Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces
aa r X i v : . [ m a t h . DG ] J u l ORIENTABILITY OF LINEAR WEINGARTEN SURFACES,SPACELIKE CMC-1 SURFACES AND MAXIMAL SURFACES
MASATOSHI KOKUBU AND MASAAKI UMEHARA
Dedicated to Professor Koichi Ogiue on the occasion of his seventieth birthday
Abstract.
We prove several topological properties of linear Weingarten sur-faces of Bryant type, as wave fronts in hyperbolic 3-space. For example, weshow the orientability of such surfaces, and also co-orientability when they arenot flat. Moreover, we show an explicit formula of the non-holomorphic hy-perbolic Gauss map via another hyperbolic Gauss map which is holomorphic.Using this, we show the orientability and co-orientability of CMC-1 faces (i.e.,constant mean curvature one surfaces with admissible singular points) in deSitter 3-space. (CMC-1 faces might not be wave fronts in general, but belongto a class of linear Weingarten surfaces with singular points.) Since both linearWeingarten fronts and CMC-1 faces may have singular points, orientability andco-orientability are both nontrivial properties. Furthermore, we show that thezig-zag representation of the fundamental group of a linear Weingarten surfaceof Bryant type is trivial. We also remark on some properties of non-orientablemaximal surfaces in Lorentz-Minkowski 3-space, comparing the correspondingproperties of CMC-1 faces in de Sitter 3-space. Introduction
Let M and N be C ∞ manifolds of dimension 2 and of dimension 3, respectively.The projectified cotangent bundle P ( T ∗ N ) has a canonical contact structure. A C ∞ map f : M → N is called a frontal if f lifts to a Legendrian map L f , i.e.,a C ∞ map L f : M → P ( T ∗ N ) such that the image dL f ( T M ) of the tangentbundle T M lies in the contact hyperplane field on P ( T ∗ N ). Moreover, f is calleda wave front or a front if it lifts to a Legendrian immersion L f . Frontals (andtherefore fronts) generalize immersions, as they allow for singular points. A frontal f is said to be co-orientable if its Legendrian lift L f can lift up to a C ∞ map into thecotangent bundle T ∗ N , otherwise it is said to be non-co-orientable . It should beremarked that, when N is Riemannian and orientable, a front f is co-orientableif and only if there is a globally defined unit normal vector field ν along f . (In[22], ‘fronts’ were implicitly assumed to be co-orientable by definition, and frontswhich are not necessarily co-orientable were distinguished as different, by callingthem ‘p-fronts’. However, in this paper, a front may be either co-orientable ornon-co-orientable and the term ‘p-front’ is not used.) Mathematics Subject Classification.
Primary: 53A35; Secondary: 53C40, 53C42.
Key words and phrases. linear Weingarten, wave front, (co-)orientability, hyperbolic space, deSitter space, zig-zag representation.The first author was supported by Grant-in-Aid for Scientific Research (C) No. 22540100 fromthe Japan Society for the Promotion of Science.The second author was supported by Grant-in-Aid for Scientific Research (A) No. 19204005from the Japan Society for the Promotion of Science.
In [14], G´alvez, Mart´ınez and Mil´an gave a fundamental framework for the theoryof linear Weingarten surfaces of Bryant type in hyperbolic 3-space H . We shallinvestigate such surfaces in H and in de Sitter 3-space S in the category of (wave)fronts which admit certain kinds of singular points. We remark that orientabilityand co-orientability are not equivalent for fronts, though they are equivalent forimmersed surfaces. We prove that non-flat linear Weingarten fronts of Bryant typein H and in S are co-orientable and orientable. Flat fronts, in contrast, areorientable but are not necessarily co-orientable (see [22]). For a co-orientable front f : M → H , a representation of the fundamental group σ f : π ( M ) → Z called the ‘zig-zag representation’ is induced (see Section 3.2), which is invariantunder continuous deformations of fronts. We shall show that the zig-zag repre-sentation of any linear Weingarten front of Bryant type is trivial (see Theorem3.9). Moreover, we give an explicit formula for the hyperbolic Gauss maps of thesesurfaces.On the other hand, we shall also study CMC-1 faces (defined by Fujimori [7]) in S . CMC-1 faces are constant mean curvature one surfaces on their regular sets,and are frontals (see Corollary 4.3), but not fronts in general. In fact, each limitingtangent plane at a singular point contains a lightlike vector. They also belong toa class of linear Weingarten surfaces of Bryant type and satisfy an Osserman-typeinequality (cf. [7] and [9]) under a certain global assumption. As an applicationof an explicit formula for hyperbolic Gauss maps, we prove the orientability andthe co-orientability of CMC-1 faces in S , which is, in fact, deeper than the abovecorresponding assertion for linear Weingarten fronts, because CMC-1 fronts in S admit only isolated singular points (see Corollary 3.4).It should be remarked that CMC-1 faces in S have quite similar properties tomaxfaces (i.e. maximal surfaces in Lorentz-Minkowski 3-space R with admissiblesingular points defined in [29]). In Section 5 (this section is a joint work withShoichi Fujimori and Kotaro Yamada), using the same method as in Section 4, weshow the existence of globally defined real analytic normal vector field on a givenmaxface in R .Throughout this paper, M denotes a smooth and connected 2-dimensional man-ifold or a Riemann surface; R denotes the Lorentz-Minkowski space of dimension4, with the Lorentz metric (cid:10) ( x , x , x , x ) , ( y , y , y , y ) (cid:11) = − x y + x y + x y + x y , and the hyperbolic 3-space H and the de Sitter 3-space S are defined by H (= H ) = n ( x , x , x , x ) ∈ R ; − x + x + x + x = − , x > o ,S = n ( x , x , x , x ) ∈ R ; − x + x + x + x = 1 o , respectively. H is a simply-connected Riemannian 3-manifold with constant sec-tional curvature −
1, and S is a simply-connected Lorentzian 3-manifold with con-stant sectional curvature 1. We occasionally consider H − := n ( x , x , x , x ) ∈ R ; − x + x + x + x = − , x < o isometric to H . RIENTABILITY OF LINEAR WEINGARTEN SURFACES 3 Linear Weingarten surfaces of Bryant type as wave fronts
Let f : M → H ( ⊂ R ) be a front in hyperbolic 3-space H .If f is co-orientable, then there exists a C ∞ map ν : M → S satisfying h f, ν i =0 and h df, ν i = 0, namely, L f = ( f, ν ) : M → T H ⊂ H × S is the Legendrian lift of f . (Here, we identify the unit tangent bundle T H withthe unit cotangent bundle T ∗ H .) We call the C ∞ map ν a unit normal field of f . Interchanging their roles, ν can be considered as a spacelike front in S with f as a unit normal field. Here, a front in S is said to be spacelike if the plane in T ν ( p ) S orthogonal to the normal vector f ( p ) ∈ H is spacelike at each point p .(Conversely, for a given spacelike co-orientable front in S , its unit normal field isa front which lies in the upper half or the lower half of the hyperboloid H ∪ H − of two sheets in R .) For each real number δ , the pair f δ = (cosh δ ) f + (sinh δ ) ν, ν δ = (cosh δ ) ν + (sinh δ ) f gives a new front f δ called a parallel front of f whose unit normal field is ν δ .From now on, we consider a front f : M → H which may not be co-orientable.The following fact implies that even when the front f has a singular point p , takinga co-orientable neighborhood U of p , we can consider a parallel front f U,δ := (cosh δ ) f + (sinh δ ) ν ( δ ∈ R )on U , where ν is a unit normal vector field of f on U . For a suitable δ ∈ R , wemay assume that f U,δ is an immersion at p . Fact 2.1.
Let f : M → H be a front. Then for each p ∈ M , taking a co-orientable neighborhood U of p , the parallel fronts f U,δ are immersions at p exceptfor at most two values δ ∈ R .First, we begin with the case f : M → H is an immersion, which is morerestrictive than the case f is a front. Definition 2.2.
An immersion f : M → H is said to be horospherical linearWeingarten if the mean curvature H (with respect to the normal field ν ) and theGaussian curvature K satisfy the relation a ( H − bK = 0 for some real constants a and b such that a + b = 0. Remark . A horospherical linear Weingarten immersion f : M → H with a = 0is just a flat immersion, hence M is orientable (see [22, Theorem B]). On the otherhand, a horospherical linear Weingarten surface with a = 0 is also orientable bythe following reasons:(1) If f is isoparametric, then it is orientable because of classification of suchsurfaces. In particular, minimal linear Weingarten surfaces orientable sinceit is isoparametric (see [5]).(2) Suppose f is not isoparametric. The opposite unit normal field − ν doesnot give a horospherical linear Weingarten immersion, that is, ( − H ) − K . Thus it cannot happen that M is non-orientable. Remark . The parallel surfaces f δ of a horospherical linear Weingarten immer-sion f are also horospherical linear Weingarten on the set R f δ of regular points of f δ (see [20]). In particular, parallel surfaces of a flat surface are also flat on R f δ ,which is a well known fact. MASATOSHI KOKUBU AND MASAAKI UMEHARA
Fact 2.1 and Remark 2.4 enable us to give the following definition.
Definition 2.5.
A front f : M → H is called a horospherical linear Weingartenfront if for each p ∈ M , there exist a co-orientable neighborhood U of p and δ ∈ R such that the parallel front f U,δ is a horospherical linear Weingarten immersion on U . (Indeed, the parallel front f U,δ is a horospherical linear Weingarten immersionat p with at most two exceptional values of δ ∈ R .)We can give two exceptional examples of horospherical linear Weingarten fronts:(1) a hyperbolic line as a degeneration of a parallel surface of a hyperboliccylinder,(2) a single point as a degeneration of a parallel surface of a geodesic sphere.These two examples have no regular points in their domains of definition. Con-versely, any horospherical linear Weingarten fronts not having regular points arelocally congruent to one of these two examples (see the appendix). So we excludethem from our study in this paper. In other words, we will assume (except in theappendix) that a horospherical linear Weingarten front has regular points .Note that a horospherical linear Weingarten front of zero Gaussian curvature isjust a flat p-front defined in [22]. (See also [23] and [21].) For horospherical linearWeingarten fronts which are not flat, the following assertion holds: Theorem 2.6.
Let f : M → H be a non-flat horospherical linear Weingartenfront which is not isoparametric. Then f is co-orientable. Moreover, there existsa unique ratio [ a : b ] of real constants ( a + b = 0) such that the parallel front f δ satisfies (2.1) a ( H δ −
1) + b δ K δ = 0 , b δ := be δ + a ( e δ − on its regular set R f δ , where H δ and K δ are the mean curvature and the Gaussiancurvature of f δ on R f δ , respectively.Proof. Fix a regular point p ∈ M arbitrarily. Let the equation a ( H −
1) + bK = 0( a + b = 0) hold for f | U with the unit normal field ν ( p ) on a neighborhood U of p . Then the equation (2.1) holds on U ∩ R f δ because of [20, (2.18)].Take another point q close to p so that there exist a neighborhood U q of q and δ ∈ R satisfying U q ∩ U = ∅ and f U q ,δ is a horospherical linear Weingartenimmersion with respect to the unit normal field ν ( q ) . Then f U q ∩ U,δ is horosphericallinear Weingarten with respect both to ν ( p ) and to ν ( q ) . It follows from Remark 2.3that ν ( q ) coincides with ν ( p ) on U q ∩ U . Moreover, since the equation (2.1) holdson U q ∩ U , it must hold on U q .Continuing this argument, we have the collection { ν ( q ) | q ∈ M } which thenfixes a global unit normal field ν . Hence the front f is co-orientable.Finally, we note that the argument above implies the equation (2.1) holds oneach U q ∩ R f δ , therefore on the regular set R f δ . (cid:3) Let f : M → H be a horospherical linear Weingarten front which satisfiesthe relation a ( H −
1) + bK = 0 for some real constants a and b ( a + b = 0).Let ν : M → S denote the unit normal field of f (assuming the co-orientability RIENTABILITY OF LINEAR WEINGARTEN SURFACES 5 whenever f is a flat front). Then f + ν is a map from M to the lightcone Λ in R . Here, the lightcone Λ is, by definition,Λ = n ( x , x , x , x ) ∈ R ; − x + x + x + x = 0 o . Except for the case where a + 2 b = 0, the map f + ν induces a pseudometricof constant Gaussian curvature ε = a/ ( a + 2 b ) (cf. [14], [20]). The equation a ( H −
1) + bK = 0 can be rewritten as 2 ε ( H −
1) + (1 − ε ) K = 0.Let W ε ( M ) denote the set of horospherical linear Weingarten fronts from M to H satisfying ( ε ( H −
1) + (1 − ε ) K = 0 (if ε ∈ R )2( H − − K = 0 (if ε = ∞ ) . • (Flat fronts) A front f ∈ W ( M ) has zero Gaussian curvature, that is, W ( M ) is the set of flat fronts defined on M in H . (In this case, a = 0.)The parallel fronts of a co-orientable flat front f also belong to W ( M ).Fundamental properties of flat fronts are given in [13], [22] and [23]. Theduality between flat surfaces in H and those in S is pointed out in [16]. • (Hyperbolic type) A front f ∈ W ε ( M ) is called a linear Weingarten frontof hyperbolic type if ε > a/ ( a + 2 b ) > f arealso in the same class S ε> W ε ( M ). (cf. [3], [28], etc., for the case ε = 1.)For each f ∈ S ε> W ε ( M ), there exists a unique δ ∈ R such that f δ is aCMC-1 (constant mean curvature one) front in H . In this case, the unitnormal vector field ν δ : M → S is HMC-1 (harmonic-mean curvatureone), that is, the mean of reciprocals of principal curvatures equals one.(See Corollary 3.3 for a related result.) • (de Sitter type) A front f ∈ W ε ( M ) is called a linear Weingarten front of de Sitter type if ε < a/ ( a + 2 b ) < f are alsoin the same class S ε< W ε ( M ). For each f ∈ S ε< W ε ( M ), there existsa unique δ ∈ R such that f δ is HMC-1 (see [20]). In this case, ν δ gives theCMC-1 front in S . (See Corollary 3.4 for a related result.) • (Horo-flat fronts) A front f ∈ W ∞ ( M ) is said to be horo-flat (cf. [17]).(In this case, a + 2 b = 0.) If f ∈ W ∞ ( M ), one of the hyperbolic Gaussmaps (see Section 3) of f degenerates. The parallel fronts of f also belongto W ∞ ( M ). Fundamental properties of horo-flat fronts are given in [17]and [27]. It should be remarked that there exist non-real analytic horo-flatsurfaces (see [2, Example 2.1]). Remark . Our terminology ‘ horospherical linear Weingarten’ comes from thefollowing reasons: The horospheres in H satisfy K = H − W ( M ), W ( M ), W − ( M ) and W ∞ ( M ) at the same time, as the degeneratecases. Recently, Izumiya et al. [15] proposed the horospherical geometry in H ,which includes all of the above classes W ε ( M ).As pointed out at the beginning of this section, if f : M → H is a co-orientablefront, there exists a unit normal field ν defined globally on M , which gives also afront ν : M → S . For f ∈ W ε ( M ), one can verify that the mean curvature ˆ H and the Gaussian curvature ˆ K of ν : M → S satisfy2 ε ( ˆ H −
1) + (1 + ε ) ˆ K = 0 . MASATOSHI KOKUBU AND MASAAKI UMEHARA
This implies that flat fronts, horo-flat fronts, linear Weingarten fronts of hyperbolictype and of de Sitter type in H correspond to those in S , respectively. Definition 2.8.
Linear Weingarten fronts of hyperbolic type, of de Sitter type andflat fronts are all called linear Weingarten fronts of Bryant type (cf. [14]). In otherwords, a linear Weingarten front of Bryant type is a non-horo-flat linear Weingartenfront.It should be remarked that Aledo and Espinar [1] recently classified completelinear Weingarten immersions of Bryant type with non-negative Gaussian curvaturein S . In this paper, almost all formulas are of linear Weingarten surfaces in H ,but one can find the several corresponding formulas of linear Weingarten surfacesin S in [1] (see also [16]).3. Orientability of linear Weingarten fronts of Bryant type and anexplicit formula for the hyperbolic Gauss map G ∗ We now concentrate on linear Weingarten fronts of Bryant type, i.e., fronts in W ε ( M ) ( ε = ∞ ), which will be denoted by f : M → H throughout this section.3.1. A representation formula and singular points.
The first, second andthird fundamental forms are denoted by I , II and III , respectively. (In general, I and III are well-defined not only for immersions but also for fronts, and II is well-defined for co-orientable fronts.) The sum I + III of the first and third fundamentalforms of f is a positive definite metric on M , because f is a front.On the other hand, one can verify that the symmetric 2-tensor εI + (1 − ε ) II for f ∈ W ε ( M ) ( ε = 0 , ∞ ) or for a co-orientable f ∈ W ( M ) is definite on M \ S f and vanishes on S f , where S f denotes the set of singular points of f .We now suppose that M is orientable. Then there is a unique complex structureon the regular set R f := M \ S f so that εI + (1 − ε ) II is Hermitian. Since thiscomplex structure is common to the whole parallel family of f , and the singularset changes when taking a parallel surface (cf. Fact 2.1), it follows that there is aunique complex structure on M so that εI + (1 − ε ) II is Hermitian. Λ ∋ p [ p ] ∈ Λ / ∼ denotes the canonical projection of the positive lightconeΛ = { ( x , x , x , x ) ∈ Λ ; x > } onto the set Λ / ∼ of positively orientedlightlike lines. Note that Λ / ∼ can be naturally identified with the Riemann sphere S ∼ = C ∪ {∞} . The hyperbolic Gauss map G := [ f + ν ] can be defined for f ∈ W ε ( M ) ( ε = 0) and for a co-orientable flat front f .From now until the end of this subsection, we assume that M is orientable(though we will later know that this assumption is not necessary). And also, throughout this section, all flat fronts considered are assumed to be co-orientable. We have already shown that non-flat linear Weingarten surfaces of Bryant type areall co-orientable. On the other hand, even if f : M → H is a non-co-orientable flatfront, we can take a double covering π : ˆ M → M such that f ◦ π is co-orientable.So this assumption of co-orientability is not so restrictive.The hyperbolic Gauss map G : M → C ∪ {∞} = S of f is a meromorphicfunction with respect to the complex structure mentioned above. On the otherhand, the metric dσ induced by f + ν : M → Λ is a Hermitian pseudometric on M of constant curvature ε which may have isolated singular points of integral order RIENTABILITY OF LINEAR WEINGARTEN SURFACES 7 (cf. [14], [20], the isolated singular points of dσ in M correspond to umbilicalpoints of f ). Then there exists a meromorphic function h on ˜ M satisfying dσ = 4 | dh | (1 + ε | h | ) , where ˜ M is the universal covering of M . ( h is the so-called developing map .)Then the fundamental forms of f are written as follows (cf. [14], [20]): I = (1 − ε ) dσ + 4 | Q | dσ + (1 − ε )( Q + ¯ Q ) , (3.1) II = ε − dσ + 4 | Q | dσ − ε ( Q + ¯ Q ) , (3.2) III = (1 + ε ) dσ − (1 + ε )( Q + ¯ Q ) + 4 | Q | dσ , (3.3)where(3.4) Q := 12 ( S ( h ) − S ( G )) = 12 (cid:0) { h : z } dz − { G : z } dz (cid:1) . The 2-differential Q is called the Hopf differential of f . (The Schwarzian derivative S ( G ) as a locally defined meromorphic 2-differential is given by [22, (2.18)]. Thedifference S ( h ) − S ( G ) does not depend on the choice of local complex coordinate z , see also [20, Lemma 3.7].) We also note that εI + (1 − ε ) II = − (1 − ε ) dσ + 4 | Q | dσ ,I + III = 1 + ε dσ + 8 | Q | dσ − ε ( Q + ¯ Q ) . The following assertion can be proved easily by a modification of [20, Proposition3.5], which is a variant of the holomorphic representation formula in [14]:
Theorem 3.1.
Let G be a meromorphic function on a Riemann surface M , and h a meromorphic function on the universal covering ˜ M such that the pseudometric dσ := 4 | dh | / (1 + ε | h | ) gives a single-valued symmetric tensor on M . Supposethat (3.5) 1 + ε dσ + 8 | Q | dσ − ε ( Q + ¯ Q ) is positive definite, where Q := ( S ( h ) − S ( G )) / { h : z } dz − { G : z } dz ) / .Define f = GAG ∗ and ν = GBG ∗ by G := i ( G h ) − / (cid:18) − GG h ( GG hh / − ( G h ) − G h G hh / (cid:19) , where G h := dG/dh, G hh := dG h /dh and A := ε | h | ε | h | − εh − εh ε | h | ! , B := − ε | h | ε | h | εhεh − − ε | h | ! . Then f : M → H = SL (2 , C ) /SU (2) is a co-orientable linear Weingarten frontof Bryant type in W ε ( M ) , and ν : M → S = SL (2 , C ) /SU (1 , is its unitnormal. Moreover, G and ( ) coincide with the hyperbolic Gauss map and thesum I + III of the first and third fundamental forms of f , respectively. Conversely, MASATOSHI KOKUBU AND MASAAKI UMEHARA any co-orientable linear Weingarten front of Bryant type in W ε ( M ) , except for thehorosphere, is given in this manner.Remark . We give some additional explanation for Theorem 3.1:(1) We often identify R with the set Herm(2) of 2 × X = ( x , x , x , x ) ↔ X = X k =0 x k e k = (cid:18) x + x x + ix x − ix x − x (cid:19) , where i = √− e = (cid:18) (cid:19) , e = (cid:18) (cid:19) , e = (cid:18) i − i (cid:19) , e = (cid:18) − (cid:19) . Since the Lorentzian inner product h , i is given by h X, Y i = −
12 trace (cid:0) Xe t Y e (cid:1) , h X, X i = − det X,H and S are given by H = { X ; X ∗ = X , det X = 1 , trace X > } ,S = { X ; X ∗ = X , det X = − } , respectively, where X ∗ denotes the transposed conjugate t ¯ X to the matrix X . Hence, they are also represented by H = { aa ∗ ; a ∈ SL (2 , C ) } = SL (2 , C ) /SU (2) ,S = { ae a ∗ ; a ∈ SL (2 , C ) } = SL (2 , C ) /SU (1 , . (2) The choice of the developing map h of dσ is not unique. However, theabove expression of f does not depend on the choice of h . We prove thishere:Suppose that we have the following two expressions dσ = 4 | dh | (1 + ε | h | ) = 4 | d ˜ h | (1 + ε | ˜ h | ) . By taking a parallel front, we may assume that f is an immersion on anopen subset U of M . Since f is real analytic, it is sufficient to showthat ˜ f associated to the pair ( G, ˜ h ) coincides with f . Since the first andsecond fundamental forms I and II can be written in terms of G, dσ and Q as in (3.1) and (3.2), ˜ f is congruent to f by the fundamental theoremof surface theory. Moreover, we can show that f = ˜ f in this situation. Infact, a = ( a ij ) i,j =1 , ∈ SL(2 , C ) isometrically acts f as af a ∗ . Then thehyperbolic Gauss map G changed as ( a G + a ) / ( a G + a ). Since ˜ f and f have the same hyperbolic Gauss map G , which implies that f = ˜ f .(3) G satisfies(3.7) G − d G = (cid:18) θdh (cid:19) , where θ = − { G : h } dh . Moreover, note that Q = θdh . (See [14] and [20]for details.) RIENTABILITY OF LINEAR WEINGARTEN SURFACES 9
Corollary 3.3.
Let f ∈ W ε ( M ) for ε > . Then there exists a unique real number δ such that f δ : M → H is a CMC-1 front whose singular set S f δ consists onlyof isolated points in M .Proof. Without loss of generality, we may assume that f is not a horosphere. Thereexists a unique real number δ such that f δ ∈ W ( M ). It follows from (3.1) with ε =1 that the first fundamental form of f δ is I f δ = 4 | Q | /dσ , which is positive definiteexcept at the zeros of the holomorphic 2-differential Q . Since Q is holomorphic, f δ is a front with only isolated singular points. (cid:3) Similarly, we get the following:
Corollary 3.4.
Let f ∈ W ε ( M ) for ε < . Then there exists a unique real number δ such that the unit normal field ν δ of f δ gives a CMC-1 front ν δ : M → S whosesingular set S ν δ consists only of isolated points in M .Proof. Without loss of generality, we may assume that f is not a horosphere. Thereexists a unique real number δ such that f δ ∈ W − ( M ), that is, f δ has constantharmonic-mean curvature one (cf. [20]). At the same time, ν δ : M → S is a CMC-1 front. It follows from (3.1) and (3.3) with ε = − ν δ is I ν δ = III f δ = 4 | Q | /dσ . Since Q is holomorphic, ν δ is a front with onlyisolated singular points. (cid:3) Remark . A meromorphic function G = z + iz and pseudometrics dσ = 4 | dh | (1 + | h | ) (resp. dσ = 4 | dh | (1 − | h | ) ) , where h = z + z , induce CMC-1 fronts in H (resp. in S ). They have an isolatedsingular point at z = 0 because S ( h ) − S ( G )(= 2 Q ) vanishes at z = 0. Remark . Let f ∈ W ( M ) and ν its unit normal field. Then one can show that f + ν is a flat front in the lightcone in R , whose singular set S f + ν consists only ofisolated points.More generally, we can show the following propositions concerning singular pointsof linear Weingarten fronts of Bryant type. Proposition 3.7.
Let f : M → H be a linear Weingarten front of Bryant typewritten in terms of ( G, dσ = 4 | dh | / (1 + ε | h | ) ) as in Theorem 3.1. Then the setof singular points equals S f = (cid:26) p ∈ M ; εI + (1 − ε ) II = 0 (cid:27) = (cid:26) p ∈ M ; 4 | Q | dσ − (1 − ε ) dσ = 0 (cid:27) , where Q is the Hopf differential (3.4) . Moreover, a singular point p ∈ S f is nonde-generate if and only if (1) f is not CMC-1, and (2) 4 εh z ¯ h + (1 + ε | h | )(ˆ θ z / ˆ θ − h zz /h z ) = 0 holds at p , where θ = ˆ θdz for a localcoordinate z , and h z = dh/dz , h zz = d h/dz . We now set ∆ := Im " √ − ε ( εh z ¯ h ε | h | + ˆ θ z ˆ θ − h zz h z ) p h z ˆ θ , where √ − ε is an imaginary number when 1 − ε < Proposition 3.8.
For a nondegenerate singular point p of a linear Weingartenfront f : M → H of Bryant type, the germ of f at p is locally diffeomorphic to (1) a cuspidal edge if and only if ∆ = 0 at p , or (2) a swallowtail if and only if ∆ = 0 at p and ddt (cid:12)(cid:12)(cid:12) t =0 ∆ ◦ γ = 0 , where γ is aparametrization of S f with γ (0) = p . The proofs of the above propositions for the case of ε = 0 are given in [21]. Theproofs for the case of ε = 0 can be followed by a quite similar argument to that in[21], though we need very lengthy calculations. We omit them here.3.2. Zig-zag representation and orientability.
For a given co-orientable front f : M → H , the zig-zag representation σ f : π ( M ) → Z is induced (cf. [24], [25] and [26]), which is invariant under the deformation of f asa wave front, and(3.8) σ f = σ ν holds for the unit normal field ν : M → S . Theorem 3.9.
Let f : M → H ( resp. f : M → S ) be a co-orientable linearWeingarten front of Bryant type. Then M is orientable. Moreover, the zig-zagrepresentation σ f is trivial.Proof. It is sufficient to prove the assertion for a co-orientable linear Weingartenfront f : M → H . (In fact, suppose that the assertion holds for linear Weingartenfront of Bryant type in H . Let f : M → S be a co-orientable linear Weingartenfront. Then its unit normal vector field ν : M → H is a co-orientable linearWeingarten front of Bryant type. Thus M must be orientable. On the other hand,the zig-zag representation is trivial by (3.8).)Suppose that f : M → H is of hyperbolic type (resp. de Sitter type). ByCorollary 3.3 (resp. Corollary 3.4), there exists δ ∈ R such that f δ (resp. ν δ ) is aCMC-1 front with only isolated singular points. Since f δ (resp. ν δ ) is an immersionof non-vanishing mean curvature on R f δ = M \ S f δ , the open submanifold R f δ isorientable. Since S f δ is discrete, we conclude that M is also orientable. On theother hand, we can take a loop γ which represents a given element of π ( M ) sothat γ does not pass through S f δ . Since the zig-zag representation is trivial whenthe corresponding loop does not meet the singular set, σ f δ ([ γ ]) = σ ν δ ([ γ ]) = σ f ([ γ ])vanishes.Next, we consider the case where f is a co-orientable flat front. The orientabilityof M has already been proved in [22]. We set (see § | ρ δ | := e − δ | θ/dh | = e − δ | Q/dh | . Then the singular set of the parallel front f δ is given by the 1-level set | ρ δ | = 1. Wefix an element [ γ ] ∈ π ( M ), where γ is a loop in M . Then we can choose γ sothat it does not pass through the zeros of dh nor of θ . Then there exists a constant c > | Q/dh | ≥ c on γ . If we take δ sufficiently small, then we have e − δ | Q/dh | > γ . Then f δ has no singular point on γ . Thus σ f δ ([ γ ]) = σ f ([ γ ])vanishes. (cid:3) RIENTABILITY OF LINEAR WEINGARTEN SURFACES 11
Remark . In contrast to the case of H , there are many flat fronts in R withnon-trivial zig-zag representations. For example, consider a cylinder over the planarcurve in Figure 1. Figure 1.
A planar curve of zig-zag number 3
Remark . An immersion f : M → R is called a linear Weingarten surfaceof minimal type if the mean curvature H and the Gaussian curvature K satisfythe relation H = aK , where a is a real constant. Such surfaces in R are thecorresponding analogue of linear Weingarten surfaces of Bryant type. As in themethod of Section 1, one can define linear Weingarten fronts of minimal type . Onecan easily prove that each linear Weingarten front f : M → R of minimal typecontains a minimal surface f : M → R as a wave front in its parallel family.Since minimal surfaces have a well-known Weierstrass representation formula, thefirst fundamental form of f is given by | Q | /dσ , where Q is the Hopf differentialand dσ := 4 | dg | / (1 + | g | ) for the Gauss map g of f (and f ). Since the singularpoints of f are isolated, the orientability of M is equivalent to the co-orientabilityof f . (This is a different phenomenon from the case of linear Weingarten frontsof Bryant type. In fact, non-orientable minimal immersions are known.) By thesame argument as in the above proof, the zig-zag representation of a given linearWeingarten front of minimal type f : M → R vanishes if f is co-orientable.3.3. An explicit formula for G ∗ . Recall that the hyperbolic Gauss map of a co-orientable front f is given by G = [ f + ν ]. On the other hand, we set G ∗ := [ f − ν ],which is also called the ( opposite ) hyperbolic Gauss map . For a co-orientable flatfront, it is known that G ∗ is also a meromorphic function, as well as G (cf. [13]).We give here an explicit formula for G ∗ : Proposition 3.12.
Under the notation of Theorem 3.1, the opposite hyperbolicGauss map G ∗ is given by (3.9) G ∗ = G − ( G h ) (1 + ε | h | ) ε ¯ hG h + ( G hh / ε | h | ) . Proof.
The matrices A and B in Theorem 3.1 split as A = ΦΦ ∗ , B = Φ e Φ ∗ , where Φ ∗ := t ¯Φ andΦ := i p ε | h | (cid:18) − − ε ¯ h ε | h | (cid:19) , e := (cid:18) − (cid:19) . Hence we obtain f − ν = G ( A − B ) G ∗ = 2 G Φ (cid:18) (cid:19) ( G Φ) ∗ . Setting G Φ = (cid:18) p qr s (cid:19) , we have f − ν = 2 (cid:18) qs (cid:19) (cid:0) ¯ q ¯ s (cid:1) . This implies G ∗ =[ f − ν ] = q/s ∈ C ∪ {∞} . Then by a straightforward calculation, we get theassertion. (cid:3)
We immediately get the following:
Corollary 3.13. G ∗ is meromorphic, as well as G , if and only if ε = 0 . Let D be the diagonal set of S × S . Then the space L ( H ) of oriented geodesicsof hyperbolic 3-space H can be identified with S × S \ D which is a parahermitiansymmetric space SO (1 , / ( SO (2) · R ∗ ) (see Kanai [18], Kaneyuki [19] and also [6]).Recently, Georgiou and Guilfoyle [12] proved that L ( H ) has a canonical neutralK¨ahler structure and the pair of hyperbolic Gauss maps( G, G ∗ ) : M → S × S \ D gives a Lagrangian surface with zero Gaussian curvature if f is a Weingarten surface.Since our explicit formula (3.9) for G ∗ is written in terms of only G , h and theirderivatives, it might be useful for constructing such surfaces with positive genus.4. Orientability and co-orientability of CMC-1 faces in S In this section, we shall give an application of our explicit formula (3.9) for G ∗ .Let M be a Riemann surface. A holomorphic map F : M → SL (2 , C ) is called null if the determinant of the derivative dF/dz with respect to each local complexcoordinate z vanishes identically. It is well known that CMC-1 surfaces in H and in S are both projections of null holomorphic curves F : ˜ M → SL (2 , C ),where ˜ M is the universal covering of M and H = SL (2 , C ) /SU (2) and S = SL (2 , C ) /SU (1 , S is given by f = −GBG ∗ for ε = − f = −GBG ∗ instead of GBG ∗ because it makes the argument below more compatible to [7] and [9]. − f and f differ merely by congruence in S , that is, they are essentially the same.) Then thenull holomorphic lift F of f is given by(4.1) F = G (cid:18) − i − i ih (cid:19) Conversely, the projection
F e F ∗ : M → S may admit singular points. Such asurface is called a CMC-1 face. More precisely, a C ∞ map f : M → S is called a CMC-1 face if the regular set is open dense in M , and for each p ∈ M , there exista neighborhood U of p and a null holomorphic immersion F : U → SL (2 , C ) suchthat it has an expression f = F e F ∗ on U (see [7] and [9]). Note that we usuallytake a null holomorphic lift F defined on the universal cover ˜ M for a CMC-1 face.We shall show in this section the orientability and the co-orientability of CMC-1faces in this setting. To show the orientability, we will need the explicit formula for G ∗ , which was a formula missing in [9]. It should also be remarked that CMC-1faces are not fronts in general but are all frontals, as seen below (see (4.4)). Onthe other hand, a CMC-1 front in S may admit isolated singular points in M , asseen in Corollary 3.4, although CMC-1 faces do not. This implies that the set ofCMC-1 fronts is not included in the set of CMC-1 faces and vice versa. RIENTABILITY OF LINEAR WEINGARTEN SURFACES 13
For a CMC-1 face f = F e F ∗ (= −GBG ∗ ), we have the following: F − dF = (cid:18) h − h − h (cid:19) Qdh , dF F − = (cid:18) G − G − G (cid:19) QdG , (4.2)where G , h and Q are corresponding to the data in Theorem 3.1. The formulas(4.2) follow from (3.7) and (4.1). Note that the Hopf differential Q and the sec-ondary Gauss map g have already been introduced in [7]. In our notation, the Hopfdifferential is denoted by the same Q , the secondary Gauss map g coincides with h , and ω := Q/dg in [7] is equal to θ = Q/dh .The singular set S f of a CMC-1 face f = F e F = −GBG ∗ is given by S f = (cid:8) p ∈ M ; | h ( p ) | = 1 (cid:9) . (See [7, Theorem 1.9].) Away from the singular set S f , we can define a unit normalfield ν : M \ S f → H ∪ H − by ν = GAG ∗ . It follows from (4.1) that(4.3) ν = F (cid:18) ih ii (cid:19) A (cid:18) ih ii (cid:19) ∗ F ∗ = ˜ ν − | h | , where(4.4) ˜ ν := F (cid:18) | h | h h | h | (cid:19) F ∗ . (See also [7, Remark 1.2].) Since the equivalence class [˜ ν ] gives a section of M into the projective tangent bundle P ( T S ), an arbitrary CMC-1 face f is a frontal.(The definition of a frontal is given in the introduction.) Generic singular pointson CMC-1 faces are cuspidal edges, swallowtails and cuspidal cross caps (see [10]).However, cuspidal cross caps never appear on fronts. Hence, CMC-1 faces are notfronts in general. Theorem 4.1.
Let f : M → S be a CMC-1 face. Then M is orientable.Proof. Suppose, by way of contradiction, that M is not orientable. There existsa double covering ˆ π : ˆ M → M such that ˆ M is orientable. Set ˆ f := f ◦ ˆ π . Let T : ˆ M → ˆ M denote the covering involution which is orientation-reversing. Wefix a regular point p ∈ ˆ M . Then there exists a simply connected neighborhood U ( ⊂ ˆ M ) of p such that the restriction ˆ f | U is an immersion. Let ˆ ν be the unitnormal field on U , and ˆ G , ˆ G ∗ denote the hyperbolic Gauss maps for ˆ f on U . Thenˆ f ◦ T = ˆ f , and either ˆ ν ◦ T ( q ) = ˆ ν ( q ) or ˆ ν ◦ T ( q ) = − ˆ ν ( q ) holds for each q ∈ U asvectors in R . First, we suppose ˆ ν ◦ T = − ˆ ν . Thenˆ G ◦ T = [ ˆ f ◦ T + ˆ ν ◦ T ] = [ ˆ f − ˆ ν ] = ˆ G ∗ holds on U . This implies that ˆ G ∗ is anti-holomorphic, which contradicts Corollary3.13. Next, we suppose ˆ ν ◦ T = ˆ ν . Then, we haveˆ G ◦ T = [ ˆ f ◦ T + ˆ ν ◦ T ] = [ ˆ f + ˆ ν ] = ˆ G, which contradicts that T is orientation-reversing. (cid:3) Since G is a holomorphic map, when the unit normal field ν of a CMC-1 face f crosses a singular curve S f = {| h | = 1 } , the image of ν moves into another sheetof the hyperboloid H ∪ H − . Thus, it is natural to expect ν to be smooth at the singular set under a certain compactification of H ∪ H − . The hyperbolic -sphere ¯ H is a 3-dimensional manifold diffeomorphic to the 3-sphere¯ H := R ∪ {∞} ∼ = S endowed with the metric 4 dx · dx/ (1 − | x | ) on S \ { equator } , where x :=( x , x , x ) ∈ R ∪ {∞} . We consider the stereographic projection(4.5) ϕ : H ∪ H − ∋ ( x , x , x , x ) ( x , x , x )1 − x ∈ ¯ H , which is an isometric embedding, and ¯ H can be considered as a compactificationof H ∪ H − . Theorem 4.2.
The unit normal field of a CMC-1 face can be considered as a realanalytic map into the hyperbolic -sphere ¯ H . In other words, the map ϕ ◦ ν , whichis defined on M \ S f , can be real analytically extended across the singular set S f .Proof. We set F = (cid:18) A BC D (cid:19) . Then
A, B, C, D are holomorphic functions thatare locally defined on a coordinate neighborhood of M . We have only to show that ϕ ◦ ν is real analytic at the point p where | h ( p ) | = 1, since S f = { p ∈ M ; | h ( p ) | =1 } .By a straightforward calculation using (3.6), (4.3), (4.4) and (4.5), one can checkthat each component of ϕ ◦ ν is a rational expression in A, B, C, D, h and theirconjugates, whose denominator r is the following: r = 2(1 − | h | ) + | A + B ¯ h | + | C + D ¯ h | + | Ah + B | + | Ch + D | . Hence, it is sufficient to show that r ( p ) = 0 when | h ( p ) | = 1. However, the equality Ah + B = A + B ¯ h = C + D ¯ h = Ch + D = 0holds at p only when F ( p ) is a singular matrix. Therefore r ( p ) cannot be zero. Thisproves the assertion. (cid:3) Corollary 4.3.
Let f : M → S be a CMC-1 face. Then there exists a realanalytic normal vector field defined on M . In particular, f is a co-orientablefrontal.Proof. Define a map ψ : H ∪ H − → R by ψ ( u ) = ( u/ | u | E if u ∈ H , − u/ | u | E if u ∈ H − , where | u | E denotes the Euclidean norm of u as a vector in R (= R ). For theunit normal field ν : M \ S f → H ∪ H − , the equality ψ ◦ ν = ψ ◦ ϕ − ◦ N holds on M \ S f . Here, N : M → ¯ H denotes the real analytic extension of ϕ ◦ ν : M \ S f → ¯ H explained in Theorem 4.2. On the other hand, we cancalculate ψ ◦ ϕ − : ¯ H \ {| x | = 1 } → R as follows: ψ ◦ ϕ − : ( x, y, z ) √ ∆ (1 + x + y + z , − x, − y, − z )where ∆ := (cid:0) x + y + z + 1 (cid:1) + 4 x + 4 y + 4 z . RIENTABILITY OF LINEAR WEINGARTEN SURFACES 15
It follows that ψ ◦ ϕ − can be smoothly extended to the whole of ¯ H . ThereforeΨ := ψ ◦ ϕ − ◦ N can be considered as a smooth map on M such that Ψ( p ) ∈ R isperpendicular to the position vector f ( p ) ∈ R and the subspace df ( T p M )( ⊂ R )at every regular point p ∈ M \ S f . Thus Ψ gives a globally defined normal vectorfield of f , which proves the co-orientability of f . (cid:3) Co-orientability of maximal surfaces in R In this section, we shall discuss the co-orientability of maximal surfaces inLorentz-Minkowski 3-space given as a joint work with Shoichi Fujimori and KotaroYamada during the authors’ stay at RIMS in Kyoto and a meeting of the secondauthor with Kotaro Yamada at Osaka University on June 2009: CMC-1 faces in S have very similar properties to maximal surfaces in Lorentz-Minkowski 3-space R of signature ( − , + , +). In [29], maxfaces (spacelike maximal surfaces with admissi-ble singular points) are defined as C ∞ maps of orientable 2-manifolds into R . If M is non-orientable, we can take a double covering ˆ π : ˆ M → M such that ˆ M isorientable. For a non-orientable 2-manifold M , a C ∞ map f : M → R is calleda maxface if f ◦ π is a maxface in the sense of [29].In contrast to CMC-1 faces in S , there are non-orientable complete maxfaceswith or without handles (see Fujimori and L´opez [8]). Figure 2 shows two differentnon-orientable maxfaces given in [8]. However, orientability and co-orientability ofmaxfaces are not discussed explicitly in [29]. Figure 2.
Non-orientable maxfaces given in [8] (the picture isproduced by Fujimori)The hyperbolic -sphere ¯ H is a Riemann sphere¯ H := C ∪ {∞} ∼ = S endowed with the metric 4 | dw | / (1 − | w | ) on S \ { equator } , where w ∈ C ∪ {∞} .The hyperboloid of two sheets in R , denoted by H ∪ H − , is H ∪ H − := (cid:8) ( x , x , x ) ∈ R ; − x + x + x = − (cid:9) . We consider the stereographic projection ϕ : H ∪ H − ∋ ( x , x , x ) x + ix − x ∈ ¯ H , which is an isometric embedding, and ¯ H can be considered as a compactificationof H ∪ H − . The following fact is well-known (cf. [29]): Fact 5.1.
Let M be a Riemann surface, f : M → R a maxface, and G : M → ¯ H = C ∪ {∞} its Lorentzian Gauss map as a meromorphic function. Then ν = ϕ − ◦ G gives a unit normal vector field defined on the regular set of f .Using this, we shall prove the following assertion regardless of whether M isorientable or not. Proposition 5.2. ([11])
Let f : M → R be a maxface. Then there exists areal analytic normal vector field defined on M . In particular, f is a co-orientablefrontal.Proof. A smooth map F : M → R is a co-orientable frontal if and only if thereexists a C ∞ vector field η defined on M such that η p ∈ T F ( p ) R is Lorentzianorthogonal to the vector space dF ( T p M ).We define a map ψ : H ∪ H − → R by ψ ( u ) = ( u/ | u | E if u ∈ H , − u/ | u | E if u ∈ H − , where | u | E denotes the Euclidean norm of u as a vector in R (= R ).Suppose that M is orientable. By Fact 5.1,(5.1) ν := ψ ◦ ϕ − ◦ G gives a Lorentzian normal vector field of f on M \ S f . On the other hand, we cancalculate ψ ◦ ϕ − : ¯ H \ {| w | = 1 } → R as follows: ψ ◦ ϕ − : w = x + iy √ ∆ (1 + x + y , − x, − y ) , where ∆ := (cid:0) x + y + 1 (cid:1) + 4 x + 4 y . It follows that ψ ◦ ϕ − can be smoothly extended to the whole of ¯ H . There-fore ν given in (5.1) is a globally defined normal vector field of f . In fact, by astraightforward calculation, we have that(5.2) ν = 1 p (1 + | G | ) + 4 | G | (cid:0) | G | , − G ) , − G ) (cid:1) . So the orientability implies the co-orientability. Next, suppose that f : M → R isa maxface and M is non-orientable. Then there exists a double covering ˆ π : ˆ M → M such that ˆ M is orientable. Set ˆ f := f ◦ ˆ π and denote by G : M → C ∪{∞} theGauss map of the maxface ˆ f . Let T : ˆ M → ˆ M be the covering involution of thedouble covering ˆ π : ˆ M → M . Then ( ϕ − ◦ G ) ◦ T ( p ) is antipodal to ( ϕ − ◦ G )( p )in H ∪ H − for every p ∈ ˆ M (cf. [8]). It follows that(5.3) G ◦ T ( p ) = 1 / G ( p ) ( p ∈ ˆ M ) . By (5.3), the vector field ν given by (5.2) is a Lorentzian normal vector field onˆ M satisfying ν ◦ T = ν . In other words, ν can be considered as a smooth normalvector field on M , which proves the assertion. (cid:3) Remark . Let f : M → R be a non-orientable maxface. Then a closed regularcurve γ : [0 , → M is called a non-orientable loop if the orientation of M reversesalong the curve. Then as pointed out in Fujimori and L´opez [8], there is at least RIENTABILITY OF LINEAR WEINGARTEN SURFACES 17 one singular point on γ . Take a double covering ˆ π : ˆ M → M such that ˆ M isorientable. Set ˆ f := f ◦ ˆ π and denote by G : ˆ M → ¯ H = C ∪ {∞} the LorentzianGauss map of the maxface ˆ f . Since the singular set S ˆ f of ˆ f is the 1-level set of | G | ,it consists of a union of regular curves outside of the discrete setΓ := n p ∈ S ˆ f ; dG ( p ) = 0 o . Here G is considered as a smooth map into S and dG is the derivative of G .A non-orientable loop γ is called generic if its lift ˆ γ on ˆ M meets the singularset only at finitely many points on S ˆ f \ Γ, where ˆ γ passes through these pointstransversally to the set S ˆ f \ Γ. We fix a generic non-orientable loop γ arbitrarily.Since T ◦ ˆ γ (0) = ˆ γ (1), by (5.3), we have | G ◦ ˆ γ (0) | = 1 / | G ◦ ˆ γ (1) | . Since the curve G ◦ ˆ γ ( t ) ( t ∈ [0 , S = { w ∈ C ∪ {∞} ; | w | = 1 } odd times. Thus the number of singular points on a generic non-orientable loop γ is odd. (In fact, the number of connected components of the singular set on thenon-orientable maxfaces in Figure 2 is one and three, respectively.) Appendix A. Degenerate Weingarten fronts
In hyperbolic 3-space H , the following facts are well-known;(1) one of the parallel surfaces of an open portion of a hyperbolic cylinderdegenerates to a hyperbolic line, and(2) one of the parallel surfaces of an open portion of a geodesic sphere (acompact totally umbilical surface) degenerates to a single point.Note that these are Weingarten fronts. (A Weingarten front is defined by Definition2.5 removing the adjective ‘horospherical linear’. This definition makes sense basedon the fact that any parallel surface of a Weingarten surface is again Weingarten.)The above fronts (1) and (2) are called a degenerate cylinder and a degeneratesphere , respectively. We wish to prove a converse statement. Before this, let usrecall the following well known lemma:
Lemma A.1.
Let f : M → H be an immersion, and κ i ( i = 1 , the principalcurvatures. Then the set of singular points of a parallel surface f δ coincides with { p ∈ M ; coth − κ i ( p ) = δ } . Lemma A.1 asserts that p ∈ M is a singular point of f δ if and only if δ is theradius coth − κ i ( p ) of the principal curvature κ i ( p ). It is easily observed that anyparallel surface is free from singular points if and only if | κ i | ≤ Proposition A.2.
Let f : M → H be a Weingarten front. Then all points in M are singular points if and only if f is locally a degenerate cylinder or a degeneratesphere.Proof. Suppose that all points in M are singular points of f . Let p ∈ M be anarbitrary point. There exist a neighborhood U of p and a real number δ such that f δ : U → H is an immersion. In other words, f : U → H is a parallel surface ofan immersion g := f δ , i.e., f = g − δ . Since all points are singular, it follows fromLemma A.1 that − δ = coth − κ ( p ) for all p ∈ U, where κ denotes one of the principal curvatures of g . It implies that κ is constanton U and satisfies | κ | >
1. Moreover, another principal curvature is also constant because g is Weingarten. Thus g is an isoparametric surface. By the classificationdue to Cartan [4], the immersion g is totally umbilical or locally a standard embed-ding of S × H into H . Under our assumption, the former is a part of geodesicsphere because | κ | >
1, the latter is a part of a hyperbolic cylinder.Finally, by the connectivity of M and by the continuity of f , we can concludethat f : M → H is a degenerate cylinder or a degenerate sphere. (cid:3) Acknowledgement
The authors thank Antonio Mart´ınez for a fruitful discussionon the geometry of linear Weingarten surfaces, when they visited the Fukuokaconference on November, 2004. They also thank Shyuichi Izumiya for his informallecture on the duality between fronts in H and S at the Karatsu workshop onOctober, 2005. The first author thanks Shoichi Fujimori for fruitful discussionsabout the proof of Proposition 3.12, after the Karatsu workshop. The secondauthor thanks Masahiko Kanai, who pointed out the importance of L ( H ) as therange of hyperbolic Gauss maps of CMC-1 surfaces in H when the second authorstayed at Nagoya University in 2001. Finally, the authors thank Kotaro Yamada,Wayne Rossman, Francisco L´opez and the referee for valuable comments. References [1] J. A. Aledo and J. M. Espinar, A conformal representation for linear Weingarten surfaces inthe de Sitter space, J. Geom. Phys. , 1669–1677 (2007).[2] J. A. Aledo and J. A. G´alvez, Complete surfaces in hyperbolic space with a constant principalcurvature, Math. Nachr. , 1111–1116 (2005).[3] R. Bryant, Surfaces of mean curvature one in hyperbolic space, Ast´erisque , 321–347(1987).[4] ´E. Cartan, Familles de surfaces isoparam´etriques dans les espaces `a courbure constante, Ann.Mat. Pura Appl. , 177–191 (1938).[5] B. Y. Chen, Minimal surfaces with constant Gauss curvature, Proc. Amer. Math. Soc. ,504–508 (1972).[6] V. Cruceanu, P. Fortuney and P. M. Gadea, A survey on paracomplex geometry, RockyMountain J. Math. , 83–115 (1996).[7] S. Fujimori, Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3-Space, Hokkaido Math.J. , 289–320 (2006).[8] S. Fujimori and F. J. L´opez, Nonorientable maximal surfaces in the Lorentz-Minkowski 3-space, Tohoku Math. J. , 311–328 (2010).[9] S. Fujimori, W. Rossman, M. Umehara, K. Yamada and S.-D. Yang, Spacelike mean curvatureone surfaces in de Sitter 3-space, Comm. Anal. Geom. , 383–427 (2009).[10] S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math.Z. , 827–848 (2008).[11] S. Fujimori, M. Kokubu, M. Umehara and K. Yamada, Personal meetings at RIMS Kyotoand at Osaka University on June, 2009.[12] N. Georgiou and B. Guilfoyle, A characterization of Weingarten surfaces in hyperbolic 3-space, Abh. Math. Semin. Univ. Hambg. , 233-253 (2010).[13] J. A. G´alvez, A. Mart´ınez and F. Mil´an, Flat surfaces in hyperbolic 3-space, Math. Ann. ,419–435 (2000).[14] J. A. G´alvez, A. Mart´ınez and F. Mil´an, Complete linear Weingarten surfaces of Bryant type.A Plateau problem at infinity, Trans. Amer. Math. Soc. , 3405–3428 (2004).[15] S. Izumiya, D. Pei, M. C. Romero-Fuster and M. Takahashi, The horospherical geometry ofsubmanifolds in hyperbolic space, J. London Math. Soc. (2) , 779–800 (2005).[16] S. Izumiya and K. Saji, The mandala of Legendrian dualities for pseudo- spheres in Lorentz-Minkowski space and “flat” spacelike surfaces, J. Singul. , 92–127 (2010).[17] S. Izumiya, K. Saji and M. Takahashi, Horospherical flat surfaces in Hyperbolic 3-space, J.Math. Soc. Japan , 789–849 (2010). RIENTABILITY OF LINEAR WEINGARTEN SURFACES 19 [18] M. Kanai, Geodesic flows of negatively curved manifolds with smooth stable and unstablefoliations, Ergodic Theory Dynam. Systems , 215–239 (1988).[19] S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J.Math. , 81–98 (1985).[20] M. Kokubu, Surfaces and fronts with harmonic-mean curvature one in hyperbolic three-space,Tokyo J. Math. , 177–200 (2009).[21] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat frontsin hyperbolic space, Pacific J. Math. , 303–351 (2005).[22] M. Kokubu, W. Rossman, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-spaceand their caustics, J. Math. Soc. Japan , 265–299 (2007).[23] M. Kokubu, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math. , 149–175 (2004).[24] R. Langevin, G. Levitt and H. Rosenberg, Classes d’homotopie de surfaces avec rebrousse-ments et queues d’aronde dans R , Canad. J. Math. , 544–572 (1995).[25] K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. , 491–529(2009).[26] K. Saji, M. Umehara and K. Yamada, A k singularities of wave fronts, Math. Proc. CambridgePhilos. Soc. , 731–746 (2009).[27] C. Takizawa and K. Tsukada, Horocyclic surfaces in Hyperbolic 3-space, Kyushu J. Math. , 269–284 (2009).[28] M. Umehara and K. Yamada, Complete surfaces of constant mean curvature-1 in the hyper-bolic 3-space, Ann. of Math. , 611–638 (1993).[29] M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space,Hokkaido Math. J. , 13–40 (2006). Department of Mathematics, School of Engineering, Tokyo Denki University, Chi-yoda-ku, Tokyo 101-8457, Japan
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