Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space
SSINGULARITIES OF SPACELIKE CONSTANT MEAN CURVATURE SURFACESIN LORENTZ-MINKOWSKI SPACE
DAVID BRANDERA
BSTRACT . We study singularities of spacelike, constant (non-zero) mean curvature (CMC)surfaces in the Lorentz-Minkowski 3-space L . We show how to solve the singular Björlingproblem for such surfaces, which is stated as follows: given a real analytic null-curve f ( x ) ,and a real analytic null vector field v ( x ) parallel to the tangent field of f , find a conformallyparameterized (generalized) CMC H surface in L which contains this curve as a singular setand such that the partial derivatives f x and f y are given by d f d x and v along the curve. Withinthe class of generalized surfaces considered, the solution is unique and we give a formula forthe generalized Weierstrass data for this surface. This gives a framework for studying the sin-gularities of non-maximal CMC surfaces in L . We use this to find the Björling data – andholomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross capsingularities.
1. I
NTRODUCTION
Spacelike constant mean curvature (CMC) surfaces in ( + ) -dimensional space-time L were studied in [5] and [12] using a generalized Weierstrass representation whereby the surfaceis represented by a holomorphic map into a loop group. This is an application of the method ofDorfmeister, Pedit and Wu (DPW) [7] for harmonic maps into symmetric spaces. In the non-compact case, the Iwasawa decomposition of the loop group, used to construct the solutions,is only valid on an open dense set, the big cell . It was shown in [5] that singularities of theCMC surface arise as the boundary of the big cell is encountered. Here we will analyze thesesingularities and show how to construct CMC surfaces with prescribed singular curves, andprescribed types of singularities, via a singular Björling formulation.One of the obstructions to the effective use of integrable systems methods for solving globalproblems in geometry has been the break-down of the loop group decompositions used toconstruct solutions. A motivating factor here is to understand and make use of the big cellboundary behaviour.1.1. Singularities of maximal surfaces and fronts.
In the context of surfaces in Euclidean3-space E , a frontal is a differentiable map f : M → E , from a surface M , which has a welldefined normal direction, that is, a map n E : M → S ⊂ E which is orthogonal to f ∗ ( T M ) .If the map ( f , n E ) is an immersion, then f is called a (wave) front . A singular point of anysmooth map f : M → E is one where f is not immersed, and singular points p and p of f : M → E and f : M → E are called diffeomorphically equivalent if there exist localdiffeomorphisms of the corresponding spaces which commute with these maps. A theory of thesingularities of fronts can be found in Arnold [3]. Geometric concepts, such as curvature andcompleteness, for surfaces with singularities have been defined by Saji, Umehara and Yamadain [17]. Mathematics Subject Classification.
Primary 53A10; Secondary 53C42, 53A35.
Key words and phrases.
Differential geometry, integrable systems, Björling problem, prescribed mean curvature. a r X i v : . [ m a t h . DG ] A ug DAVID BRANDER
In this article we will encounter three standard singularities: the cuspidal edge , given by f ( u , v ) = ( u , u , v ) , the swallowtail given by ( u + u v , u + uv , v ) and the cuspidal crosscap given by ( u , v , uv ) (Figure 1). The first two singularities are fronts, but the third is onlya frontal.F IGURE
1. Left to right: Cuspidal edge, swallowtail and cuspidal cross cap.A point to note is that if one wants a sensible theory of singularities, for example if onewould like to classify singularities for a specific type of surface, then one needs to consider generic singularities, that is singularities which persist under continuous deformations of thesurface through the appropriate class. If one considers the class of C ∞ maps of 2-manifoldsinto 3-manifolds, Whitney showed that generic singularities are cross caps [20].Fronts and frontals arise naturally within the context of integrable systems – very often itis exactly such surfaces, rather than immersions, which are produced via loop group construc-tions. Conversely, for many geometric problems, it is more or less unavoidable to considersurfaces with singularities: for example it is well known that there is no complete immersionof the hyperbolic plane into E , and for the case of spacelike maximal (mean curvature zero)surfaces in L the only complete immersion is the plane. For these two examples, genericsingularities have been classified: for constant Gauss curvature surfaces in E , Ishikawa andMachida [13] showed that they consist of cuspidal edges and swallowtails; for maximal sur-faces in L , Fujimori, Saji, Umehara and Yamada [19, 11] showed that the generic singularitiesare all three of those shown in Figure 1.Recently there have been a number of interesting studies of maximal surfaces and theirsingularities: the reader is referred to articles such as [2, 9, 8, 10, 15, 11] and the referencestherein. Most closely related to the present article are the classification of generic singularities[19, 11] already mentioned, and the work of Y.W. Kim and S.D. Yang [15] on the singularBjörling problem for maximal surfaces.1.2. The Björling problem.
The classical Björling problem for minimal surfaces in E is tofind the unique minimal surface containing a given real analytic curve with prescribed tangentplanes along the curve (see [6]). The solution is obtained from the initial data by an analyticextension and an elementary formula in terms of integrals. Since the solution is tied to theWeierstrass representation of minimal surfaces in terms of holomorphic data, one has a similarconstruction for regular maximal surfaces in L , given in [2], which also have such a holomor-phic representation. More generally, Kim and Yang [15] show that there is also a solution whenthe initial curve is null (which implies that the surface is not immersed there). Instead of pre-scribing the tangent plane along the curve, one seeks a surface which is conformally immersedexcept along the curve, with coordinates z = x + iy , and where the curve is given by { y = } ,and then prescribes the value of f y , a null vector field parallel to f x . Note that null vectorsare orthogonal if and only if they are parallel, so this makes sense in terms of the conformalcoordinates. One can then use this construction to study the singularities of maximal surfaces. INGULARITIES OF CMC SURFACES IN L As a generalization of the Weierstrass representations for minimal and maximal surfaces,one has the DPW method for CMC H (cid:54) = E and L . In [4], it was shown thatone could use this method to solve the generalization of the Björling problem to non-minimalCMC surfaces in E . It is clear that essentially the same construction works for regular CMC H (cid:54) = L , and we will show below that the singular Björling problem can also besolved for non-maximal CMC surfaces. The main obstacle which needs to be circumventedis that the DPW method depends on the use of an SU , frame (extended to the loop group)and then a loop group decomposition to go to the holomorphic data. This SU , frame isnot defined along the singular curve, because the (Lorentzian) unit normal becomes lightlikeand blows up. Below, we will get around this by defining a special SU , “frame", called the singular frame , along the curve, the definition of which is motivated by our analysis of the loopgroup construction.1.3. The DPW method.
The generalized Weierstrass representation for spacelike CMC sur-face in L follows the same logic as that for CMC surface in Euclidean 3-space: in the maximalcase, where the mean curvature H is zero, there is a Weierstrass representation in terms of a pairof holomorphic functions, just as for minimal surfaces, related to the fact that the Gauss map isholomorphic. For the non-maximal case, the Gauss map is harmonic but not holomorphic, andone can instead use the holomorphic representation for harmonic maps given in [7]. The onlyreal difference from the Euclidean case is the non-compactness of the isometry group, leadingto an incomplete picture of what is actually constructed from the given holomorphic data. Formore details and references, see [5].The DPW construction described in [5] is as follows: A CMC H immersion f : Σ → L froma Riemann surface into Minkowski 3-space can be represented by a certain type of holomorphicmap ˆ Φ : Σ → Λ SL ( , C ) σ into the twisted loop group of smooth maps from the unit circle into SL ( , C ) . The map ˆ Φ is called a holomorphic extended frame for f . In connection with theIwasawa decomposition with respect to the non-compact real form Λ SU , , the loop group Λ SL ( , C ) σ can be written as a disjoint union B , ∪ P ∪ P ∪ P ∪ ... . The set B , isopen and dense in Λ SL ( , C ) σ , and is called the (Iwasawa) big cell. As a converse to theabove statement concerning f , given a holomorphic extended frame, if we restrict to Σ ◦ : = ˆ Φ − ( B , ) , one obtains a CMC H immersion into L . Behaviour of the surface as the largesttwo small cells, P and P , are approached was examined in [5], and it was shown that theCMC surface extends continuously to ˆ Φ − ( P ) , but is not immersed there, and that the surfaceblows up as ˆ Φ − ( P ) is approached.1.4. Results of this article.
As we are interested in finite singularities, we define a generalizedCMC H surface to be a map f obtained from a holomorphic extended frame ˆ Φ , restricted to Σ s : = ˆ Φ − ( B , ∪ P ) . This includes all regular CMC H surfaces, as one can always find aholomorphic extended frame for a regular surface which takes values in the big cell B , . Weknow that the singular set C : = ˆ Φ − ( P ) , where f is not immersed, is locally given as the zeroset of a non-constant real analytic function. We say that z ∈ C is weakly non-degenerate if ˆ Φ maps some open curve containing z into P . This is simply the weakest condition needed toconsider the singular Björling construction, and holds for a generic point in C .The main results of this article can be summarized as Theorem 4.1, Theorem 5.7 and The-orem 5.9. The first of these results is the solution of the singular Björling problem for CMCsurfaces in L . It essentially says that given a real analytic curve f : J → L , from someinterval J ⊂ R ⊂ C , such that d f d x is a null vector field, and given a real analytic vector field v : J → L which is proportional to d f d x , then, for any constant H >
0, there is a unique, weaklynon-degenerate, generalized CMC H surface f satisfying f (cid:12)(cid:12) J = f and ∂ f ∂ y (cid:12)(cid:12) J = v . It also gives DAVID BRANDER a formula for the holomorphic potential for the surface in terms of analytic extensions of thedata specified along J .The other two results mentioned, Theorems 5.7 and 5.9, give the conditions on the Björlingdata for the singularity at a point z ∈ J to be diffeomorphic to a cuspidal edge, swallowtail orcuspidal cross cap. The conditions are simple: for the given Björling data, one can always write d f d x = s [ cos θ , sin θ , ] and v ( x ) = t [ cos θ , sin θ , ] , where s , t , and θ are R -valued, and weassume that s and t do not vanish simultaneously to avoid branch points. Then s ( ) (cid:54) = (cid:54) = t ( ) corresponds to a cuspidal edge at the coordinate origin; s ( ) = s (cid:48) ( ) (cid:54) = t ( ) = t (cid:48) ( ) (cid:54) = IGURE
2. Left: a CMC swallowtail singularity, computed numericallyfrom the Björling data s ( x ) = x , t ( x ) = θ ( x ) = . x . Right: aCMC cuspidal cross cap, computed from the data s ( x ) = − x , t ( x ) = x , θ ( x ) = . x . The images have been rescaled in the direction e + e .1.5. Open questions.
It appears plausible that the three types of singularities just mentionedare the generic singularities for CMC surfaces in L , just as was shown for maximal surfacesin [11]. To prove this using the constructions here, one would first need to show that genericsingularities do not occur on higher small cells P j , for j >
2. This seems likely, becausethe codimensions of the small cells P j in the loop group increase (pairwise) as j increases.Regardless of genericity, knowledge of the behaviour of the surface close to such points wouldalso be interesting to have.1.6. Alternative approaches: the Kenmotsu formula representation.
An alternative to theDPW method is the Kenmotsu formula [14] for CMC surfaces in E , adapted to spacelikeCMC surfaces in L by Akutagawa and Nishikawa in [1]. This is also a generalization ofthe Weierstrass representation for minimal/maximal surfaces, as a formula in terms of theharmonic Gauss map. In contrast to the DPW method, one is still left with the problem ofconstructing the harmonic map. The Kenmotsu-Akutagawa-Nishikawa approach has been usedby Y. Umeda [18] to study CMC surfaces with singularities in L , giving the conditions on theharmonic Gauss map corresponding to cuspidal edges, swallowtails and cuspidal cross caps, aswell as some examples. It is stated as an open problem whether or not a CMC cuspidal crosscap exists: here we give a positive answer to this question, and, in principal, construct all suchsingularities from their Björling data. INGULARITIES OF CMC SURFACES IN L
2. B
ACKGROUND MATERIAL
This section is a short summary of results in [5]. We use mostly the same notation anddefinitions here.
Notational convention:
If ˆ X is some object with values in the loop group,with loop parameter λ , then dropping the hat means the object is evaluated at λ = X : = ˆ X (cid:12)(cid:12)(cid:12) λ = . The loop group formulation for CMC surfaces in L . We use the basis e : = (cid:18) (cid:19) , e : = (cid:18) i − i (cid:19) , e : = (cid:18) i − i (cid:19) , for the Lie algebra su , . With respect to the Killing metric, (cid:104) X , Y (cid:105) = trace ( XY ) , these vectorsare orthogonal and normalized as follows: (cid:104) e , e (cid:105) = (cid:104) e , e (cid:105) = −(cid:104) e , e (cid:105) = , so we identify su , with the Lorentz-Minkowski space L = R , , and also use the notation [ a , b , c ] T = ae + be + ce for a point in L .Let G be the subgroup of SL ( , C ) consisting of elements of either SU , or of ie · SU , ,(2.1) G = (cid:26)(cid:18) a b ε ¯ b ε ¯ a (cid:19) (cid:12)(cid:12)(cid:12) a , b ∈ C , ε ( a ¯ a − b ¯ b ) = , ε = ± (cid:27) . The Lie algebra of G is g = su , .The twisted loop group U : = Λ G σ consists of maps, x : S → G , from the unit circle into G , such the diagonal and off-diagonal elements of the matrix are even and odd functions ofthe S parameter λ . All loops are of a suitable smoothness class so that the loop groups areBanach Lie groups. An element of U can again be written as in (2.1), where now a and b are respectively even and odd functions of λ . We will generally be considering loops whichextend holomorphically to an annulus around S , and for these the holomorphic extensions of¯ a and ¯ b respectively have Fourier expansions a ∗ ( λ ) : = ( a ( / ¯ λ )) and b ∗ ( λ ) : = ( b ( / ¯ λ )) . Wecan write U : = Λ G σ = U ∪ U − , where the ε in U ε corresponds to that in (2.1). We also have U = Λ SU , and U − = (cid:16) λ i λ − i (cid:17) · U . The Lie algebra, Lie ( U ) = Lie ( U ) , of U , consists of loops of matriceswith analogous properties to those in U , replacing the determinant 1 condition with the tracezero condition.The complexification of U is U C : = Λ SL ( , C ) σ , the group of loops in SL ( , C ) whichagain have the twisted condition on diagonal/off-diagonal elements mentioned above. Let D ± : = { λ ∈ C ∪ { ∞ } (cid:12)(cid:12) | λ | ± < } . Three subgroups of U C that we also use are: U C ± : = { ˆ B ∈ U C (cid:12)(cid:12) ˆ B extends holomorphically to D ± } , (cid:99) U C + : = { ˆ B ∈ U C + (cid:12)(cid:12) ˆ B (cid:12)(cid:12) λ = = (cid:16) ρ ρ − (cid:17) , ρ ∈ R , ρ > } . Let Σ be a simply connected non-compact Riemann surface, and suppose f : Σ → L is aconformal spacelike immersion with constant mean curvature H (cid:54) =
0, or an H -surface. Withoutloss of generality, we assume that H >
0, the sign being a matter of orientation. If z = x + iy is a local coordinate, there is a function u : Σ → R such that the metric is given by d s = DAVID BRANDER e u ( d x + d y ) . The coordinate frame F : Σ → SU , is well defined up to premultiplicationby ± I , by(2.2) Fe F − = f x | f x | , Fe F − = f y | f y | . Choose the conformal coordinates x and y such that the oriented unit normal is then given by N = Fe F − . The Hopf differential is defined to be Q d z , where Q : = (cid:104) N , f zz (cid:105) = −(cid:104) N z , f z (cid:105) . The Maurer-Cartan form, α , for the frame F is defined to be α : = F − d F = U d z + V d¯ z , wherethe connection coefficients U : = F − F z and V : = F − F ¯ z are given by(2.3) U = (cid:18) u z − iHe u ie − u Q − u z (cid:19) , V = (cid:18) − u ¯ z − ie − u ¯ Q iHe u u ¯ z (cid:19) . The compatibility condition d α + α ∧ α = u z ¯ z − H e u + | Q | e − u = , Q ¯ z = e u H z . The above structure for U and V are verified by a computation, using H = e − u (cid:104) f xx + f yy , N (cid:105) ,and(2.5) f z = e u F · (cid:18) (cid:19) · F − , f ¯ z = e u F · (cid:18) (cid:19) · F − . We can insert an S parameter λ into the 1-form α , defining a family ˆ α : = ˆ U d z + ˆ V d¯ z ,where(2.6) ˆ U = (cid:18) u z − iHe u λ − ie − u Q λ − − u z (cid:19) , ˆ V = (cid:18) − u ¯ z − ie − u ¯ Q λ iHe u λ u ¯ z (cid:19) . Then the assumption that H is constant is equivalent to the integrability of ˆ α for all λ . Hence itcan be integrated to obtain a map ˆ F : Σ → U . Supposing that our coordinate frame F definedabove satisfies F ( z ) = F , at some point z , we integrate ˆ α with the same initial condition,and call the map ˆ F : Σ → U thus obtained an extended frame for the H -surface f .The Sym-Bobenko formula is the map S : U → Lie ( U ) given by:(2.7) S ( ˆ F ) : = − H (cid:18) ˆ Fe ˆ F − + i λ ∂ ˆ F ∂ λ ˆ F − (cid:19) . We write S λ : U → L for the map given by evaluating this at λ ∈ S . If ˆ F : Σ → U is anextended frame for an H -surface f , then, up a translation in L , the surface is retrieved byapplying the Sym-Bobenko formula at λ = f = S ( ˆ F ) + translation . This is verified by computing S ( ˆ F ) z and S ( ˆ F ) ¯ z , using the matrices ˆ U and ˆ V . The samecomputation shows that S λ ( ˆ F ) is also an H -surface for any λ ∈ S . For such computations,note that if ˆ G − ˆ G z = (cid:18) u αλ − β λ − − u (cid:19) , ˆ G − ˆ G ¯ z = (cid:18) − ¯ u ¯ β λ ¯ αλ ¯ u (cid:19) , and we set f λ = S λ ( ˆ G ) , then one computes the following formulae:(2.8) ˆ G − f λ z ˆ G = iH (cid:18) αλ − (cid:19) , ˆ G − f λ ¯ z ˆ G = iH (cid:18) − ¯ αλ (cid:19) . INGULARITIES OF CMC SURFACES IN L One can also define a CMC surface with extended coordinate frame (cid:101) F in the other half ofthe loop group, U − , by integrating the 1-form ˆ U d z + ˆ V d¯ z with the initial condition (cid:101) F ( z ) = W = (cid:18) i λ i λ − (cid:19) . Since S ( W ˆ F ) = Ad W S ( ˆ F ) + translation – where Ad X denotes conjugation by X – and Ad W is an isometry of L , this is also a CMC surface. If ˆ F is the frame obtained with the initialcondition ˆ F ( z ) = I , then the relation between the surfaces obtained at λ = S ( (cid:101) F ) = Ad W (cid:12)(cid:12) λ = S ( ˆ F ) + translation. The coordinate frame for ˜ f = S ( (cid:101) F ) satisfies (cid:101) Fe (cid:101) F (cid:12)(cid:12) λ = = ˜ f x | ˜ f x | and (cid:101) Fe (cid:101) F (cid:12)(cid:12) λ = = ˜ f y | ˜ f y | .More generally, one can show (see, for example, the analogous argument in [4]): Lemma 2.1. If ˆ F : Σ → U = U ∪ U − is a real analytic map the Maurer-Cartan form ofwhich has the form (2.9) ˆ F − d ˆ F = α − d z λ − + ˆ β d z + ˆ γ d¯ z , where the loop-algebra valued functions ˆ β and ˆ γ extend holomorphically in λ to the unit disc,and with the regularity condition [ α − ] (cid:54) = , then the map f λ = S λ ( ˆ F ) is an H-surface in L , and the coordinate frame for this surface is given by F = ˆ F | λ D, where D : Σ → G is adiagonal matrix-valued function.
Note that the Sym-Bobenko formula is invariant under gauge transformations ˆ F (cid:55)→ ˆ FD ,where D is constant in λ and diagonal. It also follows from the fact that the 1-form ˆ F − d ˆ F ofLemma 2.1 takes values in Lie ( U ) that, in fact,ˆ F − d ˆ F = α − d z λ − + α d z + τ ( α ) d¯ z + τ ( α − ) d¯ z λ , where the involution τ that defines g = su , as a real form of sl ( , C ) is given by: τ ( X ) : = − Ad σ ¯ X t , σ = (cid:18) − (cid:19) . Construction of solutions via the DPW method.
By Lemma 2.1, the problem of con-structing a conformal spacelike CMC immersion f : Σ → L is evidently equivalent to theproblem of constructing a real analytic map ˆ F : Σ → U , such that ˆ F − d ˆ F is of the type givenby (2.9). The DPW construction does exactly that, beginning with an arbitrary holomorphic map ˆ Φ : Σ → U C which satisfies ˆ Φ − d ˆ Φ = ( β − λ − + β + ... ) d z .In order to explain this, we first need to state the Iwasawa decomposition of U C . Define,for a positive integer m ∈ Z + , ω m = (cid:18) λ − m (cid:19) , m odd; ω m = (cid:18) λ − m (cid:19) , m even. Theorem 2.2. (SU , Iwasawa decomposition [5] ) (1) The group U C is a disjoint union (2.10) U C = B , (cid:116) (cid:71) m ∈ Z + P m , where B , : = U · U C + , is called the big cell , and the n -th small cell is: (2.11) P n : = U · ω n · U C + . DAVID BRANDER (2)
In the factorization (2.12) ˆ Φ = ˆ F ˆ B , ˆ F ∈ U , ˆ B ∈ U C + , of a loop ˆ Φ ∈ B , , the factor ˆ F is unique up to right multiplication by an elementof the subgroup U of constant loops in U . Both factors are unique if we requirethat ˆ B ∈ (cid:99) U C + , and with this normalization the product map U × (cid:99) U C + → B , is a realanalytic diffeomorphism. (3) The Iwasawa big cell, B , , is an open dense subset of U C . The complement of B , in U C is locally given as the zero set of a non-constant real analytic function U C → C . It is clear from Theorem 2.2 that the big cell B , is naturally divided into two disjoint opensets corresponding to whether the element ˆ F is a loop in SU , or in ie SU , . We denote thesesubsets by B + , and B − , respectively.Now it is easy to check that if ˆ Φ : Σ → B , ⊂ U C satisfies ˆ Φ − d ˆ Φ = ( β − λ − + β + ... ) d z ,and ˆ Φ = ˆ F ˆ B is an Iwasawa factorization of ˆ Φ , with ˆ F ∈ U , then ˆ F − d ˆ F is of the requiredform (2.9). That is the essential point behind the generalized Weierstrass representation for H -surfaces which will be stated in the next theorem. Definition 2.3. A standard (holomorphic) potential on a Riemann surface Σ is a holomorphic1-form ˆ ξ ∈ Lie ( U C ) ⊗ Ω , ( Σ ) , the Fourier expansion of which begins at λ − : ˆ ξ = ∞ ∑ i = − β i λ i d z , β i : Σ → sl ( , C ) , holomorphic , and with the regularity condition on the (1,2) component of β − : [ β − ] ( z ) (cid:54) = , ∀ z ∈ Σ . Theorem 2.4. [5] . Let ˆ ξ be a standard holomorphic potential on a simply-connected Rie-mann surface Σ . Let ˆ Φ : Σ → U C be a solution of ˆ Φ − d ˆ Φ = ˆ ξ . Define the open set Σ ◦ : = ˆ Φ − ( B , ) . Assume that the map ˆ Φ , maps at least one point into B , , so that Σ ◦ is not empty, and take any G-Iwasawa splitting pointwise on Σ ◦ : (2.13) ˆ Φ = ˆ F ˆ B , ˆ F ∈ U , ˆ B ∈ U C + . Then for any λ ∈ S , the map f λ : = S λ ( ˆ F ) : Σ ◦ → L , given by the Sym-Bobenko formula (2.7) , is a conformal spacelike CMC H immersion, and is independent of the choice of ˆ F ∈ U in (2.13).Conversely, let Σ be a noncompact Riemann surface. Then any non-maximal conformalCMC spacelike immersion from Σ into L can be constructed in this manner, using a holomor-phic potential ˆ ξ that is well-defined on Σ . We call ˆ Φ a holomorphic extended frame for the family of surfaces f λ . It is also true thatif we normalize the factors in (2.13) so that ˆ B ∈ (cid:99) U C + , and define the function ρ : Σ ◦ → R byˆ B | λ = = diag ( ρ , ρ − ) , then there exist conformal coordinates ˜ z = ˜ x + i ˜ y on Σ such that theinduced metric for f is given by d s = ρ ( d ˜ x + d ˜ y ) , and the Hopf differential is given by Q d˜ z , where Q = − H b − a − . INGULARITIES OF CMC SURFACES IN L Behaviour of the surface at the boundary of the big cell.
Theorem 2.4 says that astandard holomorphic potential ˆ ξ corresponds to an H -surface, provided we restrict to Σ ◦ = ˆ Φ − ( B , ) . Now set C : = Σ \ Σ ◦ = ∞ (cid:91) j = ˆ Φ − ( P j ) , C : = ˆ Φ − ( P ) , C : = ˆ Φ − ( P ) . Theorem 2.5. [5]
Let ˆ Φ be as defined in Theorem 2.4. Then (1) Σ ◦ is open and dense in Σ . More precisely, its complement, the set C , is locally givenas the zero set of a non-constant real analytic function Σ → C . (2) The sets Σ ◦ ∪ C and Σ ◦ ∪ C are both open subsets of Σ . The sets C and C are eachlocally given as the zero set of a non-constant real analytic function Σ → R . (3) All components of any matrix F obtained by Theorem 2.4 on Σ ◦ , and evaluated at λ ∈ S , blow up as z approaches a point z in either C or C . In the limit, the unitnormal vector N, to the corresponding surface, becomes asymptotically lightlike, i.e.its length in the Euclidean space R metric approaches infinity. (4) The surface f λ obtained from Theorem 2.4 extends to a real analytic map Σ ◦ ∪ C → L , but is not immersed at points z ∈ C . (5) The surface f λ diverges to ∞ as z → z ∈ C . Moreover, the induced metric on thesurface blows up as such a point in the coordinate domain is approached. The arguments given in [5] to prove those parts of the above theorem involving C and C all depend on an explicit Iwasawa factorization of an element of the form B ω , where B is anarbitrary element of U C + . We will use this explicit factorization again several times below, andso we recall it here: Lemma 2.6. [5]
Let ˆ B = (cid:18) a bc d (cid:19) = (cid:18) ρ ρ − (cid:19) + (cid:18) µν (cid:19) λ + o ( λ ) be any element of U C + .Then there exists a factorization (2.14) ˆ B ω = ˆ X ˆ B (cid:48) , where ˆ B (cid:48) ∈ U C + and ˆ X is of one of the following three forms:k = (cid:18) u v λ ¯ v λ − ¯ u (cid:19) , k = (cid:18) u v λ − ¯ v λ − − ¯ u (cid:19) , ω θ = (cid:18) e i θ λ − (cid:19) , where u and v are constant in λ and can be chosen so that the matrix has determinant one, and θ ∈ R . The matrices k and k are in U , and their components satisfy the equation (2.15) | u || v | = | µ + ρ || ρ | . The first two forms occurs when ˆ B ω is in the big cell B , , and the third form occurs if andonly if ˆ B ω is in the first small cell, P . The three cases correspond to the cases | ( µ + ρ ) ρ | greater than, less than or equal to 1, respectively. Moreover, if ˆ B ω is given locally by a realanalytic map either from R → B , , or from R → P , then the factors ˆ X and ˆ B (cid:48) can be chosento be real analytic. Proof.
One can write down explicit expressions as follows: for the cases | ( µ + ρ ) ρ | ε > ε = ±
1, the factorization is given byˆ X = (cid:18) u v λε ¯ v λ − ε ¯ u (cid:19) , ˆ B (cid:48) = (cid:18) ε ¯ ub λ − − dv + ε ¯ ua − vc λ b ε ¯ u − vd λ − ε ¯ vb λ − + ( − ε ¯ va + ud ) λ − + uc − b ε ¯ v λ − + ud (cid:19) . (2.16)One can choose u and v so that ε ( u ¯ u − v ¯ v ) = B (cid:48) ∈ U C + , the latter conditionbeing assured by the requirement that u ¯ v = ε ( µ + ρ ) ρ . Once such choice is(2.17) v = (cid:114) ε (cid:16) | µ + ρ | | ρ | − (cid:17) , u = ε ( µ + ρ ) ρ ¯ v . It is straightforward to verify that ˆ X ˆ B (cid:48) = ˆ B ω − .For the case | ( µ + ρ ) ρ | =
1, useˆ X = (cid:18) u v λ − ¯ v λ − ¯ u (cid:19) , ˆ B (cid:48) = (cid:18) ¯ ub λ − − dv + ¯ ua − vc λ b ¯ u − vd λ ¯ vb λ − + ( ¯ va + ud ) λ − + uc b ¯ v λ − + ud (cid:19) . (2.18)and choose u ¯ v = − ( µ + ρ ) ρ . One can choose u = √ and ¯ v = − √ (( µ + ρ ) ρ ) − = − √ e i θ and (cid:18) u v λ − ¯ v λ − ¯ u (cid:19) = (cid:18) e i θ λ − (cid:19) (cid:32) √ − √ e − i θ λ √ (cid:33) . Pushing the last factor into ˆ B (cid:48) then gives the required factorization. In this case, ˆ B ω − is in P , because it can be expressed as (cid:18) e − i θ / e i θ / (cid:19) · ω · (cid:18) e i θ / e − i θ / (cid:19) ˆ B (cid:48) . The claimed analytic properties of the factors are satisfied for the explicit choices of u and v given above, because the expression ( µ + ρ ) ρ is real analytic. (cid:3)
3. T HE W EIERSTRASS REPRESENTATION FOR SURFACES WITH SINGULARITIES
Theorem 2.5 states that singularities occur at points which are mapped into P , and that theframe F is not defined at such points. In this section we define an alternative extended frameˆ F ω which does not blow up at singular points. This will be used in the next section to solve thesingular Björling problem.Let π : B , → U / U denote the projection defined by taking the equivalence class of ˆ F (under right multiplication by elements of U ) in the Iwasawa factorization ˆ Φ = ˆ F ˆ B of ˆ Φ ∈ B , . Since the Sym-Bobenko formula S is invariant under right multiplication by constantdiagonal matrices, S : U / U → Lie ( U ) is well defined, and we can extended it to a map (cid:102) S : B , → Lie ( U ) , (cid:102) S = S ◦ π . Again we define the map (cid:102) S λ : B , → L by evaluating this at λ ∈ S . The crucial factthat is exploited here and in [5] – and is proved using Lemma 2.6 – is that if ˆ Φ ∈ B , andˆ Φ ω − ∈ B , then(3.1) (cid:102) S (cid:0) ˆ Φ ω − (cid:1) = (cid:102) S (cid:0) ˆ Φ (cid:1) . INGULARITIES OF CMC SURFACES IN L Thus, if ˆ Φ : Σ → U C , and ˆ Φ ( z ) = ω ∈ P , then we can just as well consider the map ˆ Φ ω : = ˆ Φ ω − . Then ˆ Φ ω ( z ) ∈ B , in a neighbourhood of z , and if ˆ Φ is a holomorphic extendedframe, then so is ˆ Φ ω – for the same family of surfaces f λ . On the open dense set ˆ Φ − ( B , ) ∩ ˆ Φ − ω ( B , ) , we have (cid:102) S ( ˆ Φ ) = (cid:102) S ( ˆ Φ ω ) , and so it is valid to define f λ ( z ) : = (cid:102) S λ ( ˆ Φ ω ( z )) . Any element of P is of the form ˆ F ω ˆ B , and essentially the same argument can be usedto define f λ ( z ) when ˆ Φ ( z ) has this form. Hence one can define a real analytic map f λ :ˆ Φ − ( B , ∪ P ) → L which is an immersed CMC H surface on ˆ Φ − ( B , ) . Definition 3.1.
Let Σ be a simply-connected Riemann surface, ˆ ξ a standard potential, and ˆ Φ : Σ → U C the map obtained by integrating ˆ Φ − d ˆ Φ = ˆ ξ with an initial condition ˆ Φ ( z ) = ˆ Φ ∈ U C . Assume that ˆ Φ ( w ) ∈ B , for at least one point w ∈ Σ . Let Σ s ⊂ Σ be the open densesubset given by Σ s = ˆ Φ − ( B , ∪ P ) , and define, for any λ ∈ S ,f λ : Σ s → L , f λ ( z ) = (cid:102) S λ (cid:0) ˆ Φ ( z ) (cid:1) . We call the map f λ – and, more generally, any map from a Riemann surface into L which hassuch a representation locally – a generalized constant mean curvature H surface , or generalized H -surface , in L . Singular holomorphic potentials and frames.
For a typical generalized H -surface wecan expect, from Theorem 2.5 Item 2, that the singular set C = ˆ Φ − ( P ) is a curve, and wecan deduce from Item 3 that this curve must be a null curve, wherever it is regular.It is clear from the preceding discussion that one may construct a generalized H -surface witha singularity at z by integrating a standard potential ˆ ξ with the initial condition ˆ Φ ( z ) = ω , provided that the resulting complex extended frame ˆ Φ does satisfy ˆ Φ ( z ) ∈ B , for some z .Alternatively, supposing we did this, there is also the translated map ˆ Φ ω = ˆ Φ ω − – whichmay be more natural because ˆ Φ ω ( z ) = I and so this maps a neighbourhood of z into the bigcell.We first analyze the Maurer-Cartan form of ˆ Φ ω , given that ˆ ξ is a standard potential, whichhas the general form:(3.2) ˆ Φ − d ˆ Φ = (cid:26)(cid:18) a − b − (cid:19) λ − + (cid:18) c − c (cid:19) + (cid:18) a b (cid:19) λ + o ( λ ) (cid:27) d z , where a − is non-vanishing. For ˆ Φ ω = ˆ Φ ω − , the above expression is equivalent toˆ Φ − ω d ˆ Φ ω = (cid:26)(cid:18) − a − (cid:19) λ − + (cid:18) − a − a − (cid:19) λ − + (cid:18) a − b − + c − a (cid:19) λ − + (cid:18) c − a − c + a (cid:19) + (cid:18) a b (cid:19) λ + o ( λ ) (cid:27) d z . Now consider the special case that ˆ Φ ω ( z ) ∈ U for z ∈ R . Then the Iwasawa factorizationof ˆ Φ ω along R , is just ˆ Φ ω = ˆ Φ ω · I , and therefore the Iwasawa factorization of ˆ Φ for z ∈ R isjust ˆ Φ = ˆ Φ ω · ω · I . In other words, such a holomorphic frame maps the real line into P .The assumption is equivalent to demanding that ˆ Φ − ω ∂ ˆ Φ ω ∂ x ( x , ) d x has coefficients in Lie ( U ) , which implies that it must be of the form:ˆ ξ = (cid:26)(cid:18) − a (cid:19) λ − + (cid:18) − a a (cid:19) λ − + (cid:18) ab (cid:19) λ − + (cid:18) ir − ir (cid:19) (3.3) + (cid:18) b ¯ a (cid:19) λ + (cid:18) ¯ a − ¯ a (cid:19) λ + (cid:18) − ¯ a (cid:19) λ (cid:27) d x , where a and b are maps R → C while r : R → R , and all functions are restrictions to R ofholomorphic functions. Hence, the Maurer-Cartan form of ˆ Φ ω is a holomorphic extension ofthis: Definition 3.2.
Let Σ ⊂ C be a simply connected open subset which intersects the real line in aninterval: Σ ∩ R = J = ( x , x ) , and contains the origin z = . A standard singular holomorphicpotential on Σ , is a holomorphic 1-form ˆ ξ ω on Σ that can be expressed as: ˆ ξ ω = ˆ Φ − ω d ˆ Φ ω = (cid:26)(cid:18) − a (cid:19) λ − + (cid:18) − a a (cid:19) λ − + (cid:18) ab (cid:19) λ − + (cid:18) ir − ir (cid:19) (3.4) + (cid:18) b ˜ a (cid:19) λ + (cid:18) ˜ a − ˜ a (cid:19) λ + (cid:18) − ˜ a (cid:19) λ (cid:27) d z , where a, b and r are holomorphic on Σ , the restriction of r to J is real, that is r ( ¯ z ) = r ( z ) , and ˜ a and ˜ b are holomorphic extensions of the restrictions ¯ a (cid:12)(cid:12) R and ¯ b (cid:12)(cid:12) R , that is ˜ a ( z ) = a ( ¯ z ) , and ˜ b ( z ) = b ( ¯ z ) , with the regularity condition: (A) a ( z ) non-vanishing on Σ . Define the singular holomorphic frame ˆ Φ ω corresponding to ˆ ξ ω to be the map ˆ Φ ω : Σ → U C obtained by solving the equationˆ Φ − ω d ˆ Φ ω = ˆ ξ ω , ˆ Φ ω ( ) = I . Set ˆ Φ : = ˆ Φ ω ω , Σ ◦ : = ˆ Φ − ( B , ) , C : = ˆ Φ − ( P ) , Σ s : = Σ ◦ ∪ C . Note that ˆ Φ ( ) = ω / ∈ B , so it is not clear that Σ ◦ is non-empty. Theorem 3.3.
Suppose ˆ ξ ω is a standard singular holomorphic potential given by Definition3.2, and suppose that Σ ◦ is non-empty. Then (1) Σ ◦ is open and dense in Σ . (2) Σ s is also an open dense subset of Σ . For any λ ∈ S , the map f λ : Σ s → L , given byf λ = (cid:102) S λ (cid:0) ˆ Φ ω (cid:1) = (cid:102) S λ (cid:0) ˆ Φ (cid:1) , is a generalized constant mean curvature H surface. (3) The restriction f λ (cid:12)(cid:12) Σ ◦ : Σ ◦ → L is a spacelike CMC H immersion. (4) The map f λ is not immersed at points z ∈ C, and the interval J = Σ ∩ R is containedin the singular set C. Moreover, f λ (cid:12)(cid:12) J is either a single point or a real analytic nullcurve which is regular except at points where Re ( a λ − ) = . (5) A condition that ensures that Σ ◦ is non-empty is: (B) r − Im b not equivalent to zero on J = Σ ∩ R . INGULARITIES OF CMC SURFACES IN L Moreover, on a neighbourhood in Σ of a point z ∈ J, such that r ( z ) − Im b ( z ) (cid:54) = ,the sets C and J coincide.Proof. Items 1-3:
The Maurer-Cartan form of ˆ Φ = ˆ Φ ω ω is given by(3.5) ˆ Φ − d ˆ Φ = (cid:18) ir + ˜ b a λ − + ˜ b λ − ˜ a λ i (cid:0) i ( b − ˜ b ) − r (cid:1) λ − − ir − ˜ b (cid:19) d z , and we assumed a is non-vanishing, so this is a standard holomorphic potential. Since ˆ ξ ω is Lie ( U ) -valued along R , it follows that ˆ Φ ω maps J ⊂ R into U . Therefore ˆ Φ = ˆ Φ ω ω maps J into P , by definition of P . Hence items 1-3 follow from Theorem 2.5 and equation (3.1)above. Item 4:
The first statement follows from Theorem 2.5, so we are left with the second statementconcerning the regularity of f λ (cid:12)(cid:12) J .First, since ˆ Φ ω ( z ) ∈ U ⊂ B , for real values of z , it follows that the set W = ˆ Φ − ω ( B , ) isopen (and, in fact dense, see the proof of Theorem 4.1 of [5]) and contains J . Hence, pointwiseon this set, we can decomposeˆ Φ ω = ˆ F ω ˆ B ω , ˆ F ω ∈ U , ˆ B ω ∈ (cid:99) U C + ˆ F ω | J = ˆ Φ ω | J , ˆ B ω | J = I . We will call ˆ F ω a singular frame for f λ . Since ˆ B ω is normalized, the factors ˆ F ω and ˆ B ω dependreal analytically on z , and we can writeˆ B ω = (cid:18) ρ ρ − (cid:19) + (cid:18) µν (cid:19) λ + o ( λ ) , where ρ is a positive real valued function, and µ and ν are C -valued. Now on W , we haveˆ Φ = ˆ F ω ˆ B ω ω , and since ˆ B ω = I along J , we have, for z ∈ J ,ˆ F − ω d ˆ F ω = ˆ Φ − ω d ˆ Φ ω − d ˆ B ω = ˆ ξ ω − (cid:18) d ρ − ρ − d ρ (cid:19) − (cid:18) µ d ν (cid:19) λ + o ( λ ) . Because ˆ F ω is U -valued, it now follows from equation (3.4) and the reality condition defining U that, for z ∈ J ,ˆ F − ω d ˆ F ω = (cid:26)(cid:18) − a (cid:19) λ − + (cid:18) − a a (cid:19) λ − + (cid:18) ab (cid:19) λ − (cid:27) d z + (cid:18) ir − ir (cid:19) d z − (cid:18) d ρ − ρ − d ρ (cid:19) + (cid:26)(cid:18) b ¯ a (cid:19) λ + (cid:18) ¯ a − ¯ a (cid:19) λ + (cid:18) − ¯ a (cid:19) λ (cid:27) d¯ z , and it is necessary that (cid:18) b ¯ a (cid:19) λ d z − (cid:18) µ d ν (cid:19) λ = (cid:18) b ¯ a (cid:19) λ d¯ z . The (1,2) component of this matrix equation is equivalent to µ x = , µ y = i ¯ b . The reality condition for ˆ F − ω d ˆ F ω also requires that the (1,1) component of the term constantin λ is pure imaginary, so ir ( d x + i d y ) − ρ x d x − ρ y d y = i ( p d x + q d y ) , for some real functions p and q . The real part of this equation is equivalent to ρ x = , ρ y = − r . Writing the ( , ) term as ir d z − ( − r ) d y = ir d z + ir d¯ z , we have just seen that, along J , thesingular frame has Maurer-Cartan form:ˆ F − ω d ˆ F ω = ˆ U ω d z + ˆ V ω d¯ z , ˆ U ω = (cid:18) − a λ − + ir a λ − − a λ − + b λ − a λ − − ir (cid:19) , ˆ V ω = (cid:18) ir + ¯ a λ ¯ b λ − ¯ a λ ¯ a λ − ir − ¯ a λ (cid:19) . (3.6)Differentiating the Sym-Bobenko formula (2.7), we obtainˆ F − ω f λ z ˆ F ω = − H (cid:18) [ ˆ U ω , e ] + i λ ∂∂ λ ˆ U ω (cid:19) , = − ia λ − H (cid:18) − λλ − − (cid:19) , and similarly, ˆ F − ω f λ ¯ z ˆ F ω = − i ¯ a λ H (cid:18) − λλ − − (cid:19) . Adding and subtracting these equations leads toˆ F − ω f λ x ˆ F ω = − ( a λ − ) H (cid:18) i − i λ i λ − − i (cid:19) , (3.7) ˆ F − ω f λ y ˆ F ω = ( a λ − ) H (cid:18) i − i λ i λ − − i (cid:19) . Now, since ˆ F ω ( z , ¯ z , λ ) is an element of SU , , it acts by isometries on su , = L , and it followsthat f λ x and f λ y are parallel and null. Moreover, f λ x ∈ L is the zero vector if and only ifRe ( a λ − ) =
0. Since a is holomorphic, either the real part of a λ − is equivalent to zero alongthe real line, in which case f λ ( J ) is a single point, or Re ( a λ − ) has isolated zeros on J , and f λ (cid:12)(cid:12) J is regular away from these zeros. Item 5:
By Lemma 2.6, ˆ Φ is in the big cell if and only if(3.8) h : = | µ + ρ | | ρ | − (cid:54) = . Now we know that for z ∈ J , we have ρ = µ =
0, so h = | µ + ρ | | ρ | − = J asexpected. To guarantee that Σ ◦ is non-empty, we need to ensure that h is not constant, and forthis it is sufficient to require that ∂ h ∂ y (cid:54) = z ∈ J . Using the above expressionsfor ρ y and µ y , and ρ = µ =
0, one computes ∂ h ∂ y = ρ y + ( µ y + ¯ µ y )= − r + b . If this expression is non-zero at z ∈ J , then it is also non-zero on a neighbourhood N of z ,and, because h = h y (cid:54) = J ∩ N it follows that, taking N smaller if necessary, thezero set C ∩ N of h (cid:12)(cid:12) N is precisely J ∩ N . (cid:3) Note:
From here on, to simplify notation, we consider mainly f = f , rather than f λ forother values of λ ∈ S . We will also use the convention X : = ˆ X (cid:12)(cid:12) λ = , if ˆ X depends on λ .One has the following formulae for the metric and Hopf differential of the surface justconstructed: INGULARITIES OF CMC SURFACES IN L Lemma 3.4.
Let f = (cid:102) S ( ˆ Φ ω ) = (cid:102) S ( ˆ Φ ) : Σ s → L be a generalized H-surface constructedfrom a singular holomorphic frame, factored on ˆ Φ − ω ( B , ) as ˆ Φ ω = ˆ F ω ˆ B ω as in Theorem 3.3,and write the Fourier expansion of the matrix valued function ˆ B ω ∈ (cid:99) U C + as: ˆ B ω = (cid:18) ρ ρ − (cid:19) + (cid:18) µν (cid:19) λ + o ( λ ) . Let Σ ± : = ˆ Φ − ( B ± , ) . Then: (1) The metric d s , induced by f on ˆ Φ − ω ( B , ) , is given by the formula d s = g ( d x + d y ) , g = ε e u = ε χ | a | H , (3.9) ε ( z ) = ± , for z ∈ Σ ± , χ = (cid:113) || µ + ρ | − ρ − | . (3.10) The function g is real analytic on ˆ Φ − ω ( B , ) \ R , and extends as a C functionacross the real line. It has the following values at a point z ∈ R ∩ ˆ Φ − ω ( B , ) : (3.11) g = , ∂ g ∂ x = , ∂ g ∂ y = | a | ( Im b − r ) H . (2) The Hopf differential on ˆ Φ − ω ( B , ) is given by Q d z, where (3.12) Q = aH ( b − ˜ b − ir ) . Proof.
Item 1:
On ˆ Φ − ω ( B , ) ∩ ˆ Φ − ( B , ) we have, using Lemma 2.6,ˆ Φ = ˆ F ω ˆ B ω ω = ˆ F ω ˆ X ˆ B (cid:48) = ˆ F ˆ B , where ˆ F = ε ˆ F ω ˆ X , ˆ B = ε ˆ B (cid:48) , and X and ˆ B (cid:48) are given in equation (2.16). Writing the Fourierexpansion ˆ B = (cid:18) χ χ − (cid:19) + o ( λ ) , the choice of u and v in ˆ B (cid:48) given in Lemma 2.6 gives the formula (3.10) for χ . Since χ > Φ = ˆ F ˆ B with ˆ B ∈ (cid:99) U C + .Using this, and the expression (3.5) for ˆ Φ − d ˆ Φ , one obtainsˆ F − d ˆ F = ˆ B ˆ Φ − d ˆ Φ ˆ B − + ˆ B d ˆ B − = (cid:18) χ a λ − χ − ( b − ˜ b − ir ) λ − (cid:19) d z + o ( λ ) . To calculate the metric, the formulae (2.8), at λ =
1, for f z and f ¯ z then give: f x = iH F (cid:18) χ a − χ ¯ a (cid:19) F − = χ | a | H F C e F − C , where(3.13) ˆ F C : = ˆ FD , D = (cid:32) e i ( φ + π ) e − i ( φ + π ) (cid:33) , a = | a | e i φ . A well-defined choice for the function φ can be made because a is non-vanishing on the simplyconnected set Σ . Similarly we have f y = χ | a | H F C e F − C . It follows that ˆ F C is the coordinate frame defined by equations (2.2) and that 2 e u = χ | a | H (recalling that we have assumed H is positive), which gives the formula (3.9) for the metric.The factor ε is included to achieve continuity of the derivatives of g across R .The function g = ε χ | a | H is real analytic everywhere on ˆ Φ − ω ( B , ) \ J , because ρ and a arenon-vanishing and g is non-vanishing on this set. It has the limiting value zero for z → J ,because ρ (cid:12)(cid:12) J = µ (cid:12)(cid:12) J =
0. To compute the limits of the derivatives at (3.11) for real valuesof z , one can differentiate the formula χ = (cid:112) ε ( | µ + ρ | − ρ − ) , with ε = ± z ∈ Σ ± , anduse the equations µ x → = ρ x → µ y → i ¯ b , ρ y → − r , found in the proof of Theorem 3.3. Item 2:
The standard coordinate frame ˆ F C , found above, satisfiesˆ F − C d ˆ F C = (cid:18) − i χ | a | λ − i a | a | χ − ( b − ˜ b − ir ) λ − (cid:19) d z + o ( ) , = ˆ U d z + ˆ V d¯ z , where ˆ U is given at (2.6). Comparing the off-diagonal components of the above matrix withthose of ˆ U , and using χ = e u H | a | , we have i a | a | | a | He u ( b − ˜ b − ir ) = ie − u Q , which is the expression (3.12) for Q . (cid:3) The converse of Theorem 3.3.
Next we show that every generalized H -surface that con-tains a curve in the coordinate domain of its singular set can be locally represented, around thatcurve, by a standard singular holomorphic potential.If ˆ Φ : Σ → U C is a holomorphic map, and ˆ Φ maps at least one point into B , , then,according to Theorem 2.5, the singular set C = ˆ Φ − ( P ) is locally given as the zero set of anon-constant real analytic function h : R → R . In our setting, h is given by the formula (3.8), h : = | µ + ρ | | ρ | − Definition 3.5.
A point z ∈ ˆ Φ − ( P ) is said to be a non-degenerate singular point if thederivative map d h has rank 1 at z , and degenerate if d h = . If, at a point z ∈ ˆ Φ − ( P ) we have the milder condition that there exists a real analytic curve γ : ( − δ , δ ) → Σ , for some δ > , with γ ( ) = z and γ (( − δ , δ )) ⊂ ˆ Φ − ( P ) , then we call z weakly non-degenerate . Ageneralized H-surface is non-degenerate or weakly non-degenerate if all singular points havethe corresponding property. For a surface constructed via Theorem 3.3, the non-degeneracy condition is Im b − r (cid:54) = Theorem 3.6.
Let f : Σ s → L be a generalized H-surface with a corresponding standardpotential ˆ ξ and holomorphic extended frame ˆ Φ , with f = (cid:102) S ( ˆ Φ ) . Let z ∈ C = ˆ Φ − ( P ) be aweakly non-degenerate singular point. Then, on an open set Ω ⊂ Σ s , containing z , there existconformal coordinates and a standard singular holomorphic potential ˆ ξ ω , of the form (3.4),with corresponding singular holomorphic extended frame ˆ Ψ ω , such that f is represented on Ω by the surface (cid:102) S ( ˆ Ψ ω ) . INGULARITIES OF CMC SURFACES IN L Proof. If z ∈ C and ˆ Φ ( z ) = ˆ F ω ˆ B is the Iwasawa factorization, set ˆ Φ ω ( z ) = ˆ Φ ( z ) ˆ B − ω − .Then ˆ Φ ω ( z ) = ˆ F ∈ B , , so locally we can Iwasawa factorize ˆ Φ ω ( z ) = ˆ F ω ( z ) ˆ B ω ( z ) , with thetwo factors in U and (cid:99) U C + respectively. Now(3.14) ˆ Φ ( z ) = ˆ Φ ω ( z ) ω ˆ B = ˆ F ω ( z ) ˆ B ω ( z ) ω ˆ B , and this is in the big cell precisely when ˆ B ω ( z ) ω is. As z is weakly non-degenerate, there isa curve through z which is mapped by ˆ Φ into P . After a conformal change of coordinates(taking a smaller neighbourhood if necessary) we can assume that this curve is an open interval J on the line { y = } ⊂ C , and that z is the origin. By Lemma 2.6, we can, on the interval J ,writeˆ B ω ( x , ) ω = R θ ( x ) ω (cid:101) B ( x ) , R θ ( x ) : = (cid:18) e − i θ ( x ) / e i θ ( x ) / (cid:19) ∈ U , (cid:101) B ( x ) ∈ U C + , where R θ and (cid:101) B are real analytic in x . Substituting into equation (3.14), this meansˆ Φ (cid:12)(cid:12) J ( x ) = F ∗ ( x ) ω B ∗ ( x ) , F ∗ ( x ) : = ˆ F ω ( x , ) R θ ( x ) , B ∗ ( x ) : = (cid:101) B ( x ) ˆ B . Now, by extending θ ( x ) analytically, R θ has a holomorphic extension ˇ R θ : Ω → U C to someopen set Ω containing I . Similarly, since the Maurer-Cartan form of ˆ F ω (cid:12)(cid:12) J , has only a finitenumber of real analytic functions in its Fourier expansion in λ , this map also has a holomorphicextension to a map ˇ F ω : Ω → U C , taking Ω sufficiently small. Therefore B ∗ = ω − · R − θ · ˆ F − ω (cid:12)(cid:12) J · ˆ Φ (cid:12)(cid:12) J extends holomorphically to a map B ∗ : Ω → U C + , given by B ∗ ( z ) = ω − · ˇ R − θ ( z ) · ˇ F − ω ( z ) · ˆ Φ ( z ) . This allows one to define a holomorphic mapˆ Ψ ( z ) : = ˆ Φ ( z ) B − ∗ ( z )= ˇ F ω ( z ) ˇ R θ ( z ) ω . This has the property that (cid:102) S ( ˆ ψ ( z )) = (cid:102) S ( ˆ Φ ( z )) , because B − ∗ ( z ) ∈ U C + and therefore has noimpact on the Iwasawa decomposition of ˆ Φ . Moreover, it is easy to verify that ˆ Ψ − d ˆ Ψ is also a standard holomorphic potential, because right multiplication by a holomorphic map into U C + preserves the relevant properties. Finally, consider the translate, ˆ Ψ ω : = ˆ Ψ ω − . By definition,we have ˆ Ψ ω (cid:12)(cid:12) J ( x ) = F ∗ ( x ) ∈ U . Hence, as shown in Section 3.1, it follows that ˆ ξ ω : = ˆ Ψ − ω d ˆ Ψ ω is a singular holomorphicpotential of the form given by (3.4). By construction, we have, on the open set Ω , (cid:102) S ( ˆ Ψ ω ) = (cid:102) S ( ˆ Ψ ) = (cid:102) S ( ˆ Φ ) = f . (cid:3)
4. P
RESCRIBING SINGULARITIES : THE SINGULAR B JÖRLING PROBLEM
We showed that if f : Σ s → L is a generalized H -surface, and z ∈ Σ s is a weakly non-degenerate singular point, then, at least locally, f can be constructed from a singular frame ˆ F ω which satisfies the equations (3.7), which, at λ =
1, are:(4.1) F − ω f x F ω = − ( a ) H ( − e + e ) , F − ω f y F ω = ( a ) H ( − e + e ) . The singular Björling problem can be stated as the task of constructing the singular frame ˆ F ω – and hence the surface – given that we only know f (and therefore f x , if x is the parameter ofthe curve) and f y along the singular curve. So suppose we have an open set Ω ⊂ C , with coordinates z = x + iy , and such that J = Ω ∩ R = ( x , x ) is a non-empty open interval containing the origin. Suppose there exists ageneralized H -surface f : Ω → L , satisfying the Björling data along J , and with associatedholomorphic extended frame ˆ Φ , such that ˆ Φ ( J ) ⊂ P . Since the vector fields f x and f y areboth necessarily null and parallel along J , we can, on this interval, and after an isometry of L ,write f x = s (cid:18) i e i θ e − i θ − i (cid:19) , f y = t (cid:18) i e i θ e − i θ − i (cid:19) , θ ( ) = − π , where s , θ and t are all real analytic functions J → R . We assume that s and t never vanish atthe same time, so that θ is well defined on J .The equations (4.1) suggest that we choose a frame F to be the rotation about the x -axiswhich rotates [ cos θ , sin θ , ] T ∈ L so that it points in the − e direction:(4.2) F = (cid:32) e i θ + π e − i θ + π (cid:33) . The normalization of θ means that F ( ) = I . Then(4.3) F − f x F = s ( − e + e ) , F − f y F = t ( − e + e ) . Comparing this with equations (4.1), we must have, along J ,Re a = − Hs , Im a = Ht . Thus our regularity assumption on s and t is actually equivalent to the assumption that thesurface is a generalized H -surface, i.e. a is non-vanishing.To find the λ dependence of the singular frame, we know from equation (3.3) that this framesatisfies:(4.4) ˆ F − ω d ˆ F ω = (cid:26)(cid:18) − a λ − a λ − − a λ − + b λ − a λ − (cid:19) + (cid:18) ir − ir (cid:19) + (cid:18) ¯ a λ ¯ b λ − ¯ a λ ¯ a λ − ¯ a λ (cid:19)(cid:27) d x . Evaluating at λ = F − d F = (cid:18) i θ x − i θ x (cid:19) d x , and using the above formula for Im a , we obtain along J the values : r = ( θ x + Ht ) , and b = iHt . Substituting a , b and r into equation (4.4) and extending holomorphically, gives thesingular holomorphic potential ˆ ξ ω . The non-degeneracy condition r − Im b (cid:54) = θ x (cid:54) = . Theorem 4.1.
Suppose given a real analytic function f : J → L , such that d f d x is a null vectorfield, and an additional null real analytic vector field v ( x ) , such that v ( x ) is a scalar multipleof d f d x ( x ) for each x ∈ J. Suppose also that the vector fields do not vanish simultaneously atany point x ∈ J. Let s and t be defined as above. Let ˆ Φ ω be the singular holomorphic frameobtained by analytically extending the 1-form ˆ F − ω d ˆ F ω given by (4.4), witha = H ( − s + it ) , b = iHt , r = ( θ x + Ht ) , INGULARITIES OF CMC SURFACES IN L to some simply connected open set containing J, and integrating with initial condition ˆ Φ ω ( ) = I. Suppose that ˆ Φ = ˆ Φ ω ω maps at least one point into B , . Then the surfacef ( x , y ) : = (cid:102) S ( ˆ Φ ω ( x , y )) + H e + f ( ) , is the unique weakly non-degenerate generalized H-surface such that f , f x and f y coinciderespectively with f , d f d x and v along the real interval J. Uniqueness here is understood to mean that the two surfaces are both defined and agree onsome open subset of C containing the interval J . We remark that a condition that guaranteesthat ˆ Φ maps at least one point into the big cell is that d f d x is not parallel to d f d x (that is, θ x (cid:54) = J . Proof.
By construction, and with the assumption that ˆ Φ − ( B , ) is non-empty, f is a general-ized H -surface that has the required values along J , so we need to show uniqueness.Suppose ˜ f is another generalized H -surface satisfying the Björling data. It is necessar-ily weakly non-degenerate. By Theorem 3.6, there exists a standard singular holomorphicpotential ˜ ξ ω and singular holomorphic frame ˆ Ψ ω such that (cid:102) S ( ˆ Ψ ω ) = ˜ f + translation. Nocoordinate change is necessary, since the condition that ˜ f is not immersed along J implies thatthe holomorphic extended frame defining ˜ f already maps J into P .Let ˆ G ω be the singular frame obtained by the Iwasawa decomposition ˆ Ψ ω = ˆ G ω ˆ B ω , withˆ B ω ∈ (cid:99) U C + . As shown in the proof of Theorem 3.3, the map ˜ f satisfies, at points z ∈ J ,(4.6)ˆ G − ω ˜ f x ˆ G ω = − ( A λ − ) H (cid:18) i − i λ i λ − − i (cid:19) , ˆ G − ω ˜ f y ˆ G ω = ( A λ − ) H (cid:18) i − i λ i λ − − i (cid:19) , where ˆ Ψ − d ˆ Ψ = (cid:18) AB (cid:19) λ − d z + o ( λ ) , and ˆ Ψ : = ˆ Ψ ω ω . On the other hand, we have, byassumption that ˜ f x and ˜ f y satisfy the equations (4.3), namely, along J ,˜ f x = sF ( − e + e ) F − , ˜ f y = tF ( − e + e ) F − . We will first show that we can assume, without loss of generality, that Re A = − Hs and Im A = Ht as follows: comparing the equations above, it follows that, wherever s (cid:54) = (cid:54) = t we have t Im A = − s Re A = : κ . At least one of s ( x ) or t ( x ) is non-zero at each x ∈ J , and so κ : J → R is well defined andnon-vanishing. Let β be the holomorphic extension of √ κ H to a simply connected open set N ⊂ C which contains J . Set ˆ Ψ (cid:48) : = ˆ Ψ (cid:18) β − β (cid:19) . Then (cid:102) S ( ˆ Ψ (cid:48) ) = (cid:102) S ( ˆ Ψ ) because the U factor in the Iwasawa factorization is the same for bothof these. So we can replace ˆ Ψ by ˆ Ψ (cid:48) and we have ( ˆ Ψ (cid:48) ) − d ˆ Ψ (cid:48) = (cid:18) a β − B (cid:19) λ − d z + o ( λ ) , where a = H ( − s + it ) on J . The new singular frame ˆ G (cid:48) ω , which is obtained from the factoriza-tion of ˆ Ψ (cid:48) ω − = : ˆ Ψ (cid:48) ω = ˆ G (cid:48) ω ˆ B (cid:48) ω satisfies S ( ˆ G (cid:48) ω ) = S ( ˆ G ω ) = ˜ f + translation, but the frame now also satisfies, along J , the analogue of equations (4.6), replacing A with a = H ( − s + it ) .But the frame ˆ F ω constructed above for f also satisfies the same equations. This implies thatˆ F − ω ˆ G (cid:48) ω (cid:12)(cid:12) J = ˆ T , where ˆ T : J → U commutes with the matrix (cid:18) i − i λ i λ − − i (cid:19) . A computation (using that allmatrices are normalized to I at z = T must be of the formˆ T = (cid:18) − iR iR − iR + iR (cid:19) , R : J × S → R , where R depends on the loop parameter λ . Now S ( ˆ G (cid:48) ω ) = S ( ˆ F ω ˆ T )= − H (cid:18) ˆ F ω ˆ T e ˆ T − ˆ F − ω + i λ ( ∂∂ λ ˆ F ω ) ˆ F − ω + i λ ˆ F ω ( ∂∂ λ ˆ T ) ˆ T − ˆ F − ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = = S ( ˆ F ω ) − H (cid:18) ˆ F ω (cid:18) iR R − iR R + iR − iR (cid:19) ˆ F − ω + i λ ˆ F ω ∂ R ∂ λ (cid:18) − i i − i i (cid:19) ˆ F − ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = . We can use the assumption that ˜ f = f along J , that is, S ( ˆ G (cid:48) ω ) = S ( ˆ F ω ) + translation, along J . Since all maps are normalized to the identity at z =
0, this translation is actually the zerovector. It follows from this and the formula for S ( ˆ G (cid:48) ω ) that (cid:18)(cid:18) iR R − iR R + iR − iR (cid:19) + i λ ∂ R ∂ λ (cid:18) − i i − i i (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = = . This gives the pair of equations (cid:18) iR + ∂ R ∂ λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = = , (cid:18) R − iR − ∂ R ∂ λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ = = . Hence R (cid:12)(cid:12) λ = =
0, that is, G (cid:48) ω (cid:12)(cid:12) J = F ω (cid:12)(cid:12) J = F . But we already saw, in the paragraphs preceding this theorem, that, given that we know thevalue of a along J , the singular frame ˆ F ω is then uniquely determined by its value F along J .Hence ˆ G (cid:48) ω = ˆ F ω , and ˜ f = f . (cid:3) Example.
Choose I = R , and the singular curve to be the helix in L given by f ( x ) =[ sin ( x ) , − cos ( x ) , x ] T , f x = [ cos ( x ) , sin ( x ) , ] and v ( x ) = f x ( x ) . Then θ ( x ) = x , s = t = R . We have a = H ( − + i ) , b = iH and r = ( + H ) . The singular potential isˆ ξ ω = H (cid:26)(cid:18) ( − i ) λ − ( − + i ) λ − ( − i ) λ − + i λ − − ( − i ) λ − ) (cid:19) + (cid:18) i (cid:0) + H (cid:1) − i (cid:0) + H (cid:1)(cid:19) + (cid:18) − ( + i ) λ − i λ + ( + i ) λ − ( + i ) λ ( + i ) λ (cid:19)(cid:27) d z . The corresponding translated frame, ˆ Φ = ˆ Φ ω ω has, from equation (3.5), standard potential:ˆ Φ − d ˆ Φ = H (cid:18) iH ( − + i ) λ − − i λ + ( + i ) λ − i (cid:0) + H (cid:1) λ − − iH (cid:19) d z . INGULARITIES OF CMC SURFACES IN L
5. I
DENTIFYING SINGULARITY TYPES VIA THE B JÖRLING CONSTRUCTION
In this section we find the conditions on the Björling data for the surface constructed to havea cuspidal edge, swallowtail or cuspidal cross cap singularity in a neighbourhood of a singularpoint. If one considers non-degenerate H -surfaces parameterized by germs of their Björlingdata at some point, then one can see that these are the generic singularities within this class.However, see the comments in Section 1.5.We first show that every weakly non-degenerate H -surface is a frontal, and then use the cri-teria in [16] and [11] for a frontal to have these types of singularities. Examples are illustratedin Figure 2.5.1. The Euclidean normal to a generalized H -surface. The commutators of our basis ma-trices satisfy [ e , e ] = − e , [ e , e ] = e , and [ e , e ] = e , and from this it follows that theEuclidean cross-product on the vector space R corresponding to L is given by A × B = −
12 Ad e [ A , B ] , where [ , ] is the matrix commutator, and Ad X denotes conjugation by X . Let (cid:107) · (cid:107) E denote thestandard Euclidean norm on R .Let f be a generalized H -surface with holomorphic frame ˆ Φ . Since f x and f y are parallel atsingular points, the cross-product of these vanishes there. Recall that the big cell is the unionof two disjoint open sets, B , = B + , ∪ B − , . It turns out that one achieves continuity acrossthe singular set C by defining, on Σ ◦ , the Euclidean (unit) normal as follows: n E ( z ) : = ε f x × f y (cid:13)(cid:13) f x × f y (cid:13)(cid:13) E ( z ) , ε ( z ) = ± , for z ∈ ˆ Φ − ( B ± , ) . The two sets ˆ Φ − ( B ± , ) are open and disjoint, so n E is a real analytic vector field on Σ ◦ . Lemma 5.1.
Let f : Σ s → L be a weakly non-degenerate generalized H-surface. Then theEuclidean unit normal extends across C = ˆ Φ − ( P ) to give a real analytic vector field on Σ s .At a point z ∈ C, if coordinates are chosen so that the singular holomorphic frame ˆ Φ ω definedin Theorem 3.6 satisfies ˆ Φ ω ( z ) = I, then the Euclidean normal is given at z by (5.1) n E ( z ) = √ ( e + e ) . If ˆ F ω is the singular frame obtained from ˆ Φ ω then, at nearby singular values z ∈ C, the Eu-clidean normal is the unit vector in the direction of (5.2) (cid:102) n E = Ad e F ω ( − e + e ) F − ω . Proof.
On a neighbourhood, Ω ⊂ Σ , of z ∈ C we can assume by Theorem 3.6 that f is definedby a standard singular holomorphic frame ˆ Φ ω with ˆ Φ ω ( z ) = I , with coordinates such that z =
0, and that there is an interval J = Ω ∩ R containing 0 such that J ⊂ C . On an opendense subset, Ω ◦ = Ω ∩ Σ ◦ , of Ω , we can Iwasawa factorize the standard holomorphic frameˆ Φ = ˆ Φ ω ω as ˆ Φ = ˆ F ˆ B , with ˆ F ∈ U , ˆ B ∈ (cid:99) U C + . Now we have, f x = | f x | F C e F − C , f y = | f y | F C e F − C = | f x | FDe D − F − , f y = | f y | FDe D − F − where F C and D are given at equation (3.13), and so n E points in the direction of (cid:101) X = ε (cid:0) FDe D − F − (cid:1) × (cid:0) FDe D − F − (cid:1) = ε Ad e ( F e F − ) . As in the proof of Lemma 3.4, by Lemma 2.6, we haveˆ Φ = ˆ F ˆ B = ˆ Φ ω ω = ˆ F ω ˆ B ω ω = ˆ F ω ˆ K ˆ B (cid:48) , where ˆ F = ε ˆ F ω ˆ K , ˆ B ω = (cid:18) ρ ρ − (cid:19) + (cid:18) µν (cid:19) λ + o ( λ ) , and ρ : Ω → R + , and µ and ν are C -valued. We also have ˆ B ω (cid:12)(cid:12) J = I , ˆ F ω ( ) = I . On Ω ◦ we can write (cid:101) X = ε Ad e (cid:0) F ω Ke K − F − ω (cid:1) . Since F ω is real analytic on the whole of Ω , we only need to analyze (cid:101) Y : = ε Ke K − . Accordingto Lemma 2.6, we can choose ˆ K as ˆ K = (cid:18) u v λε ¯ v λ − ε ¯ u (cid:19) , v = √ ε h , u = ε ( µ + ρ ) ρ ¯ v , h : = | µ + ρ | | ρ | − , | u | − | v | = ε . Then (cid:101) Y = ε (cid:18) i ε ( u ¯ u + v ¯ v ) − iuv i ¯ u ¯ v − i ε ( u ¯ u + v ¯ v ) (cid:19) , and (cid:13)(cid:13)(cid:13)(cid:101) Y (cid:13)(cid:13)(cid:13) E = ( | u | + | v | ) + | u | | v | = ( ε + | v | ) + ( ε + | v | ) | v | = + h (cid:18) + h (cid:19) . The unit vector in the direction of (cid:101) Y is Y = (cid:0) + h − (cid:0) + h − (cid:1)(cid:1) − (cid:101) Y = i (cid:18) Z + Z − ( µ + ρ ) ρ Z ( ¯ µ + ρ ) ρ Z − Z − Z (cid:19) , (5.3)where Z : = ε h − ( + h − ( + h − ) − , lim h → Z = √ , lim h → ∂ Z ∂ y = − √ h y , (5.4) Z : = ε ( + h − ( + h − ) − , lim h → Z = , lim h → ∂ Z ∂ y = √ h y . (5.5)Thus Y is a well-defined real analytic vector field which, for real values of z , that is when h = µ = ρ =
1, has the value Y ( x ) = √ ( − e + e ) . Substituting this for ε Ke K − in the expression for (cid:101) X above, gives the stated formulae for n E ( z ) and (cid:102) n E ( x ) . (cid:3) INGULARITIES OF CMC SURFACES IN L Lemma 5.2.
Let f be a generalized H-surface constructed from the Björling data in Theorem4.1. At z = , the derivative d n E of the Euclidean unit normal is given by (5.6) d n E = − θ x √ e d x − Ht √ ( − e + e ) d y . Proof.
We showed in the previous lemma that n E = β X , for some real-valued function β and X = Ad e ( F ω Y F − ω ) , where Y is given by equation (5.3). We also have that X ( ) = n E ( ) ,which means that β ( ) =
1. Now (cid:104) n E , d n E (cid:105) E =
0, and X is parallel to n E , so it follows thatd n E = d β X + β d X = β ( d X − (cid:104) d X , n E (cid:105) E n E ) , and we need to computed X = Ad e (cid:0) F ω (cid:2) F − ω d F ω , Y (cid:3) F − ω (cid:1) + Ad e (cid:0) F ω d Y F − ω (cid:1) . At z =
0, we have, using U ω and V ω from (3.6), F − ω ( F ω ) x = i θ x (cid:18) − (cid:19) , F − ω ( F ω ) y = i ( U ω − V ω ) = (cid:18) Hs i − Ht − Hs i − Ht + Hs i − Hs i (cid:19) , and, by the formulae h y = − ( r − Im b ) , µ y = i ¯ b and ρ y = − r from the proof of Theorem 3.3, h = , h x = , h y = − θ x , µ = , µ x = , µ y = Ht , ρ = , ρ x = , ρ y = − ( θ x + Ht ) . Using these and the formulae (5.3)-(5.5) one obtains, at z = X x = − θ x √ e , X y = − Ht √ e . Together with the value n E = √ ( e + e ) at z =
0, and β ( ) =
1, this gives the expression(5.6) for β ( d X − (cid:104) d X , n E (cid:105) E n E ) (cid:12)(cid:12) z = . (cid:3) Lemma 5.3.
Let f : Σ s → L be a generalized H-surface constructed by the data in Theorem3.3, with ˆ Φ ω ( ) = ˆ Φ ( ) ω − = I. Set s : = − a ( ) H and t : = a ( ) H so thatf x = s ( − e + e ) , f y = t ( − e + e ) , Let ψ : Σ s → R be defined by ψ = ε (cid:107) f x × f y (cid:107) E , where ε ( z ) = ± , for z ∈ Φ − ( B ± , ) . Then at z = , (5.7) d ψ = | a | ( Im b − r ) H (cid:115) t + s y . In particular, d ψ ( ) = ⇔ Im b ( ) − r ( ) = .Proof. At points away from the real line, we have the decomposition ˆ Φ = ˆ F ˆ B , and the coor-dinate frame found in Lemma 3.4 is: ˆ F C : = ˆ FD , with D = diag (cid:16) e i ( φ + π ) , e − i ( φ + π ) (cid:17) , and a = | a | e i φ . The metric is given by d s = g ( d x + d y ) , with g = ε χ | a | H and χ = (cid:112) || µ + ρ | − ρ − | .And we have:(5.8) f x = ε gF C e F − C , f y = ε gF C e F − C , N = F C e F − C , where N is the Lorentzian unit normal. Now f x × f y = −
12 Ad e [ f x , f y ]= g Ad e F C e F − C , so we can write ψ = ε (cid:107) f x × f y (cid:107) E as ψ = g Γ , Γ : = ε g (cid:107) N (cid:107) E . Although g → (cid:107) N (cid:107) E → ∞ as z → R , we can get an explicit expression for the product Γ .Writing F C = (cid:16) A B ε ¯ B ε ¯ A (cid:17) , the equations (5.8) then imply that, as z →
0, we have the finite limits: g Im ( A ¯ B ) → − s , ε g ( A − B ) → − i s , g Re ( A ¯ B ) → − t , ε g ( A + B ) → − t , which imply ε gA → − ( t + is ) , ε gB → ( − t + is ) . Now N = i (cid:18) ε ( | A | + | B | ) − AB A ¯ B − ε ( | A | + | B | (cid:19) , so Γ = ε g (cid:107) N (cid:107) E = ε g (cid:0) ( | A | + | B | ) + | A | | B | (cid:1) = (cid:0) g ( | A | + | B | + | A | | B | (cid:1) , lim z → Γ = (cid:115) t + s . This limit is non-zero because a is non-vanishing.Similarly, the terms ε gA and ε gB also have well defined derivatives as z → R , followingfrom the second derivatives of f . Since ε gA and ε gB are non-zero at z =
0, their absolutevalues are also differentiable there. Hence the derivative d Γ has a well defined finite limit as z → ∈ R .Returning to ψ = g Γ , we haved ψ ( ) = lim z → ( g Γ + g d Γ ) . Lemma 3.4 informs us that lim z → g = z → ∂ g ∂ y = | a | ( Im b − r ) H , from which the claim ofthe lemma follow. (cid:3) Frontals and fronts.
Let U be a domain of R . A map f : U → E , into the three-dimensional Euclidean space, is called a frontal if there exists a unit vector field n E : U → S ,such that n E is perpendicular to f ∗ ( TU ) in E . The map L = ( f , n E ) : U → E × S is calleda Legendrian lift of f . If L is an immersion, then f is called a front . A point p ∈ U where afrontal f is not an immersion is called a singular point of f .Suppose that the restriction of a frontal f , to some open dense set, is an immersion, and forsome given Legendrian lift L of f , there exists a smooth function ψ : U → R such that, in localcoordinates ( x , y ) , f x × f y = ψ n E . INGULARITIES OF CMC SURFACES IN L Then a singular point p is called non-degenerate if d ψ does not vanish there. In this situation,the frontal f is called non-degenerate if every singular point is non-degenerate. Lemma 5.4.
Let f : Σ s → L be a weakly non-degenerate generalized H-surface. Let n E denote the Euclidean unit normal defined in Section 5.1. Let E denote the vector space L with the standard Euclidean inner product (cid:104) · (cid:105) E . Then the map f : Σ s → E , together with theLegendrian lift L = ( f , n E ) : Σ → E × S , defines a frontal. The surface is non-degenerate asan H-surface, in accordance with Definition 3.5, if and only if it is non-degenerate as a frontal.Proof. By Lemma 5.1, the map n E : Σ s → S is well defined and real analytic, and so L =( f , n E ) is a real analytic Legendrian lift of f ; in particular, f is a frontal. Regarding degeneratepoints, the map ψ above is the signed Euclidean norm ε (cid:107) f x × f y (cid:107) E , discussed in Lemma 5.3,and we showed there that d ψ vanishes at a singular point if and only Im b − r does. The latterexpression is, according to Theorem 3.3, the derivative of the function h , which was usedpreviously to define degeneracy. (cid:3) Lemma 5.5.
Let f be a non-degenerate generalized H-surface constructed from the Björlingdata in Theorem 4.1. Then f is a front on a neighbourhood of z = if and only ift ( ) (cid:54) = . Proof.
According the assumptions of the Björling construction, d f = s ( )( − e + e ) d x + t ( )( − e + e ) d y . By Lemma 5.2, d n E = − θ x e d x + Ht ( ) √ ( e − e ) d y . It follows that the mapd L = ( d f , d n E ) has rank 2 at 0 if and only if t ( ) (cid:54) = (cid:3) Cuspidal edges and swallowtails.
At a non-degenerate singular point, there is a well-defined direction, that is a non-zero vector η ∈ T p U , unique up to scale, such that d f ( η ) = null direction .A test for whether a singularity on a front is a swallowtail or a cuspidal edge is given in[16]: Proposition 5.6. ( [16] ). Let f : U → R be a front, and p a non-degenerate singular point.Suppose that γ : ( − δ , δ ) → U is a local parameterisation of the singular curve, with parameterx and tangent vector γ (cid:48) , and γ ( ) = p,. Then: (1) The image if f in a neighbourhood of p is diffeomorphic to a cuspidal edge if and onlyif η ( ) is not proportional to γ (cid:48) ( ) . (2) The image if f in a neighbourhood of p is diffeomorphic to a swallowtail if and onlyif η ( ) is proportional to γ (cid:48) ( ) and dd x det (cid:0) γ (cid:48) ( x ) , η ( x ) (cid:1) (cid:12)(cid:12)(cid:12) x = (cid:54) = . We can use this test to prove the following result:
Theorem 5.7.
Let f be a non-degenerate generalized H-surface constructed from the Björlingdata in Theorem 4.1. Then: (1) f is locally diffeomorphic to a cuspidal edge at z = if and only ift ( ) (cid:54) = s ( ) (cid:54) = . (2) f is locally diffeomorphic to a swallowtail at z = if and only ift ( ) (cid:54) = , s ( ) = dd x s ( ) (cid:54) = . Proof.
By Lemma 5.5, f is a front at z = t ( ) (cid:54) =
0, so we can use the propositionabove. We also have, along J , f x = sF ( − e + e ) F − , f y = tF ( − e + e ) F − , and the null direction is(5.9) η ( x ) = t ( x ) ∂∂ x − s ( x ) ∂∂ y . Writing x + iy = [ x , y ] T , the singular curve is given by γ ( x ) = [ x , ] T and the null direction by η ( x ) = [ t ( x ) , − s ( x )] T , and so the criteria in Proposition 5.6 imply the claim. (cid:3) Cuspidal cross caps.
From [11] (Theorem 1.4), one has the following test for whether anon-degenerate frontal is locally a cuspidal cross cap:
Theorem 5.8. ( [11] .) Let f : U → R be a frontal, with Legendrian lift L = ( f , n E ) , and let z be a non-degenerate singular point. Let X : V → R be an arbitrary differentiable function ona neighbourhood V of z such that: (1) X is orthogonal to n E . (2) X ( z ) is transverse to the subspace f ∗ ( T z ( V )) .Let x be the parameter for the singular curve, and set ˜ ψ ( x ) : = (cid:104) n E , d X ( η ) (cid:105) E (cid:12)(cid:12) x . The frontal f has a cuspidal cross cap singularity at z = z if and only: (A) η ( z ) is transverse to the singular curve; (B) ˜ ψ ( z ) = and ˜ ψ (cid:48) ( z ) (cid:54) = . Theorem 5.9.
Let f be a non-degenerate H-surface constructed from the Björling data inTheorem 4.1. Then f is locally diffeomorphic to a cuspidal cross cap around z = if and onlyif the following conditions hold:s ( ) (cid:54) = , t ( ) = dd x t ( ) (cid:54) = . Proof.
In a neighbourhood of 0, the singular curve is given by an interval J = ( x , x ) of thereal line. Recall from the proof of Lemma 5.2 that we found the following formula for n E : n E = β Ad e ( F ω Y F − ω ) Y = ae + be + ce , a = Im ( µ ) ρ Z , b = − ( Re ( µ ) + ρ ) ρ Z , c = Z + Z , and along J we have: Z = √ , Z = a = b = − / √ c = / √ Y = √ ( − e + e ) for real values of z .We will apply Theorem 5.8 with the vector field defined by the cross product: X = (cid:0) Ad e F ω e F − ω (cid:1) × (cid:0) Ad e F ω Y F − ω (cid:1) = − F ω [ e , ae + be + ce ] F − ω = − F ω ( ce + ae ) F − ω . X is orthogonal to n E because Ad e F ω Y F − ω is proportional to n E . Along J we have f x = sF ( − e + e ) F − , f y = tF ( − e + e ) F − , X = − √ F e F − , so X is transverse to f ∗ ( T z ( V )) . That is, X satisfies conditions 1 and 2 of Theorem 5.8. INGULARITIES OF CMC SURFACES IN L Now consider the conditions (A) and (B). The null direction along J is given by η = t ∂∂ x − s ∂∂ y , and this is transverse to the singular curve at z = s ( ) (cid:54) =
0, so our firstcondition is equivalent to condition (A).To investigate ˜ ψ , we need an expression for (cid:104) n E , d X (cid:105) E along J . Nowd X = − c d ( F ω e F − ω ) − a d ( F ω e F − ω ) − d c Ad F e − d a Ad F e . Along J we have d a = d ( Im µ ) · √ · =
0, because we earlier computed d µ = Ht d y whichis real. We also have a =
0, and (cid:10) n E , Ad F e (cid:11) E = (cid:68) √ ( e + e ) , e (cid:69) E =
0. We used that F takes values in SU ( ) and so preserves the Euclidean inner product. Hence only the first termin the above expression for d X contributes to (cid:104) n E , d X (cid:105) E : (cid:104) n E , d X (cid:105) E (cid:12)(cid:12) J = − √ (cid:10) n E , d ( F ω e F − ω ) (cid:11) (cid:12)(cid:12) J . To compute this, we use: F − ω ( F ω ) x = U ω + V ω , F − ω ( F ω ) y = i ( U ω − V ω ) , where, from equation (3.6), at λ = U ω + V ω = (cid:18) − i Im a + ir i Im a + ¯ b − i Im a + b i Im a − ir (cid:19) , i ( U ω − V ω ) = (cid:18) − i Re a i Re a − i ¯ b − i Re a + ib i Re a (cid:19) . With this and s = − aH , t = aH , b = iHt , and r = ( θ x + Ht ) one obtains along J ( F ω e F − ω ) x = F [ U ω + V ω , e ] F − = F (cid:18) i Im a − i Im b − i Im a + ir i Im a − ir − i Im a + i Im b (cid:19) F − = F θ x e F − , ( F ω e F − ω ) y = F [ i ( U ω − V ω ) , e ] F − = F (cid:18) i Re a − i Re b − i Re a i Re a − i Re a + i Re b (cid:19) F − = F Hs ( e − e ) F − . Hence we obtain the following expression along J ,˜ ψ = (cid:104) n E , d X ( η ) (cid:105) E (cid:12)(cid:12) x = − √ √ (cid:10) ( Ad e Ad F ( − e + e ) , tX x − sX y (cid:11) E = − (cid:10) e + e , t θ x e − s H ( e − e ) (cid:11) E = − t θ x . Condition (B) of Theorem 5.8 is thus equivalent to the pair of equations t θ x (cid:12)(cid:12)(cid:12) x = = , (cid:20) d t d x θ x + t d θ x d x (cid:21) x = (cid:54) = . Since θ x (cid:54) =
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