The Gauss map of surfaces in PSL˜2(R)

Calc. Var. (2015) 52:507–528 DOI 10.1007/s00526-014-0721-1 Calculus of Variations The Gauss map of surfaces in  PSL2(R) Benoît Daniel ·Isabel Fernández ·Pablo Mira Received: 10 May 2013 / Accepted: 1 February 2014 / Published online: 8 March 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract We defineaGaussmap forsurfacesin theuniversalcoveroftheLiegroupPSL2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is not related to the Lie group structure. We prove that the Gauss map of a nowhere vertical surface of critical constant mean curvature (CMC) is harmonic into the hyperbolic plane H2and we obtain a Weierstrass-type representation formula. This extends results in H2×Rand the Heisenberg group Nil3, and completes the proof of existence of harmonic Gauss maps for surfaces of critical CMC in any homogeneous manifold diffeomorphic to R3with isometry group of dimension at least 4. Mathematics Subject Classification (2010) Primary 53C42 ·53C30; Secondary 53A10 · 53C43 Communicated by F. Helein. B. Daniel (B) UMR 7502, Institut Élie Cartan de Lorraine, Université de Lorraine, CNRS, B.P. 70239, 54506 Vandœuvre-lès-Nancy Cedex, France e-mail: [email protected] B. Daniel School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea I. Fernández Departamento de Matemática Aplicada I, Universidad de Sevilla, 41012 Sevilla, Spain e-mail: [email protected] P. Mira Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30203 Cartagena, Murcia, Spain e-mail: [email protected] 123 508 B. Daniel et al. 1 Introduction An important property in the theory of constant mean curvature (CMC) surfaces is that the Gauss map (i.e., the unit normal vector viewed as a map into the 2-sphere S2)ofaCMC surface in R3is a harmonic map into S2. If the surface in R3is minimal, the Gauss map is holomorphic. When we consider an immersed oriented surface in a Riemannian three-manifold M, its unit normal vector is a map N:→UMthat takes values in the five-dimensional unit tangent bundle UMof M. It is then unclear how to define for aGaussmapintoS2that provides relevant information about the surface. The construction of such a Gauss map will depend on the geometric properties of M, and in general the Gauss map will not be harmonic for CMC surfaces in M. Nonetheless, for some ambient spaces Msuch a Gauss map has been constructed, and has provided key information about the global properties of certain classes of surfaces in M. An illustration of this fact is the case of CMC surfaces in hyperbolic 3-space H3.Thereis a well known way to define a “hyperbolic Gauss map” for surfaces in H3[4,11] taking values in the asymptotic boundary ∂∞H3≃¯ Cwhere ¯ C=C∪{∞}is the Riemann sphere. Bryant [4] proved that this hyperbolic Gauss map is holomorphic for CMC 1 surfaces in H3and he obtained a Weierstrass-type representation formula for these surfaces; these techniques were then developed by Umehara and Yamada [29] and many other authors, leading to very important improvements in the study of these surfaces. See also [12,18] for other applications of the hyperbolic Gauss map. Recently, much attention was drawn to surfaces in simply connected homogeneous 3- manifolds. The homogeneous spaces diffeomorphic to R3with a 4-dimensional isometry group constitute a two-parameter family; they will be denoted by E3(κ, τ) where κ⩽0and (κ, τ) = (0,0). When τ=0 we get the product space H2(κ) ×R;whenκ=0 we obtain the 3-dimensional Heisenberg group Nil3endowed with a left-invariant metric; otherwise we obtain the universal cover of the Lie group PSL2(R)endowed with a left-invariant metric; we will denote it by  PSL2(R). Thus, we see that the metrics of H2×Rand Nil3arise as limits of certain left invariant metrics on  PSL2(R). The second and third authors [14] defined a “hyperbolic Gauss map” taking values in the hyperbolic plane H2for surfaces in H2×Rhaving a regular projection on H2; they moreover proved that for CMC 1/2 surfaces this Gauss map is harmonic, and they obtained a Weierstrass-type representation formula for these surfaces. Then, in Nil3,the first author [7] studied a Gauss map defined using the Lie group structure and obtained obtained a Weierstrass-type representation formula for minimal surfaces in Nil3.These two Gauss maps have quite different definitions and properties. Moreover, the use of these Gauss maps has led to important discoveries in their respective theories, such as solutions to Bernstein type problems, classification of complete multigraphs or half-space theorems [8,15]. Let us indicate that minimal surfaces in R3and Nil3, CMC 1 surfaces in H3and CMC 1/2 surfaces in H2×Rall have critical CMC. That is, their mean curvature equals the value csuch that there exist compact CMC Hsurfaces respectively in R3, Nil3,H3,H2×Rif and only if |H|>c. By the same definition, the value cfor the critical mean curvature in  PSL2(R)=E3(κ, τ) for κ<0andτ= 0, is c=√−κ/2. Moreover, CMC surfaces with critical mean curvature in H2×R, Nil3and  PSL2(R)are related by a Lawson-type correspondence [6]. 123 The Gauss map of surfaces in PSL2(R)509 The aims of this paper are: •to define geometrically a smooth Gauss map for smooth surfaces in PSL2(R)taking values in a 2-sphere, in a way that does not depend on a choice of model and such that the Gauss map is “compatible” with ambient isometries (in a sense that will be made precise), •to try to unify the definitions of the Gauss map of surfaces in H2×R, PSL2(R)and Nil3, •to prove that the Gauss maps of CMC local graphs with critical mean curvature in PSL2(R) are harmonic maps into H2(local graphs will be defined in Sect. 2). Let us first eliminate two approaches that could seem natural to solve this problem. The first approach would be to define a left-invariant Gauss map using the Lie group structure, as it is implicitely done in R3and as it was done in Nil3[7](seealso[10]). Isometries coming from the Lie group structure in H2×Rand  PSL2(R)do not have a geometric characterization (see Remark 3.10)asinR3and Nil3(where they are “translations”). Moreover, the homogeneous manifold  PSL2(R)is isometric to another Lie group with a left-invariant metric (see Theorem 2.14 and Corollary 3.19 in [26] for a discussion of these Lie group structures), so defining a left-invariant Gauss map would depend on the choice of a Lie group structure. These left-invariant Gauss maps do not relate well to isometries. One can compute that, for surfaces with critical CMC, they satisfy some second order elliptic partial differential equations that are not harmonic map equations. The second approach would be to define a Gauss map taking values in the asymptotic boundary of the ambient manifold, as it is again implicitly done in R3and as it was done in H3.However,eveninH2×Rthis is not suitable: indeed, H2×Radmits no differentiable Hadamard compactification [22], and hence the Gauss map of a smooth surface would not necessarily be smooth. The paper is organised as follows. In Sect. 2, we introduce some preliminary material about the manifolds E3(κ, τ). In Sect. 3, we prove the existence and the uniqueness of a Gauss map satisfying some geometric conditions; this Gauss map reaches the first two of the abovementioned aims of the paper. In Sect. 4, we gather some preliminary computations related to conformal immersions and their Gauss map. Section 5is devoted to the Gauss map of surfaces with critical CMC: we prove that it is harmonic into H2for local graphs and we obtain a Weierstrass-type representation theorem; hence this Gauss map reaches the third of the above mentioned aims. We also relate the Gauss map of the surface with the Gauss map of its sister minimal surface in Nil3. In Sect. 6we give the expression of the Gauss map using the Lorentzian model for H2. 2 The manifolds E3(κ, τ) The simply connected homogeneous Riemannian 3-manifolds with a 4-dimensional isometry group constitute, together with Euclidean 3-space and round 3-spheres, a 2-parameter family (E3(κ, τ))(κ,τ)∈R2, such that there exists a Riemannian fibration π:E3(κ, τ ) →M2(κ) with bundle curvature τ,whereM2(κ) is the simply connected surface of curvature κ.The fibration is a product fibration if and only if τ=0. We refer to [6] and references therein for more details. There are different types of manifolds according to the values of κand τ. In this paper we will only consider the case where κ⩽0. 123 510 B. Daniel et al. The manifold E3(κ, τ) is diffeomorphic to R3and isometric •to Euclidean space R3when (κ, τ) =(0,0); in this case the isometry group has dimension six, and the fibration πis not unique, •to the 3-dimensional Heisenberg group endowed with a left-invariant metric when κ=0 and τ= 0; we will denote it by Nil3(τ) (up to dilations, all these metrics are isometric), •to H2(κ) ×Rwhen κ<0andτ=0, where H2(κ) is the hyperbolic plane of constant curvature κ, •to the universal cover of PSL2(R)endowed with certain left-invariant metrics when κ<0 and τ= 0; we will denote it by  PSL2(R)(up to dilations, all these metrics are isometric to one such that κ−4τ2=−1). A vector vis said to be vertical if dπ(v) =0andhorizontal if it is orthogonal to vertical vectors. We let ξbe a unit vertical field in E3(κ, τ) (the field ξis unique up to multiplication by −1). We say that a vector vis upwards pointing (respectively, downwards pointing)if v,ξ>0 (respectively, v,ξ<0). We set c:= √−κ 2. This constant is called the critical mean curvature because there exist compact CMC H surfaces in E3(κ, τ ) if and only if |H|>c. When κ<0, horocylinders, i.e. inverse images by πof horocycles of H2(κ),haveCMCc. When κ=0, vertical planes, i.e. inverse images by πof straight lines of R2, are minimal, and hence have CMC c. In particular, any E3(κ, τ) with κ⩽0 can be foliated by topological planes of critical CMC, all of them congruent to each other. A surface is said to be nowhere vertical (or a local graph)ifξis nowhere tangent to it, that is, if the restriction of πto the surface is a local diffeomorphism (in H2(κ) ×Rthis means that the surface has a regular projection on H2(κ)); it is said to be an entire graph if the restriction of πto the surface is a global diffeomorphism onto M2(κ).Theangle function of an oriented surface is the function N,ξwhere Nis the unit normal vector field to . Hence, a surface is nowhere vertical if and only if its angle function does not vanish. CMC local graphs, and in particular entire graphs, play an important role among CMC surfaces; see for instance [8,15,20,21,23–25]. For ρ>0weset D(ρ) ={z∈C;|z|<ρ}. We also set D=D(1). We use the following model for the hyperbolic plane of constant curvature κ: H2(κ) =D2 √−κ endowed with the metric given in canonical coordinates (x1,x2)by 2(dx2 1+dx2 2) where =1 1+κ 4(x2 1+x2 2)=1 1−c2|ζ|2 123 The Gauss map of surfaces in PSL2(R)511 with ζ=x1+ix2. The model we use for E3(κ, τ) is D2 √−κ×Rwhen κ<0orR3when κ=0, endowed with the metric given in canonical coordinates (x1,x2,x3)by 2(dx2 1+dx2 2)+(τ(x2dx1−x1dx2)+dx3)2. In this model we have π(x1,x2,x3)=(x1,x2). We consider the following orthonormal frame: V1=1  ∂ ∂x1−τx2∂ ∂x3,V2=1  ∂ ∂x2+τx1∂ ∂x3,V3=∂ ∂x3=ξ. (2.1) The fields V1and V2are the horizontal lifts in E3(κ, τ) by the fibration πof the fields 1  ∂ ∂x1 and 1  ∂ ∂x2in M2(κ). 3 Definition of the Gauss map In this section we set κ<0. We will define a “geometric” Gauss map for surfaces in E3(κ, τ) that coincides with the hyperbolic Gauss map [14] for graphs in H2(κ) ×Rwhen τ=0 and such that, by taking a limit when κ→0, we get the Gauss map defined for surfaces in Nil3using the Lie group structure studied in [7]. For simplicity we set E=E3(κ, τ ). We denote by Isom0(E)the connected component of the identity in the isometry group of E. All isometries in Isom0(E)leave the field ξinvariant, since the sectional curvature of the plane ξ⊥is a strict minimum (see [6] for more details); as a consequence, a fiber is mapped to another fiber. An isometry f∈Isom0(E)induces via πan orientation-preserving isometry ˜ fof H2(κ), i.e., ˜ f◦π=π◦f; we will say that ˜ fis the horizontal part of f. Moreover, the field ξis a Killing field; the isometries it generates are called vertical translations and their horizontal part is the identity. Let SU1,1=αβ ¯ β¯α;(α, β) ∈C2,|α|2−|β|2=1, PSU1,1=SU1,1/{±I}. We let Sdenote the Riemann sphere ¯ C=C∪{∞}together with a marked oriented circle Ein ¯ C.WecallEthe equator of S.ThenS\Ehas two connected components: we will call northern hemisphere and denote by S+the one whose oriented boundary is E, and we will call southern hemisphere and denote by S−the one whose oriented boundary is Ewith the opposite orientation. The connected group that acts naturally on Sis the group of (orientation-preserving) conformal diffeomorphisms of ¯ Cthat leave Einvariant and preserve the orientation of E.This group is PSU1,1≃PSL2(R), which is also the group of orientation-preserving isometries of 123 512 B. Daniel et al. H2(κ). Hence to each M∈SU1,1we associate a conformal diffeomorphism ψM:S→S as above and an orientation-preserving isometry ϕM:H2(κ) →H2(κ),sothat ψMN =ψM◦ψN,ϕ MN =ϕM◦ϕN. In this way we obtain a unique conformal diffeomorphism σ:H2(κ) ∪∂∞H2(κ) →S+∪E such that σ◦ψM=ϕM◦σ for all M. Without loss of generality we can assume that Eis the circle |z|=1oriented counter clockwise, the northern hemisphere is S+={|z|<1}, the southern hemisphere is S−={|z|>1}∪{∞}, and, for M=αβ ¯ β¯α∈SU1,1, ψMand ϕMare defined by ψM(z):= αz+β ¯ βz+¯α,ϕ M(ζ) := 1 c αcζ+β ¯ βcζ+¯α. In this way, we have σ(z)=z/c. Inspired by the properties of the “hyperbolic Gauss map” of surfaces in hyperbolic 3-space H3, we seek a Gauss map for oriented surfaces in Ewith values in Shaving the following properties: •two surfaces passing through the same point xhavethesameGaussmapatxif and only if they have the same unit normal vector at x, •applyinganisometry fofEbelongingtothe connected component of the identity switches the Gauss map gof the surface to ψM◦g,whereM∈SU1,1corresponds to the horizontal part of f, •the Gauss map at a point lies on the equator E(respectively, in the northern hemisphere) if and only if the unit normal vector at that point is horizontal (respectively, upwards pointing). The first property implies that the Gauss map of an oriented surface factorizes as g=◦Nfor some map :UE→S,whereUEdenotes the unit tangent bundle to Eand N:→UEthe unit normal vector to . Theorem 3.1 There exists a unique map :UE→S (x,Z)→ x(Z) such that (a) for any f ∈Isom0(E),ifM∈SU1,1is such that the horizontal part of f is ϕM,then ◦df=ψM◦, 123 The Gauss map of surfaces in PSL2(R)513 (b) for any (x,Z)∈UE,x(Z)∈E(respectively x(Z)∈S+,x(Z)∈S−) if and only if Z is horizontal (respectively, upwards pointing, downwards pointing), (c) for any point x ∈Eand any horizontal vector Z ∈UxE,x(Z)=σ(p)where p denotes the endpoint of the oriented geodesic of H2(κ) passing through π(x)with dπ(Z) as tangent vector, (d) for any point x ∈E,themapx:UxE→Sis a conformal diffeomorphism. Moreover the map is analytic, and its expression in the frame (V1,V2,V3)is given by x(Z)=Z1+iZ2+cζ(1+Z3) c¯ ζ(Z1+iZ2)+1+Z3 (3.1) where ζ=x1+ix2and Z =Z1V1+Z2V2+Z3V3. Proof We first notice that proving existence and uniqueness of is equivalent to proving that there exists a unique map :UE→Ssatisfying property (a) and properties (b), (c) and (d) at one point, for instance at the origin O=(0,0,0)∈E. We first prove the uniqueness of O. At the origin the frame (V1,V2,V3)is just the coordinate frame. We let s:UOE→Sdenote the stereographic projection with respect to the South Pole, i.e., if Z=Z1V1+Z2V2+Z3V3∈UOE,then s(Z)=Z1+iZ2 1+Z3. So, by property (d) there exists a conformal diffeomorphism h:S→Ssuch that O=h◦s. By property (b), h(S+)=S+. Moreover, property (a) holds for all isometries f∈ Isom0(E)preserving O. Applying this property to the vector Z=V3, we obtain that for all rotations Raround 0 ∈Swe have h(0)=R(h(0)), which implies that his a rotation around 0∈S. Finally, using property (c) we conclude that his the identity, and hence O=s. Observe that it is easy to check that O=ssatisfies (b), (c), (d) at x=Oand also (a) for all isometries f∈Isom0(E)preserving O. We now prove existence and uniqueness of , as well as the announced expression (3.1) for . We consider a point y=(y1,y2,y3)∈Eand an isometry f∈Isom0(E)such that f(y)=O.To fis associated a matrix M=αβ ¯ β¯α∈SU1,1such that β=−αcw where w=y1+iy2.Let Z=Z1V1(y)+Z2V2(y)+Z3V3(y)∈UyE and set df(Z)=Y1V1(0)+Y2V2(0)+Y3V3(0). Since isometries in Isom0(E)preserve the field ξ,wehavedf(Z), ξ(O)=Z,ξ(y), i.e., Y3=X3. On the other hand, since ϕMis holomorphic, using complex notation in H2(κ) we have dϕM=ϕ M(dx1+idx2). Consequently, since (0)=1and dπ(df(Z)) =dϕM(dπ(Z)), 123 514 B. Daniel et al. we get Y1+iY2=ϕ M(w) (w) (Z1+iZ2)=α ¯α(Z1+iZ2). From this we deduce that O(df(Z)) =Y1+iY2 1+Y3=α ¯α Z1+iZ2 1+Z3. Then by property (a) we have y(Z)=ψ−1 M(O(df(Z))) =¯αO(df(Z)) −β −¯ βO(df(Z)) +α=Z1+iZ2+cw(1+Z3) c¯w(Z1+iZ2)+1+Z3. Let us observe that this expression does not depend on the choice of the isometry f; this is because Osatisfies (a) for all isometries in Isom0(E)preserving O.Thisgivesthe uniqueness of as well as the announced expression, from which we also deduce analyticity. Conversely, by construction this map satisfies (a), (b), (c) and (d).  Definition 3.2 Let be an oriented surface in E. Then the Gauss map of is the map ◦N:→Swhere Nis the unit normal vector to . Remark 3.3 Though we have used a specific model for Eto prove Theorem 3.1, its statement about existence and uniqueness of :UE→Ssatisfying conditions (a)–(d) does not involve a model for Ebut only needs the choice of an isomorphism between the group of orientation-preservingisometries ofH2(κ) andthe groupof orientation-preservingconformal diffeomorphisms of Sthat leave Einvariant and preserve the orientation of E. In particular, Definition 3.2 is also independent of the model chosen for E. If we consider the specific model for Ewe are working with in this paper, we get the following expression for the Gauss map. Corollary 3.4 If N =N1V1+N2V2+N3V3, then the Gauss map of is g=N1+iN2+cζ(1+N3) c¯ ζ(N1+iN2)+1+N3 (3.2) with ζ=x1+ix2. Remark 3.5 In the case of H2×R, the hyperbolic Gauss map defined in [14]forCMC1/2 graphs can be written as ◦Nwith satisfying conditions (a)–(d) in Theorem 3.1,soit coincides with g. Remark 3.6 The formula given in Corollary 3.4 extends for κ=0, and in this case we get the Gauss map studied in [7]. Hence formula (3.2) provides a unified treatment for Gauss maps in all manifolds E3(κ, τ) with κ⩽0. Example 3.7 Let γbe a curve of constant curvature kin H2(κ) and let =π−1(γ ) ⊂E. Then, by property (b) of Theorem 3.1, the image of the Gauss map of lies in is the equator E, since the normal to is horizontal. When |k|>√−κ, it is the whole E;when|k|=√−κ (in which case is a horocylinder), it is a point in Eor its complement (depending on the orientation); when |k|<√−κ, it is an open arc in E. More generally, the Gauss map of a surface ⊂EtakesvaluesinEif and only if the normal to is everywhere horizontal, i.e., if and only if π() ⊂H2(κ) is a curve. 123 The Gauss map of surfaces in PSL2(R)515 Remark 3.8 By property (b) of Theorem 3.1, the Gauss map of a surface ⊂Etakes values in Dif and only if is a nowhere vertical surface with upwards pointing normal. Remark 3.9 Let r:E→Ebe the map defined by r(x1,x2,x3)=(x1,−x2,−x3).Thismap is an isometry of E(but does not lie in Isom0(E)); it is the rotation of angle πaround the x1-axis, which is a horizontal geodesic. Let x∈Eand X∈UxE.Then r(x)(dr(X)) =1 x(X). Indeed, this follows from (3.1) with (ζ, X1,X2,X3)replaced by (¯ ζ,X1,−X2,−X3). Remark 3.10 The homogeneous manifold  PSL2(R)is isometric to the universal cover of UH2(κ) endowed with a Sasaki metric (see [6] for details). Via this identification, the Riemannian fibration πis the canonical projection onto H2(κ), and isometries of  PSL2(R) coming from the Lie group structure (i.e., left-multiplications) are of the form (f,df)where fis a direct isometry of H2(κ). For example, among all screw motions around the x3-axis in  PSL2(R), those coming from the Lie group structure are exactly those whose pitch has a particular value that depends on κand τ. Vertical translations do not come from the Lie group structure. 4 Preliminary calculations This section is devoted to preliminary equations for conformal immersions in E=E3(κ, τ) where κ⩽0 and their Gauss map. 4.1 Expression of the connection of E First we compute the connection ˆ ∇of Ein the frame (V1,V2,V3).Wehave [V1,V2]=−κ 2x2V1+κ 2x1V2+2τV3,[V1,V3]=[V2,V3]=0, and so ˆ ∇V1V1=κ 2x2V2,ˆ ∇V2V1=−κ 2x1V2−τV3,ˆ ∇V3V1=−τV2, ˆ ∇V1V2=−κ 2x2V1+τV3,ˆ ∇V2V2=κ 2x1V1,ˆ ∇V3V2=τV1, ˆ ∇V1V3=−τV2,ˆ ∇V2V3=τV1,ˆ ∇V3V3=0. (4.1) 4.2 Conformal immersions Let X=(x1,x2,x3):→Ebe a conformal immersion from a Riemann surface into E. We shall denote by N:→UEits unit normal. If we fix a conformal coordinate z=u+iv in ,thenwehave Xz,X¯z=λ 2>0,Xz,Xz=0, where λis the conformal factor of the metric with respecto to z. Moreover, we will denote the coordinates of Xzand Nin the frame (V1,V2,V3)by 123 522 B. Daniel et al. However, conversely, if is simply connected and g:→¯ Csatisfies (5.1) without assuming that |g|<1, then the representation formula (5.2) only locally defines a possibly branched immersion away from points where |g|=1 (see Remark 4.2 in [7]). Remark 5.3 When κ<0, if Xis a nowhere vertical CMC immersion with critical mean curvature, it may not be possible to orient it so that its unit normal is upwards pointing, since the orientation is chosen so that the mean curvature is positive. However it is not a loss of generality to assume that the unit normal is upwards pointing. Indeed, assume that Xhas a downwards pointing unit normal and let rthe rotation of angle πaround the x1-axis (see Remark 3.9). This map ris an isometry of E, and hence r◦Xis a nowhere vertical CMC immersion with critical mean curvature cand upwards pointing normal. Then Theorem 5.1 applies to r◦X. Also, if gdenotes the Gauss map of X, then the Gauss map of r◦Xis 1/g. Remark 5.4 Theorem 5.1 implies that, when κ<0 (i.e., in H2×Rand  PSL2(R)), there generically exists a two-parameter family of non-congruent CMC immersions with critical mean curvature sharing the same Gauss map. Indeed, in Theorem 5.1,foragivenz0∈one can prescribe ζ(z0)and x3(z0); changing x3(z0)simply corresponds to a vertical translation, but changing ζ(z0)gives a new non-congruent immersion unless the Gauss map has some symmetry, by property (a) in Theorem 3.1.InH2×Rthis was noticed in Remark 5 in [14]. When κ=0 (i.e., in Nil3) this is not the case anymore (see Sect. 6 in [7]). 5.2 The Hopf differential One can associate two natural holomorphic quadratic differentials to the immersion X: •the Hopf differential Q(g)dz2of g(in the sense of harmonic maps into Dendowed with the hyperbolic metric): Q(g):= 4gz¯gz (1−|g|2)2, •the Abresch–Rosenberg differential dz2of X[1,2](seealso[13] for its expression): := 2(H+iτ)P−(κ −4τ2)η2 4=2(c+iτ)P+(c2+τ2)η2. Proposition 5.5 We have Q(g)=−. Proof By (5.3)wehave Q(g)=−(c+iτ) (1−c2|ζ|2) (¯g−c¯ ζ)(1−cζ¯g)η¯gz. We now compute . First we compute P, starting from the identity (4.4) and using (4.5), (4.6), (4.7)and(4.11): P=λ ¯η2¯ GGz+G¯ Gz (1+|G|2)2−(H−iτ)η 2 =η 2|G|2U(G,ζ)η 4+G¯ Gz−(H−iτ)(1+|G|2)2η2 8|G|2 =iτ 2η2+c2η2 4¯ G(¯ ζ+¯ G2ζ)+¯ Gzη 2¯ G. 123 The Gauss map of surfaces in PSL2(R)523 Then, using (4.18)and(4.16), we get P=iτ 2η2+c2η2 2¯ ζ+¯g2ζ (¯g−c¯ ζ)(1−cζ¯g)−c¯gη2 2 1+c2|ζ|2 (¯g−c¯ ζ)(1−cζ¯g)+η 2 1−c2|ζ|2 (¯g−c¯ ζ)(1−cζ¯g)¯gz =−c−iτ 2η2+η 2 1−c2|ζ|2 (¯g−c¯ ζ)(1−cζ¯g)¯gz. Finally we get =−Q(g). 5.3 The Gauss map of the sister minimal surface in Nil3(ˆτ) Let X:→E3(κ, τ) be a CMC immersion with critical mean curvature. Assume that is simply connected. Then, by the Lawson-type correspondence in [6], there exists a minimal immersion ˆ X:→Nil3(ˆτ) =E3(0,ˆτ) where ˆτ2=τ2+c2that is isometric to X. Moreover, the immersions Xand ˆ Xhave the same angle function (see Sect. 2)andtheir associated Abresch–Rosenberg differentials and ˆ are related by =e−2iθˆ ,where τ+ic =eiθˆτ(see for instance the end of Sect. 3 in [19] and the expression of the Abresch– Rosenberg differential given in Sect. 5.2). A pair of such isometric surfaces are called sister surfaces. The minimal sister surface in Nil3(ˆτ) of a CMC surface in E3(κ, τ) with critical mean curvature is unique up to direct isometries of Nil3(ˆτ). If g:→Dis a harmonic map for the hyperbolic metric on D,weconsidertheform μ(g)|dz|2(which is independent of the choice of the conformal parameter z)where μ(g):= 4(|gz|2+|g¯z|2) (1−|g|2)2. Two harmonic maps g1and g2are said to be associate if there exists a real constant θsuch that Q(g2)dz2=e−2iθQ(g1)dz2and μ(g2)|dz|2=μ(g1)|dz|2. We now consider a conformal immersion X:→E3(κ, τ) from a simply connected Riemann surface into E3(κ, τ ) such that X() is a local graph with positive angle function. We let ˆ X:→Nil3(ˆτ) be its sister minimal immersion. Let gbe the Gauss map of Xand ˆgthe Gauss map of ˆ X(which is unique only up to rotations around 0 ∈D, see Sect. 6 in [7]). Proposition 5.6 The Gauss maps g and ˆg are associate and Q(g)=e−2iθQ(ˆg)with θ defined by τ+ic =eiθˆτ. (5.8) Proof We keep for the immersion Xthe same notation as in the previous sections. By Proposition 5.5 we have Q(g)=−, and similarly we have Q(ˆg)=−ˆ where ˆ is theAbresch-Rosenbergdifferentialof ˆ X.Since thedifferentialsand ˆ satisfy=e−2iθˆ , we get Q(g)=e−2iθQ(ˆg). We now prove that μ(g)=μ( ˆg). Since sister surfaces have the same angle function, it follows from (4.5) and the analogous formula in Nil3(ˆτ) (see Definition 3.1 in [7]) that |G|=|ˆg|.(5.9) Denote by (ˆg,ˆη) the Weierstrass data of ˆ X(see Defintion 3.2 in [7]). By Eq. (10) in [7] (or by (4.11) with (η, H,τ,c)replaced by (ˆη, 0,ˆτ,0))wehave ˆη=4i ˆτˆgˆgz (1−|ˆg|2)2.(5.10) 123 524 B. Daniel et al. On the other hand we have (see for instance the end of Sect. 3 in [19]) η=e−iθˆη. (5.11) So using (5.3)weget gz=τ−ic τ+ic (1−c2|ζ|2)(1−|g|2)2 (¯g−c¯ ζ)(1−cζ¯g)ˆgˆgz (1−|ˆg|2)2 and so, using (5.8), (5.9)and(4.15), gz 1−|g|2=e−2iθˆgz 1−|ˆg|2ˆg(1−c¯ ζg) ¯g−c¯ ζ,(5.12) which implies, using (5.9)and(4.14), |gz| 1−|g|2=|ˆgz| 1−|ˆg|2.(5.13) Then (5.13) and the fact that |Q(g)|=|Q(ˆg)|imply that |g¯z| 1−|g|2=|ˆg¯z| 1−|ˆg|2.(5.14) Finally, we deduce from (5.13)and(5.14)thatμ(g)=μ( ˆg).Thisprovesthatgand ˆgare associate.  Remark 5.7 Formula (5.12) gives the expression of ζin terms of gand ˆgwithout the need to integrate a differential system, contrarily to the representation formula (5.2). We will use (5.12) to explicitely compute Example 5.9. We recall the followingresultfrom[9](seeCorollaries 4.6.3, 4.6.4 there; see also Theorem 6.10 in [16]). Proposition 5.8 The following conditions are equivalent for a surface X() of critical CMC in E3(κ, τ): (1) X() is an entire graph, (2) X() is a complete local graph. In particular, the sister correspondence preserves entire graphs of critical CMC (see Sect. 2 for the definition). Proof That (1)implies (2)is trivial. Conversely, let X() be a complete local graph in E3(κ, τ), which we will assume to be simply connected by passing to its universal covering if necessary. Let ds2denote the metric induced by Xon and νthe angle function of X(). Let ˆ X() denote the sister surface of Xin Nil3. Then, ds2and νcoincide with the metric induced by ˆ Xon , and with the angle function of ˆ X(), respectively. Then, in particular, ˆ X() is a complete local graph in Nil3. Consequently, by Theorem 3.1 in [7], ˆ X() is an entire graph in Nil3. By Corollary 13 in [15] (which uses a result from [5]), the metric ν2ds2 is complete, and so by Lemma 9 in [15] the surface X() is an entire graph in E3(κ, τ).This shows that (2)implies (1)and completes the proof.  123 The Gauss map of surfaces in PSL2(R)525 Example 5.9 (Surface with conformal Gauss map) We seek a local graph with critical CMC whose Gauss map g:→Dis conformal, i.e., such that the Hopf differential of gvanishes identically. Then ˆgis also conformal, so the surface will be the sister surface of that of Example 8.1 in [7]. Up to a reparametrization we may assume that =Dand that ˆg(z)=z. Then up to an isometry of E3(κ, τ) we can assume that g(z)=z. Then by (5.12)weget ζ=e2iθ−1 c z e2iθ|z|2−1. Using (5.10), (5.11)and(5.8)weget η=4i τ+ic ¯z (1−|z|2)2. Next, we compute that ζz=1−e2iθ c(e2iθ|z|2−1)2,¯ ζz=(1−e−2iθ)e−2iθ c¯z2 (e−2iθ|z|2−1)2, 1−c2|ζ|2=(1−|z|2)2 (e2iθ|z|2−1)(e−2iθ|z|2−1). Then we compute using (5.6)that (x3)z=2i τ+ic −iτ(1−e2iθ)(1−e−2iθ) 2c2|z|4−2|z|2e−2iθ+1 (e2iθ|z|2−1)(e−2iθ|z|2−1)¯z (1−|z|2)2. By (5.8)wehave(1−e2iθ)(1−e−2iθ)=4c2/(τ2+c2),soweget (x3)z=2 τ2+c2 c(|z|4+1)+2(τ sin(2θ) −ccos(2θ))|z|2 (1−|z|2)2(e2iθ|z|2−1)(e−2iθ|z|2−1)¯z =2c τ2+c2 (1+|z|2)2¯z (1−|z|2)2(|z|4−2|z|2cos(2θ)+1), and finally, up to a vertical translation, x3=−τ c2arctan |z|2−cos(2θ) sin(2θ) +2 c(1−|z|2). Hence we obtain a surface of revolution about the x3-axis; this surface is also an entire graph and x3→+∞as |ζ|→1/c. This surface is described in [28] (item 2 of Theorem 2 with 4H2+κ=0andE=0). In H2(κ) ×R(i.e., when τ=0), this is the surface described on p. 1172 in [14](seealso[1]). Example 5.10 (Surfaces with singular Gauss map) We seek complete local graphs with critical CMC whose Gauss map g:→Dis singular on an open set of .Theng() and ˆg() are geodesics of Dfor the hyperbolic metric, and these surfaces will be sister surfaces of those of Example 8.2 in [7]. These minimal surfaces in Nil3constitute a oneparameter family of surfaces that are not isometric one to another, the parameter being the hyperbolic distance between ˆg() and the origin 0 ∈D. Since sister surfaces are isometric, we get a one-parameter family of CMC surfaces in E3(κ, τ) that are not isometric one to another. Since their sister minimal surfaces in Nil3are invariant by a one-parameter family of translations, a standard argument using the integrability equations for surfaces in E3(κ, τ ) [6] 123 526 B. Daniel et al. shows that these surfaces are invariant by a one-parameter family of isometries of E3(κ, τ) (see Proposition 4.3 in [19]). Moreover, since their sister minimal surfaces in Nil3are entire graphs, it follows from Proposition 5.8 that these surfaces are also entire graphs. In H2(κ)×R (i.e., when τ=0), these are the surfaces described in Proposition 18 in [14](seealso[27]). 6 The Gauss map in the Lorentzian model In this section we assume that κ<0 and we will show an alternative expression for the Gauss map of nowhere vertical surfaces when we consider the Lorentzian model for the hyperbolic space, that is, we now consider π:E→H2(κ),where H2(κ) ={p=(p0,p1,p2)∈L3|p,p=1/κ, p0>0}⊂L3. Here p0denotes the timelike coordinate in Lorentz space L3and p,p=−p2 0+p2 1+p2 2. This construction is inspired by the one given in [14] for surfaces in H2×R. In particular, it agrees with that one when τ=0. Theorem 6.1 Let X :→Ebe a nowhere vertical surface oriented so that its unit normal vector field N points upwards. Define X∗=π◦X:→H2(κ) ⊂L3,N ∗=dπ(N): →L3and ν=N,ξ. Then the map ˜g:→H2=H2(−1)given by ˜g=1 ν(2cX∗+N∗). agrees with the Gauss map of X as defined in Corollary 3.4. Proof We first observe that ˜gactually takes values in H2. Indeed, since N∗takes values in TX∗(·)H2(κ) =X∗(·)⊥⊂L3,we have N∗,X∗=0. Also, since πis a Riemannian submersion, we have N,N=N∗,N∗+N,ξ2, from where we get N∗,N∗=1−ν2, and so ˜g,˜g=−1. Finally, since Npoints upwards, νis positive, and therefore ˜g,X∗= −1/(2cν) < 0, implying that the timelike coordinate of ˜gis positive. Let F:D→H2be the natural isometry given by F(w) =1 1−|w|21+|w|2,2Re(w), 2Im(w) and ˜ :UE→H2the map so that ˜g=˜ ◦N,thatis,forx∈Eand Z∈UxE, ˜ (x,Z)=˜ x(Z)=1 Z3(2cx∗+Z∗), where x∗=π(x)∈H2(κ),Z∗=dxπ(Z)and Z3=Z,ξ. Let us prove that ˜ =F◦,whereis the map given in Theorem 3.1.Since ˜ satisfies hypothesis (a) in above mentioned theorem, following the ideas of the proof of this result, it suffices to check that both maps agree at the origin O=1 2c,0,0,0∈E. Since Z∗=dOπ(Z)∈T1 2c,0,0H2=1 2c,0,0⊥⊂L3, then its timelike coordinate vanishes, and so ˜ O(Z)=1 Z3(1,Z1,Z2). On the other hand, since O(Z)=1 1+Z3(Z1+iZ2), it is straightforward to check that ˜ O(Z)=F(O(Z)). 123 The Gauss map of surfaces in PSL2(R)527 Remark 6.2 It is possible then to follow the computations in [14] and find an alternative Weierstrass-type representation for surfaces of critical CMC in Ein terms of the Gauss map, using this model. This method uses the fact that a nowhere antiholomorphic harmonic map into H2is (locally) the Gauss map of a CMC 1/2 surface in L3[3]. In this setting, one first needs to integrate a differential system to obtain η=2Xz,ξand then X∗is obtained from ηand the harmonic map without integration. This gives a representation formula equivalent to (5.2) but that differs significantly from it. Remark 6.3 The condition for being the Gauss map in H2×Rthat we obtain in the present paper (being nowhere antiholomorphic) is different from the one obtained in [14](theexistence of Weierstrass data). 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