obtained from these results and the sister correspondence (this was first observed in [DHM]). Constant mean curvature surfaces in 3-dimensional Thurston geometries 19 Proof. It is immediate that (1) ⇒(2). Also, by an eigenvalue estimate, the authors proved in [FeMi3] that for arbitrary surfaces in E3(κ, τ) it holds u2ds2≤gF, where F=π◦ψis the projection onto M2(κ) of ψ. Thus, if u2ds2is complete, Fis a local diffeomorphism with complete pullback metric, and by standard topological arguments, Fis a diffeomorphism, i.e. (3) ⇒(1) holds. That (1) ⇒(3) holds for minimal surfaces in Nil3was also proved in [FeMi3]: let X= (F, t) : Σ →Nil3be an entire minimal graph. By Theorem 6.7, there is an entire spacelike CMC graph f= (F, h) : Σ → L3, whose induced metric is ds2 f=u2ds2. Now we can apply a theorem by Cheng and Yau [ChYa] which says that spacelike entire CMC graphs in L3have complete induced metric. Hence u2ds2is complete, as wished. We will now prove that (2) ⇒(1) holds for minimal surfaces in Nil3. Let us observe that once this is done, we can also prove the theorem for surfaces of critical CMC in all the spaces E3(κ, τ). Indeed, as any simply connected surface of critical CMC is the sister surface of some minimal surface in Nil3, and as the correspondence preserves the metric and the angle function (therefore it preserves conditions (2) and (3) by passing to the universal covering), we can easily translate the theorem for the case of minimal surfaces in Nil3to the rest of the spaces. It is important here that we proved (3) ⇒(1) in all spaces. So, we only need to prove (2) ⇒(1) for minimal surfaces in Nil3. This was done by Daniel and Hauswirth [DaHa]. For that, they used their half-space theorem in Nil3 (Theorem 6.9) and an adaptation to Nil3of the previous proof of (2) ⇒(1) for the case of H= 1/2 surfaces in H2×Rgiven by Hauswirth-Rosenberg-Spruck [HRS]. In order to prove (2) ⇒(1) for minimal surfaces in Nil3, we argue by contradiction. Assume that there exists a complete multigraph Σ that is not entire. Then there exists an open set Σ0⊂Σ that is a graph over a disc D( R2of a function f, and a point q∈∂D such that fdoes not extend to q. Step 1: For any sequence of points {qn}in Dconverging to q, the sequence of normal vectors at the points pn= (qn, f(qn)) ∈Σ0converges to the horizontal vector orthogonal to ∂D at q. Indeed, as the surface is a multigraph, its angle function u=hN, E3i, where N denotes the unit normal vector, is a Jacobi function that does not vanish. As a result of this, Σ is (strongly) stable, and has bounded geometry. This means that locally around any pnwe can write the surface as the graph (in exponential coordinates) over a disc of radius δof its tangent plane, where δis a universal constant depending only on Σ. This neighborhood of pnwill be denoted by G(pn). The limit of the normal vectors {N(pn)} must be a horizontal vector since otherwise, the piece G(pn) of bounded geometry could be extended as a graph beyond q, which is impossible. Moreover, the limit vector must be normal to ∂D at qsince Σ0is a graph over D. Step 2: The function fdefining the graph Σ0diverges at q. Moreover, as we approach q, and after translating the surface to the origin, the surfaces converge to a piece of the (translated) vertical plane Ppassing through qand tangent to ∂D. That fdiverges at qis a consequence of the completeness of Σ, and the last part can be proved by following the ideas of Collin and Rosenberg in [CoRo]. We will assume that Pis the plane {x1=c}. Step 3: Σcontains a graph Gover a domain of the form Uǫ= (c−ǫ, c)×R⊂R2. Moreover, this graph is disjoint from Pand asymptotic to it as one approaches q. The graph Gis obtained by analytical continuation of the surfaces G(pn) used in the first step, and after a careful study of the behavior of the intersection curves of these graphs and the planes parallel to P. 20 Isabel Fern´andez and Pablo Mira Finally, the contradiction follows from the half-space theorem (Theorem 6.9). Recall that, although Ghas boundary and the theorem is formulated for surfaces without boundary, its proof applies to this case, and so we are done. Once here, we investigate the Bernstein problem for entire graphs of critical CMC in E3(κ, τ), i.e. the classification of such entire graphs (recall here that CMC graphs in E3(κ, τ) satisfy the elliptic PDE (5)). The terminology comes from the classical Bernstein theorem: entire minimal graphs in R3are planes. Equivalently, any solution to the minimal graph equation (1 + f2 y)fxx −2fxfyfxy + (1 + f2 x)fyy = 0 (13) defined on the whole plane is linear. It is interesting to compare this result with the Bernstein problem in Nil3, i.e. the classification of entire minimal graphs in Nil3. This corresponds to classifying all global solutions to the PDE (6). Observe that taking τ= 0 in (6) we obtain the classical equation (13), i.e. the classical case considered by Bernstein appears as a limit of the Heisenberg case. There exists, however, a great difference between both situations. The following result classifies the entire graphs of critical CMC in E3(κ, τ), by parametrizing the moduli space of such entire graphs in terms of holomorphic quadratic differentials. It was obtained first for minimal graphs in Nil3by the authors [FeMi3], and shortly thereafter by Daniel and Hauswirth [DaHa] for H= 1/2 graphs in H2×R. The general case follows easily from the Heisenberg case and Theorem 6.10, using the sister correspondence (this was observed first in [DHM]). Theorem 6.11 (Fern´andez-Mira, Daniel-Hauswirth). Let Qdz2denote a holomorphic quadratic differential on Σ≡Cor D, such that Q6≡ 0if Σ≡C, and let H2=−κ/4. There exists a 2-parameter family of entire CMC Hgraphs in E3(κ, τ)whose AbreschRosenberg differential agrees with Qdz2. These graphs are generically non-congruent. And conversely, these are all the entire graphs of critical CMC in E3(κ, τ). At this point, the proof for the case of minimal surfaces in Nil3is a consequence of Theorem 6.7 and the following result by Wan and Wan-Au [Wan, WaAu] on spacelike entire CMC graphs in L3:for any holomorphic quadratic differential as above, there exists a unique (up to isometries) spacelike entire CMC 1/2graph in L3with Hopf differential Qdz2.The 2-parameter family of non-congruent graphs in E3(κ, τ) comes from the loss of ambient isometries (from 6 dimensions to 4 dimensions) when passing from L3to Nil3. The remaining cases of critical CMC graphs follow since by Theorem 6.10 the sister correspondence preserves entire graphs. 6.4. Open Problems. As explained in Section 3, entire graphs are stable. It is conjectured that entire graphs and vertical cylinders are the only stable critical CMC surfaces (this has been proved for parabolic conformal type in [MaPR]). Related to this is the question of non-existence of complete stable H > 1/2 surfaces in H2×R(proved for H > 1/√3 by Nelli-Rosenberg, [NeRo2]). Also, not much is known about properly embedded surfaces of critical CMC and non-trivial topology. Can one obtain them by conjugate Plateau constructions, or by integrable systems techniques? Another remarkable problem is to establish the strong half-space theorem in Nil3: are two disjoint properly embedded minimal surfaces in Nil3 necessarily two parallel vertical planes, or two parallel entire minimal graphs? Constant mean curvature surfaces in 3-dimensional Thurston geometries 21 7. Minimal surfaces in H2×Rand S2×R Minimal surfaces in product spaces admit a special treatment, due to several reasons. One of them is the following: if ψ= (N, h) : Σ →M2×Ris a minimal surface immersed in the product space M2×R, where (M2, g) is a Riemannian surface, then the horizontal projection N: Σ →M2is a harmonic map and the height function h: Σ →Ris a harmonic function. This implies, for instance, that compact minimal surfaces in M2×R only exist if M2is compact (in particular, if M2=S2), and the only ones are the slices M2×{t0}. Another important fact about minimal surfaces in M2×Ris that there is a natural notion of minimal graph over a domain Ω ⊂M2, and that this graph satisfies a simple elliptic PDE in divergence form. This fact together with general existence results for solutions to the Plateau problem in Riemannian 3-manifolds allows a good control on the geometry of the surface. Some of the most interesting results of the theory of minimal surfaces in product spaces come from the interplay between the information provided by harmonic maps and by Plateau constructions and the minimal graph equation. Starting with the pioneer work of H. Rosenberg [Ros], and W.H Meeks and H. Rosenberg [MeRo1, MeRo2], the theory of minimal surfaces in M2×Rhas developed substantially in the last decade. We will only talk here about a few results of special relevance to the theory, and not mention many other important results. 7.1. The Collin-Rosenberg theorem. The classical Bernstein theorem in R3states that planes are the only entire minimal graphs in R3. This theorem can be extended to the case of product spaces: any entire minimal graph in M2×R, where (M2, g)is a complete surface of non-negative curvature, is totally geodesic. In contrast, in the product space H2×Rthere is a wide variety of entire minimal graphs. For instance, in [NeRo1] Nelli and Rosenberg solved the Dirichlet problem at infinity for the minimal graph equation in H2×R. They proved that any Jordan curve at the ideal boundary S1×R≡∂∞H2×Rof H2×Rwhich is a graph over S1≡∂∞H2is the asymptotic boundary of a unique entire minimal graph in H2×R(see [GaRo] for a proof of this in the more general case of entire minimal graphs in M2×R, where (M2, g) is complete, simply connected and with KM≤c < 0). All these entire minimal graphs are hyperbolic, that is, they have the conformal type of the unit disk. The problem of existence of entire minimal graphs of parabolic type (i.e. with the conformal type of C) is much harder, and was solved recently by Collin and Rosenberg [CoRo]. Theorem 7.1 (Collin-Rosenberg). There exist entire minimal graphs in H2×Rof parabolic conformal type. As the projection onto H2of a minimal graph is a harmonic diffeomorphism, the above theorem has the following consequence, which solves a major problem in the theory of harmonic maps and disproves a conjecture by R. Schoen and S.T. Yau. Corollary 7.2 (Collin-Rosenberg). There exist harmonic diffeomorphisms from Conto H2. The proof by Collin and Rosenberg is a good example of the interaction between the harmonicity properties of the minimal immersion and the use of Plateau constructions and the minimal graph equation. 22 Isabel Fern´andez and Pablo Mira The main idea in the proof is to construct first (non-entire) minimal graphs in H2×R of Scherk type over ideal geodesic polygons, having alternating asymptotic values +∞ and −∞ on the sides of the polygon. This generalizes a classical construction by Jenkins and Serrin in the case of minimal graphs over bounded domains in R3. This construction is done as follows: Let Γ be an ideal polygon of H2, so that all the vertices of Γ are at the ideal boundary of H2and Γ has an even number of sides A1, B1, A2, B2..., Ak, Bk, ordered clockwise. At each vertex ai, we consider a small enough horocycle Hiwith Hi∩Hj=∅. Each Ai (resp. Bi) meets exactly two horocycles. Denote by e Ai(resp. e Bi), the compact arc of Ai(resp Bi) which is the part of Aioutside the two horodisks. We denote by |Ai|the length of |e Ai|. Define e Biand |Bi|in the same way. Now we can consider a(Γ) = Pk i=1 |Ai|and b(Γ) = Pk i=1 |Bi|. We observe that a(Γ) −b(Γ) does not depend on the choice of the horocycle Hiat ai, since horocycles with the same point at infinity are equidistant. Keeping in mind these data, we can state the following theorem by Collin-Rosenberg [CoRo] (see also Nelli-Rosenberg [NeRo1]): Theorem 7.3. ([NeRo1], [CoRo]) There is a (unique up to additive constants) solution to the minimal surface equation in the polygonal domain P, equal to +∞on Aiand −∞ on Bi, if and only if the following conditions are satisfied: 1. a(Γ) = b(Γ), 2. For each inscribed polygon Pin Γ,P 6= Γ, and for some choice of horocycles at the vertices, one has 2a(P)<|P| and 2b(P)<|P|. All these examples have the conformal type of C. Once there, Collin and Rosenberg designed a way of enlarging a given Scherk-type graph over the interior of some Γ ⊂H2 into another one with more sides, and so that: (1) the extended surface is C2-close to the original one over an arbitrary compact set in the interior of Γ, and (2) there is a control on the conformal radius on adequate compact annuli on the surface. By passing to the limit in this sequence of minimal graphs over larger and larger domains, they obtained an entire minimal graph in H2×Rwhich, by the control on the conformal radii of these annuli, has the conformal type of C. Remark 7.4. The Collin-Rosenberg theorem has been extended by J.A. G´alvez and H. Rosenberg [GaRo] to more general product spaces M2×R:there exist entire minimal graphs of parabolic conformal type on M2×R, where (M2, g) is any complete simply connected Riemannian surface with Gaussian curvature KM≤c < 0 (KMnot constant). 7.2. Minimal surfaces of finite total curvature in H2×R.One of the most studied families among minimal surfaces in R3are the complete minimal surfaces of finite total curvature (FTC for short). A minimal surface Σ is said to have FTC if its Gaussian curvature Ksatisfies ZΣ K dA<∞. By classical theorems of Huber and Osserman, complete FTC minimal surfaces in R3are conformally equivalent to a compact Riemann surface minus a finite number of points. Moreover, the Gauss map extends meromorphically to the punctures, and the total curvature of the surface is a multiple of −4π. A key point here is that the Gauss map of a minimal surface in R3is conformal. Constant mean curvature surfaces in 3-dimensional Thurston geometries 23 In H2×Rthere is no conformal Gauss map for minimal surfaces. Nonetheless, using the global theory of harmonic maps into H2, L. Hauswirth and H. Rosenberg [HaRo] were able to prove that a similar situation holds in H2×R. Theorem 7.5 (Hauswirth-Rosenberg). Let Xbe a complete minimal immersion of Σin H2×Rwith finite total curvature. Then 1. Σis conformally equivalent to a Riemann surface punctured at a finite number of points, Σ≡Mg−{p1...., pk}. 2. Qdz2:= h2 zdz2is holomorphic on Mand extends meromorphically to each puncture. If we parameterize each puncture piby the exterior of a disk of radius r, and if Q(z)dz2=z2mi(dz)2at pi, then mi⩾−1. 3. The third coordinate uof the unit normal tends to zero uniformly at each puncture. 4. The total curvature is a multiple of 2π: ZΣ KdA = 2π 2−2g−2k− k X i=1 mi!. As a consequence, every end of a finite total curvature surface is uniformly asymptotic to a Scherk type graph described in Theorem 7.3. Proof. The first step is to prove that locally around an end, Qdz2only has at most a finite number of zeroes. Then a Huber theorem and an argument of Osserman give that the ends are conformally a punctured disk, and Qdz2extends meromorphically to the puncture. The final part of the behavior of Qdz2follows from the fact that Qdz2=h2 zdz2, where h is the height function of the surface. To prove that ugoes to 0 at the ends, take an annular neighborhood of an end where Qdz2does not vanish. Then reparameterize this annulus by w=R√Qdz. The metric conformal factor in these coordinates satisfies a sinh-Gordon equation, and the Gaussian curvature monotonically decreases to zero. Then, estimates on the growth of solutions of the sinh-Gordon equation allows one to conclude that, at a finite total curvature end, the tangent plane becomes vertical and the metric becomes flat. Finally, the expression for the total curvature follows from Gauss-Bonnet formula and the estimates for the sinh-Gordon equation obtained before. In [HaRo], the following question was also raised: are there complete non simply connected minimal surfaces with FTC in H2×R?Notice that rotational catenoids have infinite total curvature. Actually, at that time, the only known complete FTC minimal surfaces were the Scherk type graphs. This question was positively answered by J. Pyo [Pyo] and also, independently, by Rodr´ıguez and Morabito [RoMo]. Pyo constructed a 1-parameter family of genus zero properly embedded minimal surfaces in H2×Rwith kends for k≥2, similar to the k-noids in R3(although the first ones are embedded and the k-noids in R3are not). They have total curvature 4π(1 −k), and are asymptotic to vertical planes at infinity. These surfaces are obtained as the conjugate surfaces of minimal graphs over infinite geodesic triangles in H2that are asymptotic to vertical planes at infinity. Very shortly thereafter, M. Rodr´ıguez and F. Morabito discovered independently a larger family of FTC minimal surfaces, containing the previous ones. It is a (2k− 2)-parameter of properly embedded FTC minimal surfaces of genus zero with kends, obtained as the limits of simply periodic minimal surfaces called saddle towers, that are 24 Isabel Fern´andez and Pablo Mira invariant by a vertical translation of vector (0,0,2l). Taking limits when l→ ∞, they obtain genus zero minimal surfaces with kends and total curvature 4π(1 −k) that are symmetric with respect to the reflection over the slice H2×{0}. The surfaces found by Pyo appear when the ends are placed in symmetric positions. 7.3. Open problems. In [Ha], L. Hauswirth constructed a family of Riemann type minimal surfaces in H2×Rand S2×R, characterized by the property of being foliated by curves of constant curvature. It is a conjecture by W. Meeks and H. Rosenberg that in S2×Rthey are the only properly embedded minimal annuli. An approach for solving this conjecture using integrable systems techniques has been recently developed by L. Hauswirth and M. Schmidt. Another natural problem is to obtain classification results for properly embedded minimal surfaces of finite total curvature and a given simple topology in R3. Schoen and Yau proved there is no harmonic diffeomorphism from the disk to a complete surface of non-negative curvature. Can there be such a harmonic diffeomorphism onto a complete parabolic surface? This is a question by J.A. G´alvez. References [AbRo1] U. Abresch, H. Rosenberg, A Hopf differential for constant mean curvature surfaces in S2×Rand H2×R,Acta Math. 193 (2004), 141–174. [AbRo2] U. Abresch, H. Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28. [AEG1] J.A. Aledo, J.M.Espinar, J.A. G´alvez, Height estimates for surfaces with positive mean curvature in M×R.Illinois Journal of Math., 52 (2008), 203-211. [Bon] F. Bonahon, Geometric structures on 3-manifolds. In Handbook of Geometric Topology, pages 93–164. North-Holland, Amsterdam, 2002. [ChYa] S.Y. Cheng, S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. 104 (1976), 407–419. [CoRo] P. Collin, H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math., to appear (2007). [Dan1] B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87-131. [Dan2] B. Daniel, The Gauss map of minimal surfaces in the Heisenberg group, preprint, 2006, arXiv:math/0606299. [DFM] B. Daniel, I. Fern´andez, P. Mira, Surfaces of critical constant mean curvature. Work in progress. [DaHa] B. Daniel, L. Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. (3),98 no.2 (2009), 445–470. [DHM] B. Daniel, L. Hauswirth, P. Mira, Constant mean curvature surfaces in homogeneous manifolds, preprint, 2009. Published preliminarly by the Korea Institute for Advanced Study. [DaMi] B. Daniel, P. Mira, Existence and uniqueness of constant mean curvature spheres in Sol3. Preprint, 2008, arXiv:0812.3059 Constant mean curvature surfaces in 3-dimensional Thurston geometries 25 [dCF] M.P. do Carmo, I. Fern´andez, Rotationally invariant CMC disks in product space, Forum Math. 21 (2009), 951–963. [EGR] J.M. Espinar, J.A. G´alvez, H. Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv.,84 (2009), 351–386. [EsRo] J.M. Espinar, H. Rosenberg, Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv., to appear (2009). [FeMi1] I. Fern´andez, P. Mira, Harmonic maps and constant mean curvature surfaces in H2×R,Amer. J. Math. 129 (2007), 1145–1181. [FeMi2] I. Fern´andez, P. Mira, A characterization of constant mean curvature surfaces in homogeneous 3-manifolds, Diff. Geom. Appl.,25 (2007), 281–289. [FeMi3] I. Fern´andez, P. Mira, Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space. Trans. Amer. Math. Soc.,361, no 11, (2009), 5737– 5752. [FoWo] A.P. Fordy, J.C. Wood. Harmonic maps and integrable systems. Aspects of Mathematics, vol. E23, by Vieweg, Braunschweig/Wiesbaden, 1994. [GMM] J.A. G´alvez, A. Mart´ınez, P. Mira, The Bonnet problem for surfaces in homogeneous 3-manifolds, Comm. Anal. Geom. 16 (2008), 907–935. [GaRo] J.A. G´alvez, H. Rosenberg, Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface. Preprint, 2008, arXiv:0807.0997. [Ha] L. Hauswirth, Minimal surfaces of Riemann type in three dimensional product manifolds. Pacific J. Math.,224, no.1 (2006), 91–117. [HaRo] L. Hauswrith, H. Rosenberg. Minimal surfaces of finite total curvature in H×R. Mat. Contemp. 31 (2006), 65–80. [HRS] L. Hauswirth, H. Rosenberg, J. Spruck. On complete mean curvature 1 2surfaces in H2×R.Comm. Anal. Geom.,16, no.5 (2008), 989–1005. [HoMe] D. Hoffman, W. H. Meeks III. The strong halfspace theorem for minimal surfaces. Invent. Math. 101, no.2 (1990), 373–377. [HsHs] W.Y. Hsiang, W.T. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces I, Invent. Math. 98 (1989), 39–58. [Lee] H. Lee. Extension of the duality between minimal surfaces and maximal surfaces. Preprint, 2009. [MaPR] M. Manzano, J. P´erez, M. Rodr´ıguez. Parabolic stable surfaces with constant mean curvature. Preprint, 2009, arXiv:0910.5373. [MPR] W. H. Meeks III, J. P´erez, A. Ros. Stable constant mean curvature surfaces, Handbook of Geometric Analysis no.1 (2008). [Mee] W.H. Meeks. Constant mean curvature surfaces in homogeneous 3-manifolds. Preprint (2009). [MeRo1] W.H. Meeks, H. Rosenberg, The theory of minimal surfaces in M×R,Comment. Math. Helv. 80 (2005), 811–858. [MeRo2] W.H. Meeks, H. Rosenberg, Stable minimal surfaces in M×R,J. Differential Geom. 68 (2004), 515–534.