and positive (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 respect to the induced metrics from E3 and L3 have opposite signs, we can prove the following statement for minfaces:
Theorem D. The Gaussian curvature with respect to the induced metric from L3 near a cuspidal edge on a minface and the singular curvature have the same sign.
In fact we will prove a stronger result (Theorem 4.5.12) than TheoremD. By The-orems C and D, we obtain criteria for the sign of the Gaussian curvature near any non-degenerate singular point on a minface. Moreover we should remark that, by (ii) of TheoremC, we obtain a class of surfaces with swallowtails near which the Gaussian curvature with respect to the Euclidean metric (we denote it byKE) is positive. There are a few explicitly known examples of such swallowtails, although there are many known examples of swallowtails on surfaces with negative Gaussian curvatureKE (see Remark 4.5.6).
This chapter is organized as follows: in Section 4.2 we describe some notions of timelike surfaces and null regular curves. We also give the definition of minfaces as a class of timelike minimal surfaces with singular points by using a representation formula derived in [54]. In Section 4.3 we investigate the behavior of the Gaussian curvature near regular points. Finally, in Section4.5we discuss the sign of the Gaussian curvature near singular points on minfaces and prove our main results: Theorem Cand Theorem 4.5.12. In AppendixCwe review a precise description of geometry of minfaces given in Takahashi’s Master thesis [54].
where
ε=⟨ν, ν⟩= {
1 iff is timelike,
−1 iff is spacelike.
Similarly to the case of surfaces inE3, the shape operatorS of any spacelike surface inL3 is always diagonalizable over Rand we can take real principal curvatures λ1 and λ2 of such surface. Therefore the discriminant of S of a spacelike surface satisfies the following inequality (see [39] for details)
H2+K =
(λ1+λ2 2
)2
−λ1λ2 =
(λ1−λ2 2
)2
≥0. (4.1)
A spacelike surface inL3 whose mean curvature vanishes identically is called a maximal surface, and the Gaussian curvatureK of the surface satisfies K ≥0 by the inequality above. A pointp∈Σ of a timelike or spacelike immersionf is called anumbilic pointif the second fundamental form II is a multiple of the first fundamental form I atp. Flat points of a maximal surface inL3 consist of umbilic points by (4.1).
On the other hand, one of the most important differences between spacelike surfaces and timelike surfaces is the diagonalizability of the shape operator, that is, the shape operator of a timelike surface is not always diagonalizable even overC. Therefore there are three possibilities of the diagonalizability of the shape operator of a timelike surface inL3 as follows:
(i) The shape operator is diagonalizable over R. In this case H2−K ≥ 0 with the equality holds on umbilic points.
(ii) The shape operator is diagonalizable overC\R. In this case H2−K <0.
(iii) The shape operator is non-diagonalizable over C. In this case H2−K= 0.
About the case (iii), Clelland [13] introduced the following notion:
Definition 4.2.1 ([13]). A pointpon a timelike surface Σ is calledquasi-umbilicif the shape operator of Σ is non-diagonalizable overC.
A timelike surface in L3 whose mean curvature vanishes identically is called a timelike minimal surface. By the above arguments, the diagonalizability of the shape operator of a timelike minimal surface is determined by the sign of the Gaussian curvature, and flat points of the surface consist of umbilic points and quasi-umbilic points.
4.2.2 Timelike minimal surfaces and minfaces
For a timelike surfacef: Σ−→L3, near each point, we can take aLorentz isothermal coordinate system(s, t), that is, the first fundamental form I is written as I =E(−ds2+ dt2) with a non-zero function E, and anull coordinate system(u, v) that is, I is written as I = 2Λdudv. A regular curveγ inL3 whose velocity vector fieldγ′ is lightlike is called a null regular curve, and a null coordinate system is a coordinate system on which the image of coordinate curves are null regular curves. Moreover, up to constant multiple, there is a one-to-one correspondence between these coordinate systems as follows:
s= u−v
2 , t= u+v 2 .
On each null coordinate system (u, v), an immersion f and its mean curvature H satisfy Hν = Λ2∂u∂v∂2f . Therefore, we obtain the following well-known representation formula by McNertney.
Fact 4.2.2 ([44]). If φ(u) and ψ(v) are null regular curves in L3 such that φ′(u) and ψ′(v) are linearly independent for all u and v, then
f(u, v) = φ(u) +ψ(v)
2 (4.2)
gives a timelike minimal surface. Conversely, any timelike minimal surface can be writ-ten locally as the equation (4.2) with two null regular curves φ andψ.
We note that the decomposition (4.2) is unique in the following sense:
Lemma 4.2.3. If we take another null coordinate system (ξ, η) whose intersection with (u, v) is not empty andf can be written as
f(ξ, η) =φ(ξ) + ˜˜ ψ(η)
2 on(ξ, η).
Then either there are smooth functionsξ˜and η˜such that ξ = ˜ξ(u), η= ˜η(v) and φ(u) = ˜φ( ˜ξ(u)) +c1 and ψ(v) = ˜ψ(˜η(v)) +c2
for some real constants c1 and c2, or there are smooth functions ξ˜ and η˜ such that ξ= ˜ξ(v), η= ˜η(u) and
φ(u) = ˜ψ(˜η(u)) +c3 and ψ(v) = ˜φ( ˜ξ(u)) +c4 for some real constants c3 and c4.
Proof. By the chain rule, we haveφu = ˜φξξu+ ˜ψηηu and ψv = ˜φξξv+ ˜ψηηv, and hence 0 =⟨φu, φu⟩= 2ξuηu⟨φ˜ξ,ψ˜η⟩ and 0 =⟨ψv, ψv⟩= 2ξvηv⟨φ˜ξ,ψ˜η⟩. (4.3) Since f is immersed, ⟨φ˜ξ,ψ˜η⟩ ̸= 0 holds at each point. If ξv = 0 at a point, then ξu ̸= 0 and ηu = 0 on an open neighborhood containing this point by the equations (4.3). Hence we obtainξv = 0 on the open neighborhood as above. Therefore, there are smooth functions ˜ξ and ˜η such that ξ= ˜ξ(u), η= ˜η(v) and
φu = ˜φξξ˜u and ψ(v) = ˜ψ˜ηv.
By integrating these equations, we obtain the desired result. The proof for the case that ηv = 0 at a point is same as before.
Lemma 4.2.3 says that null coordinate transformations on a timelike minimal sur-face correspond to reparametrizations of the two null regular curves which generate the surface.
For a timelike minimal immersion, as the classical minimal surface theory in Eu-clidean 3-spaceE3, there exists a one parameter family of isometric immersions and the conjugate surface as follows:
Figure 4.1: The timelike elliptic helicoid (the blue part) and the timelike elliptic catenoid.
Definition 4.2.4(cf. [29,46,57]). Theassociated family of a timelike minimal immer-sion f written as (4.2) is the family{fλ}λ∈R\{0} consists of
fλ(u, v) = λφ(u) +λ−1ψ(v)
2 ,
and theconjugate surfacefˆis
fˆ(u, v) = φ(u)−ψ(v)
2 .
Remark 4.2.5. As pointed out in [29], the conjugate surface ˆf is not contained in the associated family{fλ}λ∈R\{0}. The conjugate surface ˆf is anti-isometric tof, that is, ˆf satisfies
fˆ∗⟨,⟩=−f∗⟨,⟩.
Example 4.2.6(The timelike elliptic helicoid and catenoid). Let us take the null regular curveγ(t) = (cost,sint, t) andφ(u) =γ(u),ψ(v) =γ(v). The timelike minimal surfaces
f(u, v) = φ(u) +ψ(v)
2 , fˆ(u, v) = φ(u)−ψ(v) 2
are called the timelike elliptic helicoid and the timelike elliptic catenoid, respectively (see, for example, [31]). These surfaces have singular points, that is, points on which the maps are not immersed (see Figure 4.1). The elliptic helicoid f is an inner part of the usual helicoid inE3.
In this chapter, we consider the following class of timelike minimal surfaces with sin-gular points of rank one, which are calledminfacesintroduced in [54] (see also Definition C.2in AppendixC):
Definition 4.2.7. A smooth map f: Σ −→ L3 is called a minface if at each point of Σ there exist a local coordinate system (u, v) in a domain U, functions g1 = g1(u), g2 =g2(v), and 1-forms ω1 = ˆω1(u)du, ω2 = ˆω2(v)dv with g1(u)g2(v) ̸= 1 on an open
dense set of U and ˆω1 ̸= 0, ˆω2 ̸= 0 at each point on U such that f can be decomposed into two null regular curves
f(u, v) = 1 2
∫ u
u0
(1−(g1)2,2g1,−1−(g1)2) ω1
+1 2
∫ v
v0
(1−(g2)2,−2g2,1 + (g2)2)
ω2+f(u0, v0). (4.4) We denote these two null regular curves by φ = φ(u) and ψ = ψ(v). The quadruple (g1, g2, ω1, ω2) is called real Weierstrass data.
Asingular pointof a minfacef is a point of Σ on whichfis not immersed, and the set of singular points onU of a minfacef corresponds to the set {(u, v)∈U |g1(u)g2(v) = 1}.
Remark 4.2.8. In [54], Takahashi originally gave the notion of minfaces as Definition C.2 in Appendix C by using the notion of para-Riemann surfaces. To study the local behavior of the Gaussian curvature near singular points of timelike minimal surfaces, we adopt the above definition. In Appendix C, we prove the representation formula (4.4) from the original definition of minfaces (FactC.7) and give a precise description of geometry of minfaces.
4.2.3 Null regular curves
In this subsection, we describe some notions of null regular curves.
Definition 4.2.9(cf. [16,50]). A null regular curveγ:I ⊂R−→L3is calleddegenerate ornon-degenerate at u ∈I ifγ′×γ′′ = 0 or γ′×γ′′̸= 0 at u, respectively. If γ is non-degenerate everywhere, it is called anon-degeneratenull regular curve.
A null regular curve which is degenerate everywhere is a straight line with a lightlike direction. As pointed out in [50, Section 2], the non-degeneracy of a null regular curve is characterized by the following conditions.
Lemma 4.2.10 (cf. [50]). For a null regular curveγ:I ⊂R−→L3, the following are equivalent:
(i) γ is non-degenerate at u∈I,
(ii) γ′′(u) is a non-zero spacelike vector, that is, ⟨γ′′(u), γ′′(u)⟩>0, (iii) det (γ′(u) γ′′(u) γ′′′(u))̸= 0.
By Lemma 4.2.10, we can introduce the following notions for non-degenerate null regular curves.
Definition 4.2.11 ([9, 56]). For a non-degenerate null regular curve γ = γ(u), a pa-rameteru is called apseudo-arclength parameter ofγ if⟨γ′′(u), γ′′(u)⟩ ≡1.
Definition 4.2.12. We define thesign of a non-degenerate null regular curve γ by the sign of det (γ′ γ′′ γ′′′).
Remark 4.2.13. If we take a pseudo-arclength parameters, then det (γ′ γ′′ γ′′′) =±1, which represents the sign ofγ. Moreover, the sign of a non-degenerate null regular curve has the following geometric meaning: If we consider the projection of γ′, which is on the lightcone Q2, into the time slicez= 1, then the projected curve on S1 ={(x, y,1)| (x, y,1)∈Q2} is anticlockwise if the sign is positive, or clockwise if the sign is negative asz increases. See Figure4.2 and Remark4.3.6.
Figure 4.2: Examples of non-degenerate null regular curves with positive sign (the left figure) and negative sign (the right figure).