Type changes of spacelike maximal surfaces in Minkowski 3-space to - - PowerPoint PPT Presentation

Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces

8th International Meeting on Lorentzian Geometry, M´ alaga Kotaro Yamada

Tokyo Institute of Technology kotaro@math.titech.ac.jp

• 23. Sept. 2016
• K. Yamada

GELOMA 2016

• 23. Sept. 2016

1 / 50

This talk is based on the Joint works. . .

• S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, KY

Entire zero mean curvature graphs of mixed type in Lorentz- Minkowski 3-space preprint, arXiv:1511.07954, to appear.

• A. Honda, M. Koiso, M. Kokubu, M. Umehara and KY

Mixed type surfaces with bounded mean curvature in 3-dimensional space-times preprint, arXiv:1508.02514.

• S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin,
• M. Umehara, S.-D. Yang and KY

Zero mean curvature surfaces in Lorentz-Minkowski 3-space and 2-dimensional fluid mechanics

• Math. J. Okayama Univ., vol. 57 (2015).
• K. Yamada

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Surfaces in L3

A surface in the Lorentz-Minkowki 3-space L3 := ( (R3; t, x, y), ⟨ , ⟩ = −dt2 + dx2 + dy2 ) is said to be Spacelike if all tangent planes are spacelike. Timelike if all tangent planes are timelike. spacelike timelike type change

• K. Yamada

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• 23. Sept. 2016

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What kind of surfaces can change types?

Theorem (Honda-Koiso-Kokubu-Umehara-Y)

f : R2 ⊂ D → L3: an immersion such that U+ := {spacelike points} ̸= ∅, U− := {timelike points} ̸= ∅, and the mean curvature function H is bounded on U+ ∪ U−. Then for ∀ p ∈ U+ ∩ U−, ∃ a sequence {pn} ⊂ U+ ∪ U− such that pn → p (n → ∞), and H(pn) → 0 (n → ∞).

Corollary

Surfaces of non-zero constant mean curvature cannot change causal types.

• K. Yamada

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ZMC surfaces

Question

Can zero mean curvature (ZMC) surfaces change types?

• K. Yamada

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ZMC surfaces

A surface in L3 is said to be Zero Mean Curvature (ZMC) if the union of the spacelike part and the timelike part is dense on the surface, and the mean curvature vanishes identically on spacelike/timelike part. a spacelike ZMC surface is called a (spacelike) maximal surface. a timelike ZMC surface is called a (timelike) minimal surface. This talk deals Description of type-change phenomenon for ZMC surfaces from the viewpoint of maximal surfaces. Examples of embedded ZMC surfaces. Examples of entire ZMC graphs.

• K. Yamada

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Maximal surfaces

A maximal surface is A spacelike surface in L3 with vanishing mean curvature. A critical point of the area functional. A Weierstrass-type representation formula (Osamu Kobayashi, 1983) Written in terms of holomorphic data on the surface.

• cf. The Weierstrass representation for minimal surfaces in R3.

A complete maximal surface in L3 is a spacelike plane (Calabi, cf. Rubio’s talk) It is natural to consider maximal surfaces with singularities.

• O. Kobayashi

Maximal surfaces in the 3-dimensional Minkowski space L3 Tokyo J. Math., 6 (1983), 297–309.

• O. Kobayashi

Maximal surfaces with conelike singularities

• J. Math. Soc. Japan, 36 (1984), 609–617.
• K. Yamada

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• 23. Sept. 2016

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The Weierstrass-type representation

• O. Kobayashi (1983):

A spacelike maximal immersion f : M 2 → L3 is expressed as f = Re F; F = ∫ ( −2g, 1 + g2, √ −1(1 − g2) ) ω where (g, ω) = ( a meromorphic fct., a holomorphic 1-form )

• n M2 (with complex structure induced by the induced metric).

The induced metric is ds2 = (1 − |g|2)2 |ω|2 (|g| = 1: singularity) Remark: f∗ = Im F is also a maximal surface called the conjugate of f.

• cf. Yasumoto’s talk
• K. Yamada

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The Weierstrass-type representation: Gauss maps

The unit normal vector of f is expressed by the Weierstrass data (g, ω) as ν = (|g|2 + 1 |g|2 − 1, 2 Re g 1 − |g|2 , 2 Im g 1 − |g|2 ) : M2 → H2

± ⊂ L3,

that is, g is the unit normal vector ν composed with the stereographic projection (the Gauss map).

• K. Yamada

GELOMA 2016

• 23. Sept. 2016

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Maxfaces—Maximal surfaces with singularities

Definition (Umehara-Y, 2006)

A map f : M 2 → L3 is a maxface (M2: a Riemann surface) ⇔ ∃W ⊂ M 2: open dense, such that fW is a conformal spacelike maximal immersion, and d f(p) ̸= 0 (∀p ∈ M2). Estudillo and Romero (1992) introduced a notion of and investigated global properties of them. Our notion of maxfaces is equivalent to non-branched generalized maximal surfaces in the sense of Estudillo-Romero.

• K. Yamada

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• 23. Sept. 2016

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Conjugate Maxfaces

Weierstrass representation f = Re F; F = ∫ ( −2g, 1 + g2, √ −1(1 − g2) ) ω gives a maxface ⇔ ds2

# := (1 + |g|2)2|ω|2 is positive definite.

• cf. the induced metric: ds2 = (1 − |g|2)2|ω|2

The singular set of f is {p ∈ M2 ; |g| = 1}. The conjugate f∗ = Im F corresponds to (g, −√−1ω). The singular set of f∗ coincides with that of f.

• K. Yamada

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• 23. Sept. 2016

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Singularities of Maxfaces

Theorem (Fujimori-Saji-Umehara-Y, 2008)

Generic singular points of maxfaces are cuspidal edges, swallowtails, and cuspidal crosscaps. The duality Maxface f The conjugate f∗ cuspidal edge cuspidal edge swallowtail cuspidal crosscap cuspidal crosscap swallowtail

Cuspidal edge swallowtail cuspidal crosscap Note: Similar duality properties hold for timelike minimal surfaces. (cf. H. Takahashi, Master Thesis, 2012);

• K. Yamada

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Example (the catenoid)

M2 = C \ {0}, (g, ω) = ( z, dz

z2

) M 2 = C \ {0}, (g, ω) = ( z, dz

z2

)

1 1

Lorentzian-Catenoid in L3 Euclidean-Catenoid in R3 |z| = 1: cone-like singularities non-generic

• K. Yamada

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• 23. Sept. 2016

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Example (the helicoid)

The conjugate surface of the catenoid: Lorentz-Helicoid in L3 Helicoid in R3 |z| = 1 corresponds to the fold singularities. the image of fold singularities consists of a null (light-like) curve.

• K. Yamada

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• 23. Sept. 2016

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Fold singularities

− → − →

Fact (cf. Fujimori-Kim-Koh-Rossman-Shin-Umehara-Yang-Y, 2015)

The conjugate of a cone-like singularity is a fold singularity. For a maxface obtained by the Weierstrass data (g, ω), the singular curve γ is a fold singularity ⇔ |g| = 1, dg ̸= 0, and Re dg g2ω = 0 hold on γ.

• K. Yamada

GELOMA 2016

• 23. Sept. 2016

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The analytic extension of the helicoid

The image of the Lorentz helicoid can be extended across the singular curve: Lorentz helicoid in L3 The analytic extension The dark part is a timelike minimal surface. Remark: The image of the right-hand figure coincides with the Euclidean helicoid (cf. Albujer’s talk).

• K. Yamada

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Analytic extensions of along fold singularities

Fact

f : M 2 → L3: a maxface with fold singularities γ(t). ⇒ The image ˆ γ(t) = f ◦ γ(t) is a null curve in L3 which is non-degenerate (i.e. ˙ γ(t) and ¨ γ(t) are linearly independent for all t). The map ˜ f(u, v) := ˆ γ(u + v) + ˆ γ(u − v) 2 gives a timelike minimal surface. The image of ˜ f gives the analytic extension of the image of f along ˆ γ. The union of the images of f and ˜ f is an immersed surface near ˆ γ.

• C. Gu (1985); V. A. Klyachin (2003); Kim-Koh-Shin-Yang (2006). (cf. [3])
• K. Yamada

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• 23. Sept. 2016

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The analytic extension

Take ˆ γ(t): a null curve, which is nondegenerae, i.e. ¨ ˆ γ(t) is not proportional to ˙ ˆ γ(t). Take the midpoint of two points: f(s, t) := 1 2 ( ˆ γ(s) + ˆ γ(t) ) f gives a timelike minimal surface.

• K. Yamada

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Example (Scherk-type surface)

M 2 = C \ {±1, ±√−1} M 2 = C \ {±1, ±√−1} (g, ω) = ( z,

√−1dz 1−z4

) (g, ω) = ( z,

dz 1−z4

) Lorentz-Scherk in L3 Scherk R3 The graph x3 = log cos x1 cos x2 .

• K. Yamada

GELOMA 2016

• 23. Sept. 2016

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The analytic extension of Scherk-type surface

t = log cosh x cosh y Entire ZMC graph (O. Kobayashi (1983))

• K. Yamada

GELOMA 2016

• 23. Sept. 2016

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Embedded Examples: Schwarz type (Euclidean case)

Ma := { (z, w) ∈ (C ∪ {∞})2 ; w2 = z8 + (a4 + a−4)z4 + 1 } , (0 < a < 1) Schwarz P g = z ω = dz w Schwarz D g = z ω = √−1dz w

Figure: Shoichi Fujimori (Okayama Univ.)

• K. Yamada

GELOMA 2016

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Embedded Examples: Schwarz type (Lorentzian case)

Ma := { (z, w) ∈ (C ∪ {∞})2 ; w2 = z8 + (a4 + a−4)z4 + 1 } , (0 < a < 1)

Figure: Fujimori

Schwarz P g = z ω = dz w Schwarz D g = z ω = √−1dz w

• K. Yamada

GELOMA 2016

• 23. Sept. 2016

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The analytic extension of the Schwarz D-type maxface

→ Embedded (Fujimori-Rossman-Umehara-Yang-Y (2014))

Figure: Fujimori

• K. Yamada

GELOMA 2016

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Kobayashi surfaces

Our Aim: To produce examples of maxfaces with fold singularities. To investigate global properties of their analytic extensions.

Definition (Fujimori-Kawakami-Kokubu-Rossman-Umehara-Y [1])

A Kobayashi surface is a weakly complete maxface f : C ∪ {∞} \ {p1, . . . , pN} − → L3 with Weierstrass data (g, ω) such that The Gauss map g is meromorphic on C ∪ {∞}, ∃I : C ∪ {∞} → C ∪ {∞}: anti-holo. involution such that f ◦ I = f, p1,. . . , pN ∈ Σ := {fixed points of I}, Σ \ {p1, . . . , pN} consists of fold singularities. The maxface is weakly complete ⇔ ds2 = (1 + |g|2)2|ω|2 is complete.

• K. Yamada

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Weierstrass data for Kobayashi Surfaces

Theorem (Fujimori-Kawakami-Kokubu-Rossman-Umehara-Y [1])

A Kobayashi surface is expressed by the following Weierstrass data up to scaling and congruence: g =

n−1

∏

i=1

z − bi 1 − biz , ω = √−1λ

n−1

∏

i=1

(1 − biz)2

2n−1

∏

j=0

( z − e

√−1αj

) dz where 0 = α0 ≤ α1 ≤ · · · ≤ α2n−1 < 2π |bi| < 1 (i = 1, 2, . . . , n − 1), λ = exp  √ −1

2n−1

∑

j=0

αj 2   n = deg g + 1: the order of the Kobayashi surface

• K. Yamada

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• 23. Sept. 2016

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Weierstrass data for Kobayashi Surfaces

0 = α0 ≤ α1 ≤ · · · ≤ α2n−1 < 2π (angular data) bj ∈ C, |bj| < 1 (j = 1, 2, . . . , n − 1) (zeros of the Gauss map g in the unit disk) pj = e

√−1αj,

I(z) = 1 ¯ z

Proposition ([1])

The set of Kobayashi surfaces of order n has 3 degrees of freedom if n = 2. 4n − 7 degrees of freedom if n ≥ 3. By a change of parameter z, we can set b1 = b2 = 0 (resp. b1 = 0) for n ≥ 3 (resp. n = 2).

• K. Yamada

GELOMA 2016

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Analytic extension of Kobayashi surfaces

The singularities of a Kobayashi surface consists of fold singularities. ⇒ ∃ an analytic extension to timelike minimal surface. Take the maximal (non-extendable) analytic extension ˜ f of a Kobayashi surface f.

Question

When ˜ f is properly immersed? When the image of ˜ f is an entire graph over the xy-plane? Can ˜ f be properly embedded (but not a graph)?

• K. Yamada

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Example

n = 2, α0 = α1 = α2 = α3 = 0: Properly embedded; The Ruled Enneper surface (O. Kobayashi, 1983)

• K. Yamada

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Example

n = 2, α0 = α1 = 0, α2 = α3 = π: An entire graph t = x tanh y (O. Kobayashi, 1983)

• K. Yamada

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Example

n = 2, α0 = 0, α1 = π

2 , α2 = π, α3 = 3π 2 :

An entire graph t = log cosh x

cosh y (O. Kobayashi, 1983)

• K. Yamada

GELOMA 2016

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Kobayashi surfaces with properly immersed extensions

Q: When the analytic extension is properly immersed?

Theorem (Fujimori-Kawakami-Kokubu-Rossman-Umehara-Y [1])

The maximal analytic extension of Kobayashi surface of order n is properly immersed if b1 = . . . bn−1 = 0 and |αj − αj+1| ≤ π n − 1 (j = 0, . . . , 2n − 1), (∗) properly immersed if |αj − αj+1| < π n − 1 (j = 0, . . . , 2n − 1) |bj| (j = 1, . . . , n − 1) are sufficiently small. Under the assumption that b1 = · · · = bn−1 = 0, (∗) is the necessary and sufficient condition for ˜ f to be properly immersed.

• K. Yamada

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cf.

The Weierstrass data of a Kobayashi surface is g =

n−1

∏

i=1

z − bi 1 − biz , ω = √−1λ

n−1

∏

i=1

(1 − biz)2

2n−1

∏

j=0

( z − e

√−1αj

) dz where 0 = α0 ≤ α1 ≤ · · · ≤ α2n−1 < 2π |bi| < 1 (i = 1, 2, . . . , n − 1), λ = exp  √ −1

2n−1

∑

j=0

αj 2   .

• K. Yamada

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Kobayashi surfaces as ZMC graphs

Q: When the analytic extension is a graph over the xy-plane?

Theorem (Fujimori-Kawakami-Kokubu-Rossman-Umehara-Y [1])

The maximal analytic extension of Kobayashi surface of order n is an entire graph over xy-plane if b1 = · · · = bn−1 = 0 and |αj − αj+1| ≤ π 2(n − 1) (j = 0, . . . , 2n − 1) an entire graph over xy-plane if |bj| (j = 1, . . . , n − 1) are sufficiently small and |αj − αj+1| < π 2(n − 1) (j = 0, . . . , 2n − 1)

• cf. The extension is properly immersed if bj = 0 and

|αj − αj+1| < π n − 1 (j = 0, . . . , 2n − 1).

• K. Yamada

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ZMC graphs

Corollary

For each n ≥ 3 (resp. n = 2) there exists 4n − 7 (resp. 3) parameter family of entire ZMC graphs over the xy-plane whose spacelike parts are Kobayashi surfaces of order n. Remark: For the proof of proper immersedness and graph properties, we need the explicit expression of analytic extension, rather than the “mid-point” for- mula.

• K. Yamada

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Example

n = 2, α0 = α1 = 0, α2 = α3 = π: An entire graph t = x tanh y (O. Kobayashi, 1983)

• K. Yamada

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Example

n = 2, α0 = 0, α1 = π

2 , α2 = π, α3 = 3π 2 :

An entire graph t = log cosh x

cosh y (O. Kobayashi, 1983)

• K. Yamada

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Example

n = 2, α0 = α1 = 0, α2 = π/2, α3 = 3π/2: An entire graph (FKKRUY)

• K. Yamada

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Example

n = 2, α0 = α1 = α2 = 0, α3 = π: An entire graph, foliated by parabolas (FKKRUY [1], see also S. Akamine, arXiv:1510.07451)

• K. Yamada

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Example

An entire graph (FKKRUY)

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The ZMC equation

Fact (ZMC equation)

A graph t = f(x, y) is a ZMC surface if and only if (1 − f2

y )fxx + 2fxfyfxy + (1 − f2 x)fyy = 0.

(∗) if 1 − f2

x − f2 y > 0: the graph is spacelike and (∗) is elliptic.

(in a divergence form, cf. Rubio’s talk) if 1 − f2

x − f2 y < 0: the graph is timelike and (∗) is hyperbolic.

Corollary

There are infinitely many entire solutions of (∗).

• cf. Minimal surface equation

(1 + f2

y )fxx − 2fxfyfxy + (1 + f2 x)fyy = 0.

The only entire solutions are linear functions, Bernstein (1910)

• K. Yamada

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• cf. 2-dimensional fluid dynamics

Equations in 2-D fluid dynamics

(c2 − ϕ2

x)ϕxx − 2ϕxϕyϕxy + (c2 − ϕ2 y)ϕyy = 0

(1) (c2ρ2 − ψ2

y)ψxx + 2ψxψyψxy + (c2ρ2 − ψ2 x)ψyy = 0

(2) Let v(x, y) = (u(x, y), v(x, y)) be a velocity vector field of 2-dimensional flow of gas on the xy-plane, which is steady: time-independent barotropic: p = p(ρ), where p: the pressure, ρ: the density without external forces. irrotational: rot v = 0. Then there exist functions ϕ(x, y) such that ϕx = u, ϕy = v, (the velocity potential), ψ(x, y) such that ψx = −ρv, ψy = ρu (the stream function).

• K. Yamada

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• cf. 2-dimensional fluid dynamics

Equations in 2-D fluid dynamics

(c2 − ϕ2

x)ϕxx − 2ϕxϕyϕxy + (c2 − ϕ2 y)ϕyy = 0

(1) (c2ρ2 − ψ2

y)ψxx + 2ψxψyψxy + (c2ρ2 − ψ2 x)ψyy = 0

(2) The velocity potential ϕ satisfies (1), and the stream function ψ satisfies (2), where c := dp/dρ is the local speed of sound.

• K. Yamada

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• cf. 2-dimensional fluid dynamics

Equations in 2-D fluid dynamics

(c2ρ2 − ψ2

y)ψxx + 2ψxψyψxy + (c2ρ2 − ψ2 x)ψyy = 0

(2) The case ρc = 1 (virtual gas): (2) is the ZMC equation. Remark: ρc = 1 ⇔ p(ρ) = p0 − 1

ρ.

For actual gas, p(ρ) = ργ, (γ > 0, γ = 1.4 for the air).

Observation (Fujimori-Kim-Koh-Rossman-Umehara-Yang-Y, [3])

A ZMC graph with type-change is interpreted as a stream function of transonic 2D flow of virtual gas.

• K. Yamada

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Kobayashi surfaces with embedded extensions

Q: Can ˜ f be properly embedded (but not a graph)? Properly embedded; The Ruled Enneper surface (O. Kobayashi, 1983)

• K. Yamada

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Example: Jorge-Meeks’ type maxfaces

Euclidean Lorentzian

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Example: Jorge-Meeks’ type maxfaces

Properly Embedded (Fujimori-Kawakami-Kokubu-Rossman-Umehara-Y, 2015)

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Type changes for ZMC graph

Fact (Klyachin (2003))

A type change of ZMC graph occurs along either a non-degenerate null curve (← fold singularities of maxfaces)

• r a null line.

The second case?

Theorem (Fujimori-Kim-Koh-Rossman-Umehara-Yang-Y, 2014)

There exists a ZMC graph which changes type along a null line. f(x, y) = y +

∞

∑

k=3

bk(y) k xk

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