Behavior of the Gaussian curvature of timelike minimal surfaces with singularities
BBEHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKEMINIMAL SURFACES WITH SINGULARITIES
SHINTARO AKAMINE
Abstract.
We prove that the sign of the Gaussian curvature of any timelike minimalsurface in the -dimensional Lorentz-Minkowski space is determined by the degeneracyand the orientations of the two null curves that generate the surface. We also investigatethe behavior of the Gaussian curvature near singular points of a timelike minimal surfacewhich admits some kind of singular points. Introduction
A timelike surface in the 3-dimensional Lorentz-Minkowski space L is a surface whosefirst fundamental form is a Lorentzian metric. In contrast with surfaces in the 3-dimensionalEuclidean space E and spacelike surfaces in L , timelike surfaces do not always have realprincipal curvatures, that is, their shape operators are not always diagonalizable even overthe complex number field C . In general the diagonalizability of the shape operator of atimelike surface is determined by the discriminant of the characteristic equation for theshape operator, which is H − K where H is the mean curvature and K is the Gauss-ian curvature of the considered timelike surface. In this paper we study the case that H vanishes identically.A timelike surface whose mean curvature vanishes identically is called a timelike minimalsurface , and McNertney [16] proved that any such surface can be expressed as the sum oftwo null curves (see also Fact 2.2), where a null curve is a regular curve whose velocityvector field is lightlike. Based on the studies [7, 22] for spacelike case, Takahashi [21]introduced a notion of timelike minimal surfaces with some kind of singular points ofrank one, which are called minfaces (see Definition 2.3 and Definition A.2). He also gavecriteria for cuspidal edges, swallowtails and cuspidal cross caps which appear frequentlyon minfaces.The diagonalizability of the shape operator of a timelike minimal surface is determinedby the sign of the Gaussian curvature K . More precisely the shape operator is diago-nalizable over the real number field R on points with negative Gaussian curvature anddiagonalizable over C \ R on points with positive Gaussian curvature. Flat points consistof umbilic points and quasi-umbilic points (see Definition 2.1). Therefore the problem ofthe diagonalizability of the shape operator of a timelike minimal surface is reduced to theproblem of the sign of the Gaussian curvature. This would be quite different from minimalsurfaces in E , which have non-positive Gaussian curvature, and from maximal surfacesin L , which have non-negative Gaussian curvature. Hence, to determine the sign of the Mathematics Subject Classification.
Primary 53A10; Secondary 57R45, 53B30.
Key words and phrases.
Lorentz-Minkowski space, timelike minimal surface, Gaussian curvature, wavefront, singularity. a r X i v : . [ m a t h . DG ] M a y S. AKAMINE
Gaussian curvature of timelike minimal surfaces is an important problem. In this paperwe investigate how to determine the sign of the Gaussian curvature of a timelike minimalsurface near regular and singular points.To achieve our goal, we first give a characterization of flat points of a timelike minimalsurface by the notion of non-degeneracy of null curves which generate the surface (Propo-sition 3.1). Near non-flat points, we also prove that the sign of the Gaussian curvature isdetermined only by the orientations of two generating null curves (see Definition 2.8). Inaddition to this result, by using the notion of pseudo-arclength parameters of null curves,we can also give a construction method of conformal curvature line coordinate systems andconformal asymptotic coordinate systems near non-flat points according to the sign of theGaussian curvature of a timelike minimal surface (Theorem 3.4).About the behavior of the Gaussian curvature near singular points of surfaces in anarbitrary 3-dimensional Riemannian manifold, some notions of curvatures along singularpoints of frontals and wave fronts (or fronts for short, and the definitions of frontals andfronts are given in Section 4) were introduced in [15, 20], and many relations between thebehaviors of these curvatures and the Gaussian curvature along singular points of frontalsand fronts were revealed in [7, 15, 20]. On the other hand, in L , Takahashi [21] provedthat any minface is a frontal and gave a necessary and sufficient condition for a minfaceto be a front (see Fact 4.2). Based on these backgrounds, we prove the following result: Theorem A.
Let f : Σ −→ L be a minface and p ∈ Σ a singular point of f . (i) If p is a cuspidal edge, then there is no umbilic point near p . (ii) If f is a front at p and p is not a cuspidal edge, then there is no umbilic and quasi-umbilic points near p . Moreover the Gaussian curvature K is negative near p and lim q → p K ( q ) = −∞ . (iii) If f is not a front at p and p is a non-degenerate singular point, then there is noumbilic and quasi-umbilic points near p . Moreover the Gaussian curvature K ispositive near p and lim q → p K ( q ) = ∞ . On the Gaussian curvature near cuspidal edges, Saji, Umehara and Yamada pointedout in [20] that the shape of singular points is restricted when the Gaussian curvature isbounded. In [20], they introduced the singular curvature on cuspidal edges, and provedthat if the Gaussian curvature with respect to the induced metric from E is bounded andpositive (resp. non-negative) near a cuspidal edge, then the singular curvature is negative(resp. non-positive). Noting that for a timelike surface the Gaussian curvatures with respectto the induced metrics from E and L have opposite signs, we can prove the followingstatement for minfaces: Theorem B.
The Gaussian curvature with respect to the induced metric from L near acuspidal edge on a minface and the singular curvature have the same sign. In fact we will prove a stronger result (Theorem 4.8) than Theorem B. By Theorems Aand B, we obtain criteria for the sign of the Gaussian curvature near any non-degeneratesingular point on a minface.This article is organized as follows: In Section 2 we describe some notions of timelikesurfaces and null curves. We also give the definition of minfaces as a class of timelikeminimal surfaces with singular points by using a representation formula derived in [21].
EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 3
In Section 3 we investigate the behavior of the Gaussian curvature near regular points.Finally, in Section 4 we discuss the sign of the Gaussian curvature near singular pointson minfaces and prove our main results: Theorem A and Theorem 4.8. In Appendix Awe review a precise description of geometry of minfaces given in Takahashi’s Master thesis[21].
Acknowledgement.
The author is grateful to Professor Atsufumi Honda for his valu-able comments and fruitful discussion. He is also grateful to Professor Miyuki Koiso forher encouragement and suggestions. This work was supported by Grant-in-Aid for JSPSFellows Number 15J06677. 2.
Preliminaries
Timelike surfaces and their shape operators.
We denote by L the 3-dimensionalLorentz-Minkowski space, that is, the 3-dimensional real vector space R with the Lorentzianmetric (cid:104) , (cid:105) = − ( dx ) + ( dx ) + ( dx ) ,where ( x , x , x ) are the canonical coordinates in R . In L , a vector v has one of thethree causal characters : it is spacelike if (cid:104) v, v (cid:105) > or v = 0 , timelike if (cid:104) v, v (cid:105) < , and lightlike or null if (cid:104) v, v (cid:105) = 0 and v (cid:54) = 0 . We denote the set of null vectors by Q := (cid:8) v = ( v , v , v ) ∈ L | (cid:104) v, v (cid:105) = 0 , v (cid:54) = 0 (cid:9) and call it the lightcone . Let Σ := Σ be a two-dimensional connected and oriented smooth manifold and f : Σ −→ L be an immersion.An immersion f is said to be timelike (resp. spacelike ) if the first fundamental form, thatis, the induced metric I = f ∗ (cid:104) , (cid:105) is Lorentzian (resp. Riemannian) on Σ .For a timelike immersion f and its spacelike unit normal vector field ν , the shapeoperator S and the second fundamental form II are defined as df ( S ( X )) = −∇ X ν, II(
X, Y ) = (cid:104)∇ df ( X ) df ( Y ) − df ( ∇ X Y ) , ν (cid:105) , where X and Y are smooth vector fields on Σ , and ∇ , ∇ are the Levi-Civita connections on Σ and L , respectively. The mean curvature H and the Gaussian curvature K are definedas H = (1 /
2) tr II and K = det S . Let ˜ K be the sectional curvature of the Lorentzianmanifold (Σ , I) . Then the Gauss equation ˜ K = K implies that the Gaussian curvature K is intrinsic.One of the most important differences between spacelike surfaces and timelike surfacesis the diagonalizability of the shape operator, that is, the shape operator of a timelikesurface is not always diagonalizable even over C . For surfaces in E and spacelike surfacesin L , the Gaussian curvature K and mean curvature H satisfy H − K ≥ , and theequality holds on umbilic points, where an umbilic point of a surface is a point on whichthe second fundamental form II is a scalar multiple of the first fundamental form I . Onthe other hand, there are three possibilities of the diagonalizability of the shape operatorof a timelike surface in L as follows:(i) The shape operator is diagonalizable over R . In this case H − K ≥ with theequality holds on umbilic points.(ii) The shape operator is diagonalizable over C \ R . In this case H − K < .(iii) The shape operator is non-diagonalizable over C . In this case H − K = 0 . S. AKAMINE
About Case (iii), Clelland [4] introduced the following notion:
Definition 2.1 ([4]) . A point p on a timelike surface Σ is called quasi-umbilic if the shapeoperator of Σ is non-diagonalizable over C .2.2. Timelike minimal surfaces and minfaces.
For a timelike surface f : Σ −→ L ,near each point, we can take a Lorentz isothermal coordinate system ( x, y ) , that is, thefirst fundamental form I is written as I = E ( − dx + dy ) with a non-zero function E , anda null coordinate system ( u, v ) that is, I is written as I = 2Λ dudv . A curve γ in L whosevelocity vector field γ (cid:48) is lightlike is called a null curve , and a null coordinate system isa coordinate system on which the image of coordinate curves are null curves. Moreover,up to constant multiple, there is a one-to-one correspondence between these coordinatesystems as follows: x = u − v , y = u + v . On each null coordinate system ( u, v ) , an immersion f and its mean curvature H satisfy Hν = ∂ f∂u∂v . Therefore, we obtain the following well-known representation formula. Fact 2.2 ([16]) . If ϕ ( u ) and ψ ( v ) are null curves in L such that ϕ (cid:48) ( u ) and ψ (cid:48) ( v ) arelinearly independent for all u and v , then f ( u, v ) = ϕ ( u ) + ψ ( v )2 (1) gives a timelike minimal surface. Conversely, any timelike minimal surface can be writtenlocally as the equation (1) with two null curves ϕ and ψ . In this paper, we consider the following class of timelike minimal surfaces with singularpoints of rank one, which was introduced in [21] (see also Definition A.2 in Appendix A):
Definition 2.3.
A smooth map f : Σ −→ L is called a minface if at each point of Σ thereexists a local coordinate system ( u, v ) in a domain U , functions g = g ( u ) , g = g ( v ) , and1-forms ω = ˆ ω ( u ) du , ω = ˆ ω ( v ) dv with g ( u ) g ( v ) (cid:54) = 1 on an open dense set of U and ˆ ω (cid:54) = 0 , ˆ ω (cid:54) = 0 at each point on U such that f can be decomposed into two null curves: f ( u, v ) = 12 (cid:90) uu (cid:0) − − ( g ) , − ( g ) , g (cid:1) ω + 12 (cid:90) vv (cid:0) g ) , − ( g ) , − g (cid:1) ω + f ( u , v ) . (2)We denote these two null curves by ϕ = ϕ ( u ) and ψ = ψ ( v ) . The quadruple ( g , g , ω , ω ) is called the real Weierstrass data .A singular point of a minface f is a point of Σ on which f is not immersed, and the setof singular points on U of a minface f corresponds to the set { ( u, v ) ∈ U | g ( u ) g ( v ) = 1 } . Remark 2.4.
In [21], Takahashi originally gave the notion of minfaces as Definition A.2in Appendix A by using the notion of para-Riemann surfaces. To study the local behaviorof the Gaussian curvature near singular points of timelike minimal surfaces, we adopt theabove definition. In Appendix A, we prove the representation formula (2) from the originaldefinition of minfaces (Fact A.7) and give a precise description of geometry of minfaces.
EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 5
Null curves.
In this subsection, we describe some notions of null curves.
Definition 2.5 (cf. [6, 19]) . A null curve γ = γ ( t ) in L is called degenerate or non-degenerate at t if γ (cid:48) × γ (cid:48)(cid:48) = 0 or γ (cid:48) × γ (cid:48)(cid:48) (cid:54) = 0 at t , respectively. If γ is non-degenerateeverywhere, it is called a non-degenerate null curve.A null curve which is degenerate everywhere is a straight line with a lightlike direction.As pointed out in Section 2 in [19], the non-degeneracy of a null curve is characterized bythe following conditions. Lemma 2.6 (cf. [19]) . For a null curve γ = γ ( t ) in L the following (i), (ii) and (iii) areequivalent: (i) γ is non-degenerate at t , (ii) γ (cid:48)(cid:48) ( t ) is a non-zero spacelike vector, that is, (cid:104) γ (cid:48)(cid:48) ( t ) , γ (cid:48)(cid:48) ( t ) (cid:105) > , (iii) det ( γ (cid:48) ( t ) γ (cid:48)(cid:48) ( t ) γ (cid:48)(cid:48)(cid:48) ( t )) (cid:54) = 0 . By Lemma 2.6, we can introduce the following notions for non-degenerate null curves.
Definition 2.7 ([3, 23]) . For a non-degenerate null curve γ = γ ( t ) , a parameter t is calleda pseudo-arclength parameter of γ if (cid:104) γ (cid:48)(cid:48) ( t ) , γ (cid:48)(cid:48) ( t ) (cid:105) ≡ . Definition 2.8.
We define the orientation of a non-degenerate null curve γ by the sign of det ( γ (cid:48) γ (cid:48)(cid:48) γ (cid:48)(cid:48)(cid:48) ) . Remark 2.9.
If we take a pseudo-arclength parameter s , then det ( γ (cid:48) γ (cid:48)(cid:48) γ (cid:48)(cid:48)(cid:48) ) = ± ,which represents the orientation of γ . Moreover, the orientation of a non-degenerate nullcurve has the following geometric meaning: If we consider the projection of γ (cid:48) , whichis on the lightcone Q , into the time slice x = 1 , then the projected curve on S = { (1 , x , x ) | (1 , x , x ) ∈ Q } is anticlockwise if the orientation is positive, and clockwiseif the orientation is negative as x increases. See Figure 1 and Remark 3.5. Figure 1.
Examples of non-degenerate null curves with positive (the leftfigure) and negative orientation (the right figure).3.
The sign of the Gaussian curvature and orientations of null curves
In this section we give a characterization of flat points and investigate the sign of theGaussian curvature of minfaces by using the notions of degeneracy and orientations of nullcurves.
S. AKAMINE
A characterization of flat points.
As we saw in Section 2, flat points on eachminface consist of umbilic and quasi-umbilic points. First, we give a characterization offlat points of a minface from a viewpoint of null curves.
Proposition 3.1.
Let p be a regular point in a minface f . Then the following statementshold: (i) p is an umbilic point of f if and only if the two null curves in the equation (1) aredegenerate at p . (ii) p is a quasi-umbilic point of f if and only if only one of the two null curves in theequation (1) is degenerate at p .Proof. If we take a null coordinate system ( u, v ) on which f is written as (1), then the firstand the second fundamental forms can be written as follows: I = 2Λ dudv and
II =
Qdu + Rdv .Therefore, the shape operator is S = I − II = (cid:18) (cid:19) (cid:18) Q R (cid:19) = (cid:18) R Λ Q Λ (cid:19) . (3)On the other hand, we can see that there exists a real number a such that f uu ( p ) = ϕ (cid:48)(cid:48) ( p ) = aϕ (cid:48) ( p ) + 2 Q ( p ) ν ( p ) .Therefore, ϕ is degenerate at p if and only if Q ( p ) = 0 . By using (3), we obtain the desiredresult. (cid:3) Remark 3.2.
The differential coefficients Q and R are called (coefficients of) Hopf differ-entials on a timelike surface, which was introduced in [9]. The degenerate points of twonull curves ϕ and ψ correspond to zeros of these Hopf differentials Q and R , respectively. Example 3.3.
Let us take the two null curves ϕ and ψϕ ( u ) = (cid:18) u + u , u , u − u (cid:19) , ψ ( v ) = (cid:18) − v − v , v , v − v (cid:19) , which are degenerate at the origin, and consider the timelike minimal surface constructedby the equation (1). The Gaussian curvature K of this surface is K = − uv (1+ u v ) . Propo-sition 3.1 states that the set of flat points of this surface consists of quasi-umbilic pointsexcept the intersection and the intersection is an umbilic point. See Figure 2. As this ex-ample, the quasi-umbilic points (and also the umbilic points) of a timelike minimal surfaceare not isolated in general. Figure 2.
An example on which the sign of the Gaussian curvaturechanges along quasi-umbilic curves (black curves except the intersection).
EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 7
The sign of the Gaussian curvature near non-flat points.
In the previoussubsection we gave a characterization of flat points using the notion of degeneracy of nullcurves of a minface. In this subsection we investigate how to determine the sign of theGaussian curvature and give a construction method of conformal curvature line (resp.conformal asymptotic) coordinate systems near non-flat points of a minface based on thestudy by Takahashi [21].First we consider the two null curves ϕ = ϕ ( u ) and ψ = ψ ( v ) in the equation (2).Away from flat points, the two null curves ϕ and ψ are non-degenerate by Proposition3.1, and hence we can take pseudo-arclength parameters of ϕ and ψ near non-flat points.Since (cid:104) ϕ (cid:48)(cid:48) , ϕ (cid:48)(cid:48) (cid:105) = 4 g (cid:48) ˆ ω , ˆ ω (cid:54) = 0 , and (ii) of Lemma 2.6, g (cid:48) (cid:54) = 0 near each non-flat point.Moreover, the parameter u is a pseudo-arclength parameter of ϕ if and only if ˆ ω and g satisfy g (cid:48) ˆ ω ( u ) = − ε ϕ , ε ϕ = ± . After a straightforward calculation, we obtain the equation det( ϕ (cid:48) , ϕ (cid:48)(cid:48) , ϕ (cid:48)(cid:48)(cid:48) ) = ε ϕ , thatis, ε ϕ is nothing but the orientation of ϕ which was introduced in Definition 2.8. Similarly,the parameter v is a pseudo-arclength parameter of ψ if and only if ˆ ω and g satisfy g (cid:48) ˆ ω ( v ) = − ε ψ , ε ψ = ± , and ε ψ also represents the orientation of ψ . Therefore we obtain the following formula nearnon-flat points f ( u, v ) = 12 (cid:90) uu (cid:0) − − ( g ) , − ( g ) , g (cid:1) − ε ϕ g (cid:48) du + 12 (cid:90) vv (cid:0) g ) , − ( g ) , − g (cid:1) − ε ψ g (cid:48) dv + f ( u , v ) . (4)From now on, we consider the Lorentz isothermal coordinate system ( x, y ) = ( u − v , u + v ) associated to the null coordinate system ( u, v ) constructed from pseudo-arclength param-eters of ϕ and ψ . On the coordinate system, the first and the second fundamental forms I and II can be written as follows: I = ε ϕ ε ψ g (cid:48) g (cid:48) (1 − g g ) ( − dx + dy ) , II = ( ε ϕ − ε ψ dx + dy ) + ( ε ϕ + ε ψ ) dxdy. We denote the conformal factor ε ϕ ε ψ g (cid:48) g (cid:48) (1 − g g ) by E . Then the Gaussian curvature K ofthe minface is written as K = ε ϕ ε φ E . (5)Therefore, the sign of the Gaussian curvature of the non-flat points of a minface is deter-mined only by the orientations of two null curves ϕ and ψ . In summary, we have obtainedthe following theorem, which also gives a construction method of conformal curvatureline coordinate systems and conformal asymptotic coordinate systems by using pseudo-arclength parameters. Theorem 3.4.
Away from flat points, each minface f : Σ → L can be written locally asthe equation (4). The Gaussian curvature K is positive (resp. negative) if and only if ϕ and ψ have the same orientation (resp. different orientations). In this case, the Lorentzisothermal coordinate system ( x, y ) = ( u − v , u + v ) associated to the null coordinate system ( u, v ) in (4) is a conformal asymptotic (resp. conformal curvature line) coordinate system. S. AKAMINE
Remark 3.5.
In Remark 1 in [17], Milnor normalized null coordinates u, v so that u and v are Euclidean arclength parameters of ϕ/ and ψ/ in the equation (1), that is, on thiscoordinate system a timelike minimal surface f can be written as f ( u, v ) = 1 √ (cid:18) u − u , (cid:90) uu cos A ( τ ) dτ, (cid:90) uu sin A ( τ ) dτ (cid:19) + 1 √ (cid:18) v − v , (cid:90) vv cos B ( τ ) dτ, (cid:90) vv sin B ( τ ) dτ (cid:19) + f ( u , v ) , where, A and B are called the Weierstrass functions . By using these functions, Milnorgave the following formula giving control over the sign of the Gaussian curvature K : sgn K = sgn( A (cid:48) B (cid:48) ) (6)After a straightforward calculation, we get det( ϕ (cid:48) , ϕ (cid:48)(cid:48) , ϕ (cid:48)(cid:48)(cid:48) ) = ( A (cid:48) ) , and hence sgn( A (cid:48) B (cid:48) ) = ε ϕ ε ψ . About the sign of the Gaussian curvature, the equation (5) is nothing but (6).4. Behavior of the Gaussian curvature near singular points
In this section we investigate the behavior of the Gaussian curvature near non-degeneratesingular points on a minface by using some notions about null curves given in Section 2.3and results for the Gaussian curvature near regular points given in Section 3.4.1.
Frontals and fronts.
First we recall the singularity theory of frontals and fronts,see [1, 7, 20, 22] for details. Let U be a domain in R and u , v are local coordinates on U .A smooth map f : U −→ R is called a f rontal if there exists a unit vector field n on U such that n is perpendicular to df ( T U ) with respect to the Euclidean metric (cid:104) , (cid:105) E of R .We call n the unit normal vector field of a frontal f . Moreover if the Legendrian lift L ofa frontal f L = ( f, n ) : U −→ R × S is an immersion, f is called a f ront . A point p ∈ U where f is not an immersion is calleda singular point of the frontal f , and we call the set of singular points of f the singularset . We can take the following smooth function λ on Uλ = det( f u , f v , n ) = (cid:104) f u × E f v , n (cid:105) E , where × E is the Euclidean vector product of R . The function λ is called the signed areadensity function of the frontal f . A singular point p is called non-degenerate if dλ p (cid:54) = 0 .The set of singular points of the frontal f corresponds to zeros of λ . Let us assumethat p is a non-degenerate singular point of a frontal f , then there exists a regular curve γ = γ ( t ) : ( − ε, ε ) −→ U such that γ (0) = p and the image of γ coincides with the singularset of f around p . We call γ the singular curve and the direction of γ (cid:48) = dγdt the singulardirection . On the other hand, there exists a non-zero vector η ∈ Ker( df p ) because p isnon-degenerate. We call η the null direction .Let U i , i = 1 , be domains of R and p i , i = 1 , be points in U i . Two smooth maps f : U −→ R and f : U −→ R are A -equivalent (or right-left equivalent ) at the points p ∈ U and p ∈ U if there exist local diffeomorphisms Φ of R with Φ( p ) = p and Ψ of R with Ψ( f ( p )) = f ( p ) such that f = Ψ ◦ f ◦ Φ − . A singular point p of a map f : U −→ R is called a cuspidal edge , swallowtail or cuspidal cross cap if the map f at p EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 9 is A -equivalent to the following map f C , f S or f CCR at the origin, respectively (see Figure3): f C ( u, v ) = ( u , u , v ) , f S ( u, v ) = (3 u + u v, u + 2 uv, v ) , f CCR ( u, v ) = ( u, v , uv ) .Cuspidal edges and swallowtails are non-degenerate singular points of fronts, and thesetwo types of singular points are generic singularities of fronts (cf. [2]). In addition to thesesingular points, cuspidal cross caps often appear on minfaces, which are not singular pointsof fronts but are non-degenerate singular points of frontals. Figure 3.
The cuspidal edge, swallowtail and cuspidal cross cap.4.2.
Singular points on minfaces.
In [7], Fujimori, Saji, Umehara and Yamada provedthat the singular points of spacelike maximal surfaces in L generically consist of cuspidaledges, swallowtails and cuspidal cross caps. Similarly, these singular points frequentlyappear on timelike minimal surfaces. By using Facts as mentioned above, Takahashi gavethe following criteria for cuspidal edges, swallowtails and cuspidal cross caps of minfaces byusing their real Weierstrass data ( g , g , ω , ω ) . Now, we identify the Lorentz-Minkowskispace L with the affine space R . Fact 4.1 ([21]) . Let f : U −→ L be a minface and p ∈ U a singular point. If we take thereal Weierstrass data ( g , g , ˆ ω du, ˆ ω dv ) on U , then f is A -equivalent to (i) a cuspidal edge at p if and only if g (cid:48) g ˆ ω − g (cid:48) g ˆ ω (cid:54) = 0 and g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:54) = 0 at p, (ii) a swallowtail at p if and only if g (cid:48) g ˆ ω − g (cid:48) g ˆ ω (cid:54) = 0 , g (cid:48) g ˆ ω + g (cid:48) g ˆ ω = 0 , and (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g − (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g (cid:54) = 0 at p, (iii) a cuspidal cross cap at p if and only if g (cid:48) g ˆ ω − g (cid:48) g ˆ ω = 0 , g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:54) = 0 , and (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g + (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g (cid:54) = 0 at p. To prove Fact 4.1, Takahashi used the following fact. We shall recall the proof in [21]which will be helpful to prove our main results.
Fact 4.2 ([21]) . Let f : U −→ L be a minface with the real Weierstrass data ( g , g , ˆ ω du, ˆ ω dv ) .Then (i) a point p is a singular point of f if and only if g ( p ) g ( p ) = 1 . (ii) f is a frontal at any singular point p . (iii) f is a front at a singular point p if and only if g (cid:48) g ˆ ω − g (cid:48) g ˆ ω (cid:54) = 0 at p . Moreover inthis case, p is automatically a non-degenerate singular point.Proof. Let u , v be local coordinates on U . Since f u = ˆ ω − − g , − g , g ) , f v = ˆ ω g , − g , − g ) , it holds that f u × f v = ˆ ω ˆ ω − g g )( − g − g , g − g , − − g g ) , where × denotes the Euclidean outer product. Since f is a minface, we obtain ˆ ω (cid:54) = 0 and ˆ ω (cid:54) = 0 at any point, and hence p is a singular point if and only if g ( p ) g ( p ) = 1 . Moreover f is a frontal with unit normal vector field n = 1 (cid:112) (1 − g g ) + 2( g + g ) ( − g − g , g − g , − − g g ) . Next we prove (iii). Since df p and dn p are written as df p = − g ω + g ω g + g , g − g , − , dn p = (cid:16) − dg g + dg g (cid:17) ( g + g ) (cid:112) g + g ) (0 , , g − g ) , where ω = ˆ ω du and ω = ˆ ω dv , we obtain the following two vector fields η and µ suchthat df p ( ν ) = 0 and dn p ( µ ) = 0 : η = 1 g ˆ ω (cid:18) ∂∂u (cid:19) p + 1 g ˆ ω (cid:18) ∂∂v (cid:19) p , µ = g (cid:48) g (cid:18) ∂∂u (cid:19) p + g (cid:48) g (cid:18) ∂∂v (cid:19) p . (7)On the other hand, the minface f is a front at p if and only if the directions η and µ arelinearly independent (see, for example, proof of Lemma 3.3 in [22]) and by the equations(7) we get det( η, µ ) = g (cid:48) g ˆ ω − g (cid:48) g ˆ ω at p, which proves the first part of the conclusion. Moreover, the signed area density function λ can be written as λ = − ˆ ω ˆ ω − g g ) (cid:112) (1 − g g ) + 2( g + g ) , and hence its derivative at p can be written as follows: dλ p = ˆ ω ˆ ω √ | g + g | (cid:18) dg g + dg g (cid:19) . (8)Therefore a singular point p is non-degenerate if and only if g ( p ) (cid:54) = 0 or g ( p ) (cid:54) = 0 , andhence we have proved the desired result. (cid:3) EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 11
Behavior of the Gaussian curvature near singular points.
Now we are in theposition to investigate the behavior of the Gaussian curvature near singular points ofminfaces by using the facts given above.
Proof of Theorem A.
We use the representation (2) for a minface f . Since ϕ in (2) satisfies (cid:104) ϕ (cid:48)(cid:48) , ϕ (cid:48)(cid:48) (cid:105) = 4 g (cid:48) ˆ ω and ˆ ω (cid:54) = 0 on the minface f , ϕ is degenerate at p if and only if g (cid:48) ( p ) = 0 .Similarly, ψ is degenerate at p if and only if g (cid:48) ( p ) = 0 . By (i) of Fact 4.1, near a cuspidaledge g (cid:48) (cid:54) = 0 or g (cid:48) (cid:54) = 0 . Hence there is no umbilic point near p by Proposition 3.1. Nextwe prove (ii). If we assume that one of g (cid:48) ( p ) and g (cid:48) ( p ) vanishes, then the other one alsovanishes by (i) of Fact 4.1. By (iii) of Fact 4.2, it contradicts the assumption that f is afront, that is, there is no flat point near p . By Proposition 3.1 and Lemma 2.6, we cantake pseudo-arclength parameters of ϕ and ψ , that is, g (cid:48) ˆ ω ( u ) = − ε ϕ , g (cid:48) ˆ ω ( v ) = − ε ψ , (9)and hence g (cid:48) g ˆ ω − g (cid:48) g ˆ ω = − ε ϕ g ˆ ω + ε ψ g ˆ ω and g (cid:48) g ˆ ω + g (cid:48) g ˆ ω = − (cid:18) ε ϕ g ˆ ω + ε ψ g ˆ ω (cid:19) . (10)Since f is a front at p , the quantity g (cid:48) g ˆ ω − g (cid:48) g ˆ ω does not vanish at p by (iii) of Fact 4.2.On the other hand, if we assume that the singular point p is not a cuspidal edge, then thequantity g (cid:48) g ˆ ω + g (cid:48) g ˆ ω vanishes at p by (i) of Fact 4.1. Therefore by the second equationof (10), the orientations of ϕ and ψ are different. Hence, by Theorem 3.4, the Gaussiancurvature K is negative and K diverges to −∞ at p . Finally if we assume that f is not afront at p and p is a non-degenerate singular point, then the quantity g (cid:48) g ˆ ω − g (cid:48) g ˆ ω vanishesat p . Hence, if one of g (cid:48) ( p ) and g (cid:48) ( p ) vanishes, then the other one also vanishes, whichcontradicts the assumption that p is non-degenerate and the equation (8). Therefore, thereis no flat point near p . By taking pseudo-arclength parameters of ϕ and ψ with (9) againand considering the first equation of (10), we conclude that the orientations of ϕ and ψ are the same. By Theorem 3.4, the Gaussian curvature K is positive and K diverges to ∞ at p , which completes the proof. (cid:3) Remark 4.3.
In general, the sign of the Gaussian curvature near cuspidal edges of aminface cannot be determined. If we take the real Weierstrass data g ( u ) = u , g ( v ) = 1 + v , ω ( u ) = du and ω ( v ) = dv ,in the equation (2), then the singular set Σ f is determined by the equation g ( u ) g ( v ) = u (1 + v ) = 1 , that is, Σ f = { ( v , v ) ∈ R | v ∈ R } and quantities in (i) of Fact 4.1 arecomputed as follows g (cid:48) g ˆ ω ± g (cid:48) g ˆ ω = u ± v (1+ v ) = (1+ v ) ± v (1+ v ) = (1 ± v ) +3 v +6 v +4 v + v (1+ v ) > on Σ f .Therefore, the set of singular points Σ f consists of only cuspidal edges. On the other hand,the Gaussian curvature is K ( u, v ) = v ( u (1+ v ) − . Hence the sign of the Gaussian curvaturecannot be determined near cuspidal edges in general. Moreover, the Gaussian curvatureof this example does not diverge along the quasi-umbilic curve v = 0 (the curve appearsas the boundary of black and gray parts in Figure 4). Figure 4.
A minface with cuspidal edges on which the sign of the Gaussiancurvature changes along a quasi-umbilic curve.
Remark 4.4.
In [22], Umehara and Yamada introduced the notion of maxfaces in L , andproved that any maxface f is locally represented as f ( z ) = (cid:60) (cid:90) zz ( − G, G , i (1 − G )) η, where ( G, η ) is a pair of a meromorphic function and a holomorphic -form on a simplyconnected domain in C containing a base point z such that (1 + | G | ) | η | (cid:54) = 0 on thedomain. Moreover, the first fundamental form of f is given by I = (1 − | G | ) | η | , andhence a point z is a singular point of f if and only if | G ( z ) | = 1 . By using ( G, η ) , theintrinsic Gaussian curvature K of f can be written as K = 4 | dG | (1 − | G | ) | η | , (11)where the non-degeneracy of a singular point p means dG p (cid:54) = 0 (Lemma 3.3 in [22]).Therefore at a non-degenerate singular point p of any maxface, the Gaussian curvature K always diverges to ∞ . Example 4.5 ([9, 13, 21]) . If we take the real Weierstrass data g ( u ) = u , g ( v ) = − v , ω ( u ) = du and ω ( v ) = dv in the equation (2), we obtain the following two null curves ϕ ( u ) = 12 ( − u − u , u − u , u ) and ψ ( v ) = 12 ( v + v , v − v , v ) . The surface obtained by these two null curves is called the timelike Enneper surface ofisothermic type or an analogue of Enneper’s surface . Since g (cid:48) ˆ ω = 1 and g (cid:48) ˆ ω = − , ϕ and ψ are parametrized by pseudo-arclength parameters and have negative and positiveorientations, respectively. Hence, Theorem 3.4 states that the Gaussian curvature K isnegative. Moreover the singular set is Σ f = { ( u, v ) ∈ R | uv = − } and the quantitiesin Fact 4.1 are computed as g (cid:48) g ˆ ω ± g (cid:48) g ˆ ω = 2( v ± u ) on Σ f . Therefore, Σ f consists ofcuspidal edges Σ f \ { (1 , − , ( − , } and swallowtails { (1 , − , ( − , } . By (ii) of FactA.12 in Appendix A, the conjugate minface of the timelike Enneper surface of isothermictype f ∗ defined by ( A. has cuspidal edges Σ f \ { (1 , − , ( − , } and cuspidal cross caps { (1 , − , ( − , } , see Figure 5. EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 13
Figure 5.
The timelike Enneper surface of isothermic-type and its conjugate.As Example 3.3 umbilic points and quasi-umbilic points on a timelike minimal surfaceare not isolated in general. We also saw an example in Remark 4.3 where a curve of quasi-umbilic points on a minface accumulates to a cuspidal edge. As a corollary of Theorem A,we obtain the following:
Corollary 4.6.
The following (i) and (ii) hold: (i)
Umbilic points do not accumulate to a non-degenerate singular point of a minface. (ii)
If quasi-umbilic points accumulate to a non-degenerate singular point p of a minface,then p is a cuspidal edge.Proof. The claim (i) follows from the equation(8) and (i) of Proposition 3.1. To prove theclaim (ii), if we assume that quasi-umbilic points accumulate to a non-degenerate singularpoint p which is not a cuspidal edge, then the condition K ( p ) = 0 contradicts to (ii) or(iii) of Theorem A. (cid:3) Remark 4.7.
In contrast with the corollary as above, umbilic points on a maxface do notaccumulate to a non-degenerate singular point by the equation (11) and the non-degeneracyof a singular point in Remark 4.4.In the end of this section, we give a criterion for the sign of the Gaussian curvature nearcuspidal edges on minfaces. Let f : U −→ R be a front with the unit normal vector field n and γ = γ ( t ) a singular curve on U consists of cuspidal edges. By (i) of Fact A.10, γ isa regular curve and we can take the null vector fields η such that ( γ (cid:48) ( t ) , η ( t )) is positivelyoriented with respect to the orientation of U . The singular curvature κ s of the cuspidaledge γ was defined in [20] as κ s ( t ) = sgn( dλ ( η )) det(ˆ γ (cid:48) ( t ) , ˆ γ (cid:48)(cid:48) ( t ) , n ) | ˆ γ (cid:48) ( t ) | , where ˆ γ = f ◦ γ and | ˆ γ (cid:48) ( t ) | = (cid:104) ˆ γ (cid:48) ( t ) , ˆ γ (cid:48) ( t ) (cid:105) / E . The singular curvature is an intrinsicinvariant of cuspidal edges, and related to the behavior of the Gaussian curvature asstated in Introduction. For minfaces, the singular curvature characterizes the sign of theGaussian curvature near cuspidal edges: Theorem 4.8.
Let f : U −→ L be a minface with the real Weierstrass data ( g , g , ˆ ω du, ˆ ω dv ) and γ ( t ) the singular curve passing through a cuspidal edge p = γ (0) . Then the Gaussiancurvature K and the singular curvature κ s have the same sign. In particular, zeros of κ s correspond to either zeros of g (cid:48) or g (cid:48) . Proof.
By the proofs of Facts 4.1 (see Appendix A) and 4.2, we can compute det(ˆ γ (cid:48) ( t ) , ˆ γ (cid:48)(cid:48) ( t ) , n ) = ˆ ω ˆ ω g (cid:48) g (cid:48) (cid:114) ( g + g ) (cid:18) g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:19) , | ˆ γ (cid:48) | = (cid:114) ˆ ω ˆ ω ( g + g ) (cid:18) g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:19) and sgn( dλ ( η )) = sgn(ˆ ω ˆ ω ) , where we take the null vector field η satisfying the condition det( γ (cid:48) , η ) = (cid:18) g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:19) > . Therefore, the singular curvature κ s is written as κ s = 2 g (cid:48) g (cid:48) ˆ ω ˆ ω ( g + g ) (cid:16) g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:17) . (12)Hence, zeros of κ s correspond to either zeros of g (cid:48) or g (cid:48) . On the other hand, the Gaussiancurvature K of the minface f which is represented as (2) is written as K = 4 g (cid:48) g (cid:48) ˆ ω ˆ ω (1 − g g ) . (13)By (12) and (13), we obtain the desired result. (cid:3) Appendix A. Geometry of minfaces
In this appendix we give a precise description of the notion of minfaces and their repre-sentation formulas based on the work by Takahashi [21].First we shall recall the notion of paracomplex algebra. For a more detailed expositionon paracomplex numbers, see [5, 9, 13] and their references. Let C (cid:48) be the -dimensionalcommutative algebra of the form C (cid:48) = R ⊕ R j with multiplication law: j · · j = j, j = 1 . An element of C (cid:48) is called a paracomplex number and C (cid:48) is called the paracomplex algebra .Some authors use the terminology split-complex numbers or Lorentz numbers instead ofparacomplex numbers. For a paracomplex number z = x + jy , we call (cid:60) z := x , (cid:61) z := y and ¯ z := x − jy the real part, the imaginary part and the conjugate of z , respectively. Theparacomplex algebra C (cid:48) can be identified with the Minkowski plane L = ( R , (cid:104) , (cid:105) L = − dx + dy ) as follows: C (cid:48) (cid:51) z = x + jy ←→ z = ( x, y ) ∈ L . Under the identification, the scalar product (cid:104) z , z (cid:105) L of L can be written as −(cid:60) (¯ z z ) .In particular, (cid:104) z , z (cid:105) L = − z ¯ z and we define (cid:104) z (cid:105) := z ¯ z . We also define the n -dimensionalparacomplex space as C (cid:48) n := { ( z , z , · · · , z n − ) | z , z , · · · , z n − ∈ C (cid:48) } .A (1 , -tensor field J on a 2-dimensional oriented manifold Σ is called an almost para-complex structure if J satisfies J = id and dim( V − ) = dim( V + ) = 1 , where V − and V + are ± -eigenspaces for J . As pointed out in [9, 24] every almost complex structure J on Σ isintegrable, that is, there exists a coordinate system ( u, v ) compatible with the orientationof Σ such that J ( ∂∂u ) = ∂∂u and J ( ∂∂v ) = − ∂∂v near each point. We also call ( u, v ) a null EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 15 coordinate system , ( x = u − v , y = u + v ) a Lorentz isothermal coordinate system on (Σ , J ) and (Σ , J ) a para-Riemann surface .A smooth map ϕ between para-Riemann surfaces ( M, J ) and ( N, J (cid:48) ) is called paraholo-morphic if dϕ ◦ J = J (cid:48) ◦ dϕ . Paraholomorphicity of maps locally can be characterized asfollows: Fact A.1 ([21]) . Let D ⊂ C (cid:48) be a domain with a coordinate z = x + jy = u − v + j u + v . Afunction ϕ = ϕ + jϕ is paraholomorphic if and only if there exist functions f = f ( u ) and g = g ( v ) such that ϕ ( z ) = f ( u )+ g ( v )2 + j f ( u ) − g ( v )2 . It follows directly from the observations that ϕ satisfies dϕ ( J ( ∂∂u )) = J ( dϕ ( ∂∂u )) if andonly if there exists a function g = g ( v ) such that ϕ − ϕ = g , and dϕ ( J ( ∂∂v )) = J ( dϕ ( ∂∂v )) if and only if there exists a function f = f ( u ) such that ϕ + ϕ = f , where J is thecanonical paracomplex structure on C (cid:48) . A -form ω is called paraholomorphic if ω canbe written as ω = ˆ ωdz in any local paracomplex coordinate z with a paraholomorphicfunction ˆ ω .In [21], Takahashi introduced the notion of timelike minimal surfaces with some kind ofsingular points of rank one, which are called minfaces as follows: Definition A.2 ([21]) . Let (Σ , J ) be a para-Riemann surface. A smooth map f : Σ −→ L is a minface if there is an open dense set W ⊂ Σ such that f is a conformal timelike minimalimmersion on W , and on each null coordinate system ( u, v ) , f u (cid:54) = 0 and f v (cid:54) = 0 at eachpoint. A point p ∈ Σ is called a singular point of f if f is not an immersion at p . Remark A.3.
In [11], Kim, Koh, Shin and Yang defined the notion of generalized timelikeminimal surfaces as follows: Let Σ be a -dimensional C -manifold. A non-constant map f : Σ −→ L is called a generalized timelike minimal surface if at each point of Σ thereexists a local coordinate system ( x, y ) such that (i) (cid:104) f x , f x (cid:105) ≡ −(cid:104) f y , f y (cid:105) ≥ , (cid:104) f x , f y (cid:105) ≡ ,(ii) f xx − f yy ≡ and (iii) (cid:104) f x , f x (cid:105) = −(cid:104) f y , f y (cid:105) > almost everywhere on Σ . A singularpoint of such a surface is in either A := { p | f x or f y is lightlike } or B := { p | df p vanishes } .By definition, a minface is a generalized timelike minimal surface without singular pointsbelonging to B . However, the converse is not true, that is, we can construct an exampleof a generalized timelike minimal surface with only singular points belonging to A whichis not a minface by taking only one of two generating null curves with a singular point.A paraholomorphic map F = ( F , F , F ) : Σ −→ C (cid:48) is called a Lorentzian null map if F z · F z := − ( F z ) + ( F z ) + ( F z ) ≡ holds on Σ , where z = u − v + j u + v is a local paracomplex coordinate in a domain U ⊂ Σ and F z = ∂F∂z = (cid:2)(cid:0) ∂F∂u − ∂F∂v (cid:1) + j (cid:0) ∂F∂u + ∂F∂v (cid:1)(cid:3) . By the paraholomorphicity of F and FactA.1, we can take the decomposition of F : U −→ C (cid:48) as F ( z ) = ϕ ( u ) + ψ ( v )2 + j ϕ ( u ) − ψ ( v )2 . (A.1)Since F z · F z = 12 (cid:2) (cid:104) ϕ (cid:48) ( u ) , ϕ (cid:48) ( u ) (cid:105) + (cid:104) ψ (cid:48) ( v ) , ψ (cid:48) ( v ) (cid:105) + 2 j (cid:0) (cid:104) ϕ (cid:48) ( u ) , ϕ (cid:48) ( u ) (cid:105) − (cid:104) ψ (cid:48) ( v ) , ψ (cid:48) ( v ) (cid:105) (cid:1)(cid:3) , we have the following: Fact A.4 ([21]) . Let U be a domain in C (cid:48) and z = x + jy = u − v + j u + v be the canonicalcoordinate on C (cid:48) . If we take the decomposition (A.1), then the following conditions areequivalent: (i) F is a Lorentzian null map, (ii) ϕ and ψ satisfy (cid:104) ϕ (cid:48) ( u ) , ϕ (cid:48) ( u ) (cid:105) = 0 and (cid:104) ψ (cid:48) ( v ) , ψ (cid:48) ( v ) (cid:105) = 0 . Remark A.5.
The condition (ii) above does not mean that ϕ and ψ are null curves becausethere may be a point p such that ϕ (cid:48) ( p ) = 0 or ψ (cid:48) ( p ) = 0 . Since the Jacobi matrix of F canbe written as J F = 12 (cid:18) ϕ (cid:48) ψ (cid:48) ϕ (cid:48) − ψ (cid:48) (cid:19) , a necessary and sufficient condition that the Lorentzian null map F as above is an immer-sion is both of ϕ and ψ are null curves.Similar to the case of maxfaces (see Proposition 2.3 in [22]), any minface can be writtenby using a paraholomorphic Lorentzian null immersion as follows: Fact A.6 ([21]) . Let (Σ , J ) be a para-Riemann surface and f : Σ → L be a minface.Then there is a paraholomorphic Lorentzian null immersion F : (cid:101) Σ −→ C (cid:48) such that f ◦ π = F + F , where π : (cid:101) Σ −→ Σ is the universal covering map of Σ . Conversely, if F : (cid:101) Σ −→ C (cid:48) is a paraholomorphic Lorentzian null immersion which gives a timelikeminimal immersion f = F + F on an open dense set, then f is a minface.Proof. By Definition A.2, there exists an open dense set W ⊂ Σ such that f | W is aconformal timelike minimal immersion. Then if we take a paracomplex coordinate z in adomain U , we obtain f z ¯ z ≡ on U ∩ W . Since W is a dense set, the equality as aboveholds on U , and hence ∂f = f z dz is a paraholomorphic 1-form on Σ . We can take aparaholomorphic map F : ˜Σ −→ C (cid:48) such that dF = ∂ ( f ◦ π ) . Since ∂ ( F + ¯ F ) = dF , thereexists a real number c such that F + ¯ F = f ◦ π + c . In particular, we can take c = 0 . Letus take null coordinates u , v in a domain U ⊂ Σ near any point p ∈ Σ , and consider nullcoordinates ˜ u , ˜ v in each connected component of π − ( U ) such that π ◦ ˜ u = u and π ◦ ˜ v = v .By Fact A.1, we can take the following decomposition F (˜ u, ˜ v ) = ϕ (˜ u ) + ψ (˜ v )2 + j ϕ (˜ u ) − ψ (˜ v )2 . Since F + ¯ F = f ◦ π + c , we obtain f u ( π (˜ u, ˜ v )) = ϕ (cid:48) (˜ u ) , f v ( π (˜ u, ˜ v )) = ψ (cid:48) (˜ v ) . By the assumption that f | W is a conformal timelike minimal immersion and Fact A.4, F is a Lorentzian null map on ˜Σ . By Remark A.5, we conclude that F is an immersion on ˜Σ . Next we prove the converse. By the assumption, f = F + ¯ F satisfies the condition (i)of Definition A.2, and the condition (ii) of Definition A.2 follows from Remark A.5. (cid:3) In particular, any minface f can be written as the equation (1). We call the paraholo-morphic Lorentzian null immersion F as above the paraholomorphic lift of the minface f .Moreover the following Weierstrass-type representation formula for minfaces is known. EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 17
Fact A.7 (Local version of the Weierstrass representation formula in [21]) . Let f : Σ → L be a minface. For each point p ∈ Σ , after a rotation with respect to the time axis, there exista paraholomorphic function g and a paraholomorphic 1-form ω = ˆ ωdz which are definednear p such that f can be written as follows f ( z ) = (cid:60) (cid:90) zz (cid:0) − − g , j (1 − g ) , g (cid:1) ω + f ( z ) . (A.2) Moreover if we decompose paraholomorphic functions g and ˆ ω into g ( z ) = g ( u ) + g ( v )2 + j g ( u ) − g ( v )2 , (A.3) ˆ ω ( z ) = ˆ ω ( u ) + ˆ ω ( v )2 + j ˆ ω ( u ) − ˆ ω ( v )2 , z = x + jy = u − v j u + v , (A.4) then, a minface f can be decomposed into two null curves as the equation (2).Proof. Let us take a paracomplex coordinate z near p and consider the followings ω = ˆ ωdz = ( − f z + jf z ) dz, g = f z ˆ ω . Here, we shall prove that after a rotation with respect to the time axis, we can take (cid:104) ˆ ω ( p ) (cid:105) (cid:54) = 0 , that is, g is locally paraholomorphic. Let us assume that (cid:104) ˆ ω ( p ) (cid:105) = 0 and take the paraholomorphic lift F of f as equation (A.1). Since ˆ ω = − f z + jf z and f z = ϕ u − ψ v + j ϕ u + ψ v , we obtain (cid:104) ˆ ω (cid:105) = ( − f z + jf z )( − f z − jf z )= ( − ϕ u + ϕ u )( ψ v + ψ v ) . For arbitrary θ ∈ R , let us define ˜ f := θ − sin θ θ cos θ f and ˆ ω θ := − ˜ f z + j ˜ f z . Next we prove that there exists a θ such that (cid:104) ˆ ω θ (cid:105) (cid:54) = 0 at p . A similar computation asabove shows that (cid:104) ˆ ω θ (cid:105) = ( − ϕ u + ϕ u cos θ − ϕ u sin θ )( ψ v + ψ v cos θ − ψ v sin θ ) . Note that the assumption (cid:104) ˆ ω ( p ) (cid:105) = 0 is equivalent to the condition − ϕ u ( p ) + ϕ u ( p ) = 0 or ψ v ( p ) + ψ v ( p ) = 0 . In the former case, ϕ u ( p ) = 0 because F is a Lorentzian null map. Since f is a minface,we get ϕ u ( p ) = ϕ u ( p ) (cid:54) = 0 and (cid:104) ˆ ω θ ( p ) (cid:105) = ϕ u ( p )( − θ )( ψ v ( p ) + ψ v ( p ) cos θ − ψ v ( p ) sin θ ) . (A.5)Let us consider the quantity (cid:104) ˆ ω π ( p ) (cid:105) = − ϕ u ( p )( ψ v ( p ) − ψ v ( p )) . If it is non-zero then theproof is completed. We consider the case (cid:104) ˆ ω π ( p ) (cid:105) = 0 . Again, we can see that ψ v ( p ) = 0 and ψ v ( p ) = ψ v ( p ) (cid:54) = 0 . The equation (A.5) can be written as (cid:104) ˆ ω θ ( p ) (cid:105) = ϕ u ( p )( − θ ) ψ v ( p )(1 + cos θ ) , and hence we can take a θ such that (cid:104) ˆ ω θ ( p ) (cid:105) (cid:54) = 0 . The proof for the case that ψ v ( p ) + ψ v ( p ) = 0 is similar. Therefore we can take ω and g as paraholomorphic -form andparaholomorphic function near p .Next let us prove the equations (A.2) and (2). By a straightforward computation, weobtain f z dz = 12 ( − − g , j (1 − g ) , g ) ω, (A.6)and hence we obtain the equation (A.2). For the null coordinates u and v , f can be writtenas f z = f u − f v + j f u + f v . By (A.6), we get the relation f u − f v = (cid:60) ( − − g , j (1 − g ) , g )ˆ ω. By using the decompositions (A.3) and (A.4), the equation above can be written f u − f v = ˆ ω (cid:0) − − ( g ) , − ( g ) , g (cid:1) + ˆ ω (cid:0) − − ( g ) , − g ) , g (cid:1) , and hence f u = ˆ ω (cid:0) − − ( g ) , − ( g ) , g (cid:1) and f v = ˆ ω (cid:0) g ) , − ( g ) , − g (cid:1) . By integrating the derivative df = f u du + f v dv , we obtain the desired equation (2). (cid:3) Remark A.8.
The formula as mentioned above is valid locally, that is, we cannot choosethe function g in Fact A.7 as a paraholomorphic function globally. However, the notionof parameromorphic function was introduced in [21] and by using it, Takahashi gave thesame formula as (A.2) with a paraholomorphic 1-form ω and a parameromorphic function g which are defined on the universal cover ˜Σ of Σ . In this paper we only need the formulas(A.2) and (2) near each point to discuss the local behavior of the Gaussian curvature, andhence we can always take the function g as a paraholomorphic function locally. Remark A.9.
It should be remarked that Magid [14] originally proved a representationformula using null curves similar to (2) away from singular points.In [21], the pair ( g, ω ) and the quadruple ( g , g , ω , ω ) were called ( paraholomorphic ) Weierstrass data and real Weierstrass data , respectively. The imaginary part f ∗ ( z ) := (cid:61) (cid:90) zz (cid:0) − − g , j (1 − g ) , g (cid:1) ω (A.7)also gives a minface which is called the conjugate minface of f . The conjugate minface isdefined on ˜Σ and corresponding to a minface with the Weierstrass data ( g, jω ) or the realWeierstrass data ( g , g , ω , − ω ) .In the end of the paper, we give a proof of Fact 4.1 and dualities of singular points onminfaces, which were given in [21]. To prove Fact 4.1, we use the following criteria forcuspidal edges, swallowtails and cuspidal cross caps: Fact A.10 ([12]) . Let f : U −→ R be a front and p ∈ U a non-degenerate singular pointof f . Take a singular curve γ = γ ( t ) with γ (0) = p and a vector field of null directions η ( t ) . Then (i) p is a cuspidal edge if and only if det ( γ (cid:48) (0) , η (0)) (cid:54) = 0 . (ii) p is a swallowtail if and only if det ( γ (cid:48) (0) , η (0)) = 0 and ddt det ( γ (cid:48) ( t ) , η ( t )) (cid:12)(cid:12) t =0 (cid:54) = 0 . EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 19
Fact A.11 ([7]) . Let f : U −→ R be a frontal and p ∈ U a non-degenerate singular pointof f . Take a singular curve γ = γ ( t ) with γ (0) = p and a vector field of null directions η ( t ) . Then p is a cuspidal cross cap if and only if det ( γ (cid:48) (0) , η (0)) (cid:54) = 0 , det ( df ( γ (cid:48) (0)) , n (0) , dn ( η (0))) = 0 and ddt det ( df ( γ (cid:48) ( t )) , n ( t ) , dn ( η ( t ))) (cid:12)(cid:12) t =0 (cid:54) = 0 .Proof of Fact 4.1. Since the singular set on U is written by { p ∈ U | g ( p ) g ( p ) = 1 } ,the singular curve γ ( t ) = ( γ ( t ) , γ ( t )) near the non-degenerate singular point p = γ (0) satisfies g ( γ ( t )) g ( γ ( t )) = 1 . Taking the derivative, we obtain g (cid:48) γ (cid:48) g + g g (cid:48) γ (cid:48) = 0 . By using the equality g g = 1 on the singular set, we can parametrize γ as γ (cid:48) ( t ) = g (cid:48) g (cid:18) ∂∂u (cid:19) γ ( t ) − g (cid:48) g (cid:18) ∂∂v (cid:19) γ ( t ) and by the first equation (7), we obtain det( γ (cid:48) ( t ) , η ( t )) = g (cid:48) g ˆ ω + g (cid:48) g ˆ ω . (A.8)By (i) of Fact A.10 and (iii) of Fact 4.2, we have proved the claim (i). By (A.8), we cancompute ddt det (cid:0) γ (cid:48) ( t ) , η ( t ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) γ (cid:48) + (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) γ (cid:48) = (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g − (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g . (A.9)By the equations (A.8), (A.9), (ii) of Fact A.10 and (iii) of Fact 4.2, we obtain the claim(ii). Next we prove the claim (iii). After a straightforward computation we obtain det (cid:0) df ( γ (cid:48) ( t )) , n ( t ) , dn ( η ( t )) (cid:1) = α (cid:18) g (cid:48) g ˆ ω − g (cid:48) g ˆ ω (cid:19) , (A.10)where α = α ( t ) = − ˆ ω ˆ ω (cid:16) g (cid:48) g ˆ ω + g (cid:48) g ˆ ω (cid:17) . Since cuspidal cross caps are non-degeneratesingular points on frontals, we always assume that α (0) (cid:54) = 0 .Moreover, ddt det (cid:0) df ( γ (cid:48) ( t )) , n ( t ) , dn ( η ( t )) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = α (cid:48) (0) (cid:18) g (cid:48) g ˆ ω − g (cid:48) g ˆ ω (cid:19) + α (0) (cid:20)(cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g + (cid:18) g (cid:48) g ˆ ω (cid:19) (cid:48) g (cid:48) g (cid:21) . (A.11)By the equations (A.8), (A.10), (A.11) and Fact A.11, we have proved the claim (iii). (cid:3) As a corollary of Fact 4.1, we obtain the following dualities of singular points corre-sponding to results for maxfaces in [7] and [22]. It is known that these kinds of dualitiesalso hold for other surfaces, see also [8, 10, 18].
Fact A.12 ([21]) . Let f : Σ −→ L be a minface and p ∈ Σ a singular point. (i) A singular point p of a minface f is a cuspidal edge if and only if p is a cuspidaledge of its conjugate minface f ∗ . (ii) A singular point p of a minface f is a swallowtail (resp. cuspidal cross cap) if andonly if p is a cuspidal cross cap (resp. swallowtail) of its conjugate minface f ∗ . References [1] V.I. Arnol’d,
Singularities of caustics and wave fronts , Math. and its Appl. . Kluwer AcademicPublishers Group, Dordrecht (1990).[2] V.I. Arnol’d, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps , Vol. 1,Monogr. Math. , Birkhäuser Boston, Inc., Boston, MA, 1985.[3] W.B. Bonnor, Null curves in a Minkowski space-time , Tensor N.S. , 229–242 (1969).[4] J.N. Clelland, Totally quasi-umbilic timelike surfaces in R , , Asian J. Math. , 189–208 (2012).[5] V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry , Rocky Mountain J. , 83–115 (1996).[6] S. Fujimori, Y.W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada and S.-D. Yang, Zero mean curvature surfaces in Lorentz-Minkowski -space and -dimensional fluid mechanics , Math.J. Okayama Univ. , 173–200 (2015).[7] S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces , Math. Z. ,827–848 (2008).[8] A. Honda,
Duality of singularities for spacelike CMC surfaces , to appear in Kobe Journal of Mathe-matics.[9] J. Inoguchi and M. Toda,
Timelike minimal surfaces via loop groups , Acta Appl. Math. , 313–355(2004).[10] S. Izumiya and K. Saji, The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowskispace and “flat” spacelike surfaces , J. Singul. , 92–127 (2010).[11] Y. W. Kim, S.-E. Koh, H. Shin and S.-D. Yang, Spacelike maximal surfaces, timelike minimal surfaces,and Björling representation formulae , J. Korean Math. Soc. , 1083–1100 (2011).[12] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyper-bolic -space , Pacific J. Math. , 303–351 (2005).[13] J. Konderak, A Weierstrass representation theorem for Lorentz surfaces , Complex Var. Theory Appl. (5), 319–332 (2005).[14] M.A. Magid, Timelike surfaces in Lorentz -space with prescribed mean curvature and Gauss map ,Hokkaido Math. J. , 447–464 (1991).[15] L.F. Martins, K. Saji, M. Umehara and K. Yamada, Behavior of Gaussian curvature and mean cur-vature near non-degenerate singular points on wave fronts , Geometry and Topology of Manifolds,Springer Proc. Math. Stat. , 247–281 (2016).[16] L. McNertney,
One-parameter families of surfaces with constant curvature in Lorentz -space , Ph.D.thesis, Brown University (1980).[17] T.K. Milnor, Entire timelike minimal surfaces in E , , Michigan Math. J. , 163–177 (1990).[18] Y. Ogata and K. Teramoto, Duality between cuspidal butterflies and cuspidal S − singularities onmaxfaces , preprint.[19] Z. Olszak, A note about the torsion of null curves in the -dimensional Minkowski spacetime and theSchwarzian derivative , Filomat , 553–561 (2015).[20] K. Saji, M. Umehara and K. Yamada, The geometry of fronts , Ann. of Math. , 491–529 (2009).[21] H. Takahashi,
Timelike minimal surfaces with singularities in three-dimensional spacetime (in Japan-ese), Master thesis, Osaka University (2012).[22] M. Umehara and K. Yamada,
Maximal surfaces with singularities in Minkowski space , Hokkaido Math.J. , 13–40 (2006).[23] E. Vessiot, Sur les curbes minima , C. R. Acad. Sci., Paris , 1381–1384 (1905).[24] T. Weinstein,
An Introduction to Lorentz Surfaces , de Gruyter Exposition in Math. , Walter deGruyter, Berlin (1996). EHAVIOR OF THE GAUSSIAN CURVATURE OF TIMELIKE MINIMAL SURFACES 21
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