Math. J. Okayama Univ. 57 (2015), 173–200

ZERO MEAN CURVATURE SURFACES IN
LORENTZ-MINKOWSKI 3-SPACE AND
2-DIMENSIONAL FLUID MECHANICS
S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara,
K. Yamada and S.-D. Yang
Abstract. Space-like maximal surfaces and time-like minimal surfaces
in Lorentz-Minkowski 3-space R31 are both characterized as zero mean
curvature surfaces. We are interested in the case where the zero mean
curvature surface changes type from space-like to time-like at a givennon-degenerate null curve. We consider this phenomenon and its interesting connection to 2-dimensional fluid mechanics in this expository
article.

1. Introduction
We denote by R31 := {(t, x, y) ; t, x, y ∈ R} the Lorentz-Minkowski 3space with the metric h , i of signature (−, +, +). Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R31 areboth characterized as zero mean curvature surfaces. This is an expository
article about type changes of zero mean curvature surfaces in R31 . Klyachin[17] showed, under a sufficiently weak regularity assumption, that a zeromean curvature surface in R31 changes its causal type only on the following
two subsets:• null curves (i.e., regular curves whose velocity vector fields are lightlike) which are non-degenerate (cf. Definition 2.1), or• light-like lines, which are degenerate everywhere.
Recently, actual occurrence of the second case was shown in the authors’
work [5]. So we now pay attention to the former possibilities: Given a nondegenerate null curve γ in R31 , there exists a zero mean curvature surface
which changes its causal type across this curve from a space-like maximal
surface to a time-like minimal surface (cf. [10, 11, 12], [17], [16] and [14]).
This construction can be accomplished using the Björling formula for the
Mathematics Subject Classification. Primary 53A10; Secondary 53B30, 35M10.
Key words and phrases. maximal surface, type change, zero mean curvature, subsonicflow, supersonic flow, stream function.
Kim was supported by NRF 2009-0086794, Koh by NRF 2009-0086794 and NRF 20110001565, and Yang by NRF 2012R1A1A2042530. Fujimori was partially supported by the
Grant-in-Aid for Young Scientists (B) No. 21740052, Rossman was supported by Grant-inAid for Scientific Research (B) No. 20340012, Umehara by (A) No. 22244006 and Yamadaby (B) No. 21340016 from the Japan Society for the Promotion of Science.
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Figure 1. Hyperbolic catenoids C+ , C− .
Weierstrass-type representation formula of maximal surfaces. By unifying
the results of Gu [10], Klyachin [17], and [14], we explain the mechanism forhow zero mean curvature surfaces change type across non-degenerate nullcurves, and give ‘the fundamental theorem of type change for zero mean
curvature surfaces’ (cf. Theorem 2.19) in the second section of this paper.
Locally, such a surface is the graph of a function t = f (x, y) satisfying(1.1)(1 − fy2 )fxx + 2fx fy fxy + (1 − fx2 )fyy = 0.

We call this and its graph the zero mean curvature equation and a zero mean
curvature surface or a zero mean curvature graph, respectively.
As pointed out in [4], the space-like hyperbolic catenoid(1.2)

C+ = {(t, x, y) ∈ R31 ; sin2 x + y 2 − t2 = 0}

and the time-like hyperbolic catenoid(1.3)

C− = {(t, x, y) ∈ R31 ; sinh2 x + y 2 − t2 = 0}

are both typical examples of zero mean curvature surfaces containing singular light-like lines as subsets (cf. Figure 1). The space-like hyperbolic
catenoid C+ is singly periodic.
Also, both the space-like Scherk surface (cf. [4])(1.4)

S+ = {(t, x, y) ∈ R31 ; cos t = cos x cos y}and the time-like Scherk surface of the first kind (cf. [4])(1.5)

S− = {(t, x, y) ∈ R31 ; cosh t = cosh x cosh y}contain singular light-like lines as subsets (cf. Figure 2). As seen in theleft-hand side of Figure 2, S+ is triply periodic.
As an application of the results in Section 2, we show in Section 3 thatC+ and C− induce a common zero mean curvature graph (cf. Figure 3, left)(1.6)

C0 = {(t, x, y) ∈ R31 ; t = y tanh x}

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via their conjugate surfaces. The graph C0 changes type at two non-degenerate
null curves. Similarly, we also show that the Scherk-type surfaces S+ andS− induce a common zero mean curvature graph via their conjugate surfaces(cf. Figure 3, right)(1.7)

S0 = {(t, x, y) ∈ R31 ; et cosh x = cosh y},which changes type at four non-degenerate null curves. These two phenomena were briefly commented upon in [4]. The entire zero-mean curvaturegraphs C0 and S0 were discovered by Osamu Kobayashi [18]. On the other
hand, the above three examples (1.4), (1.5) and (1.7) are particular casesof the general families presented in Sergienko and Tkachev [20, Theorem 2].
Moreover, several doubly periodic mixed type zero mean curvature graphswith isolated singularities are given in [20]. Space-like maximal surfaces
frequently have singularities. See the references [2], [21] and [9] for generaltreatment of these singularities.

Figure 2. Scherk-type surfaces S+ and S− .
In Section 4, we remark on an interesting connection between zero mean
curvature surfaces in R31 and irrotational two-dimensional barotropic steadyflows, where the fluid is called barotropic if the pressure p is a function
depending only on the density ρ. In fact, the stream function ψ(x, y) satisfies(cf. [19], see also Proposition 4.1 in Section 4)(1.8)(ρ2 c2 − ψy2 )ψxx + 2ψx ψy ψxy + (ρ2 c2 − ψx2 )ψyy = 0,

where c is the local speed of sound given by c2 = dp/dρ (cf. (4.1)). We
choose the units so that ρ = 1 and c = 1 when ψx = ψy = 0. Then the
product ρc is equal to 1 if
1(1.9)
p = p0 − ,ρ176

FUJIMORI ET. AL.

Figure 3. Zero mean curvature graphs C0 and S0 (the curves
where the surfaces change type are also indicated).
for a constant p0 , which implies that the graphs of zero mean curvaturesurfaces can be interpreted as stream functions of a virtual gas with (1.9).
(In fact, p is approximately proportional to ρ1.4 for air.) As an applicationof the singular Björling formula for zero mean curvature surfaces (cf. Theorem 2.19), we can construct a family of stream functions which change frombeing subsonic to supersonic at a given locally convex curve in the xy-plane.
The velocity vector fields of these gas flows diverge at the convex curve,
although the streaming functions are real analytic.
2. Type change of zero mean curvature surfaces
In this section, we discuss type change for zero mean curvature surfaces,by unifying the results of Gu [10, 11, 12], Klyachin [17] and four of theauthors here [14].
A regular curve γ : (a, b) → R31 is called null or isotropic if γ ′ (t) :=dγ(t)/dt is a light-like vector for all t ∈ (a, b).

Definition 2.1. A null curve γ : (a, b) → R31 is called degenerate or nondegenerate at t = c if γ ′′ (c) is or is not proportional to the velocity vector
γ ′ (c), respectively. If γ is non-degenerate at each t ∈ (a, b), it is called anon-degenerate null curve.
We now give a characterization of zero mean curvature surfaces thatchange type across a real analytic non-degenerate null curve. Given an
arbitrary real analytic null curve γ : (a, b) → R31 , we denote the uniquecomplex analytic extension of it by γ again throughout this article, by aslight abuse of notation. We consider the two surfaces
Φ(u, v) :=γ(u + iv) + γ(u − iv),
2

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177

and
γ(u + v) + γ(u − v),
2which are defined for v sufficiently close to zero. We recall the following
assertion:Ψ(u, v) :=Proposition 2.2 ([10, 11, 12], [17] and [14]). Given a real analytic nondegenerate null curve γ : (a, b) → R31 , the union of the images of Φ and Ψ
given as above are subsets of a real analytic immersion, and the intersection
is γ. Moreover, Φ gives a space-like maximal surface and Ψ gives a time-likeminimal surface if v is sufficiently close to zero. Furthermore, this analytic
extension of the curve γ as a zero mean curvature surface does not depend
upon the choice of the real analytic parametrization of the curve γ.
Proof. We give here a proof for the sake of the readers’ convenience. We
have that∞∞X
X(−1)n γ (2n) (u)v 2n
γ (2n) (u)v 2nΦ(u, v) =,Ψ(u, v) =
(2n)!
(2n)!
n=0

n=0

near v = 0, where γ (j) = dj γ/dtj . In particular, if we set∞
X
γ (2n) (u)v n
F (u, v) :=
,
(2n)!
n=0

then it gives a germ of a real analytic function satisfying
F (u, −v 2 ) = Φ(u, v),

F (u, v 2 ) = Ψ(u, v),

which prove that the images of Φ and Ψ lie on a common real analytic
surface. Since γ is non-degenerate, the two vectors
γ ′′ (u)Fu (u, 0) = γ ′ (u),Fv (u, 0) =2
are linearly independent, and F gives an immersion which contains γ.
Moreover, it can be easily checked that Φ gives a space-like maximal
surface (cf. Lemma 2.16) and Ψ gives a time-like minimal surface.
We now show the last assertion: Since the surface is real analytic, it issufficient to show that given an arbitrary real analytic diffeomorphism µfrom (a, b) onto its image in R,γ̃(u + iv) + γ̃(u − iv)
γ(u + iv) + γ(u − iv)and Ψ̃(u, v) =2
2
induce the same surface as their graphs, where γ̃(t) := γ(µ(t)). We define
A, B byΨ(u, v) =A = (µ(u + v) + µ(u − v))/2,

B = (µ(u + v) − µ(u − v))/2.

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Thus it is sufficient to show that the map(u, v) 7→ (A, B)
is an immersion at (u, 0). In fact, the Jacobian of the map is given by
 ′

µ (u)
0
J = det6= 0.
0µ′ (u)

Definition 2.3. Let Ω2 be a domain in R2 and f : Ω2 → R a C ∞ -function
satisfying (1.1). We set
B := 1 − fx2 − fy2 .

A point p on Ω2 is called a non-degenerate point of type change 1 with respect
to f if
B(p) = 0,∇B(p) 6= 0,
where ∇B := (Bx , By ) is the gradient vector of the function B.

Since ∇B does not vanish at p, the function f actually changes type atthe non-degenerate point p.
Proposition 2.4 ([11, 12]). Under the assumption that B(p) vanishes, thefollowing two assertions are equivalent.
(1) p is a non-degenerate point of type change.
(2) p is a dually regular point in the sense of [11], that is, fxx fyy −(fxy )2
does not vanish at p.
T
T
T
Proof.
Note that (∇B) = −2H(∇f ) , where is the transpose and H :=
fxx fxy. Note also that B(p) = 0 implies that ∇f (p) 6= 0.
fxy fyyNow suppose that (2) holds. Then det H(p) 6= 0, which with ∇f (p) 6= 0
implies that H(p)(∇f (p))T 6= 0, that is, (1) holds.
Suppose on the contrary that (2) does not hold. By a suitable linear
coordinate change of (x, y), we may assume without loss of generality thatfxy (p) = 0. Then either fxx (p) = 0 or fyy (p) = 0. Also, the zero mean
curvature equation

0 = (1 − fy2 )fxx + 2fx fy fxy + (1 − fx2 )fyy
with B(p) = 0 and fxy (p) = 0 imply thatfx (p)2 fxx (p) + fy (p)2 fyy (p) = 0.
1In Gu [11], ‘dual regularity’ for points of type change is equivalent to our notion.

Klyachin [17] did not define this particular notion, but used it in an essential way.

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This with fxx (p) = 0 or fyy (p) = 0 implies that

  
f
(p)f
(p)
0
x
xx
H(p)(∇f (p))T ==,fy (p)fyy (p)0so (1) does not hold.

Moreover, the following assertion holds:
Proposition 2.5 ([11, 12], [17]). Let γ be a real analytic non-degenerate nullcurve, and let fγ be the real analytic function induced by γ as in Proposition
2.2, which satisfies (1.1). Then the image of γ consists of non-degenerate
points of type change with respect to fγ .
Note that the conclusion is stronger than that of Proposition 2.2.
Proof. Let γ be a non-degenerate null curve. Without loss of generality, we
may take the time-component t as the parameter of γ. Then we have theexpression
γ(t) = (t, x(t), y(t))(a < t < b)such thatx′ (t)2 + y ′ (t)2 = 1.

(2.1)

Since γ is non-degenerate, it holds that
0 6= γ ′′ (t) = (0, x′′ (t), y ′′ (t)).

(2.2)

Differentiating the relation t = f (x(t), y(t)), we have thatx′ (t)fx (x(t), y(t)) + y ′ (t)fy (x(t), y(t)) = 1.

(2.3)

On the other hand, the relation B = 0 implies thatfx (x(t), y(t))2 + fy (x(t), y(t))2 = 1.

(2.4)

Then by (2.1), (2.3) and (2.4), it holds thatx′ (t) = fx ,y ′ (t) = fy .

Thus we have that
(2.5)



d
(x , y ) =fx (x(t), y(t)), fy (x(t), y(t))
dt= (x′ fxx + y ′ fxy , x′ fxy + y ′ fyy )
′′

′′

1
= (fx fxx + fy fxy , fx fxy + fy fyy ) = − ∇B.
2
By (2.2), we get the assertion.
Conversely, we can prove the following.



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Proposition 2.6 ([11, 12], [17, Lemma 2]). Let f : Ω2 → R be a C ∞ function satisfying the zero mean curvature equation (1.1), and let p =(x0 , y0 ) ∈ Ω2 be a non-degenerate point of type change. Then there exists
a non-degenerate C ∞ -regular null curve in R31 with image passing through(f (x0 , y0 ), x0 , y0 ) and contained in the graph of f .
Proof. By the implicit function theorem, there exists a unique C ∞ -regularcurve σ(t) = (x(t), y(t)) in the xy-plane with p = σ(0) so that B = 0 along
the curve. Since B = 0 on σ, the velocity vector σ ′ is perpendicular to∇B. Since ∇f is also perpendicular to ∇B, we can conclude that ∇f is
proportional to σ ′ . In fact
1
fx fxx + fy fxy= fx2 fxx + 2fx fy fxy + fy2 fyy− ∇f · ∇B = (fx , fy )fx fxy + fy fyy
2= (1 − fy2 )fxx + 2fx fy fxy + (1 − fx2 )fyy

− (1 − fx2 − fy2 )(fxx + fyy )= 0.

Since fx2 + fy2 = 1, by taking an arclength parameter of σ, we may set
x′ = f x ,y ′ = fy ,and thenB = 1 − fx2 − fy2 = 1 − (x′ )2 − (y ′ )2 = 0
holds along σ, which implies that t 7→ (t, x(t), y(t)) is a null curve. Sincedf (x(t), y(t)) = x′ fx + y ′ fy = fx2 + fy2 = 1,
dt
there exists a constant c such that f (x(t), y(t)) = t + c. By translating thegraph vertically if necessary, we may assume thatf (x(t), y(t)) = t
holds for each t. Then we obtain the identity (2.5) in this situation, which
implies that ∇B(p) = (x′′ (0), y ′′ (0)) 6= 0, namely,(a, b) ∋ t 7→ (f (x(t), y(t)), x(t), y(t)) = (t, x(t), y(t)) ∈ R31

gives a non-degenerate null curve near t = 0 lying in the graph of f .

Definition 2.7 ([3]). Let Σ2 be a Riemann surface. A C ∞ -map ϕ : Σ2 →R31 is called a generalized maximal surface if there exists an open dense
subset W of Σ2 such that the restriction ϕ|W of ϕ to W gives a conformal(space-like) immersion of zero mean curvature. A singular point of ϕ is apoint at which ϕ is not an immersion. A singular point p satisfying dϕ(p) = 0
is called a branch point of ϕ. Moreover, ϕ is called a maxface if ϕ does nothave any branch points. (A maxface may have singular points in general).

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181

Remark 2.8. The above definition of maxfaces is given in [3], which is simpler
than the definition given in [21] and [9]. However, this new definition isequivalent to the previous one, as we now explain. Suppose that ϕ|W is a
conformal (space-like) immersion of zero mean curvature. Then ∂ϕ = ϕz dz
is a C 3 -valued holomorphic 1-form on W , where z is a complex coordinate ofΣ2 . Since ϕ is a C ∞ -map on Σ2R, ∂ϕ can be holomorphically extended to Σ2 .
z
Then the line integral Φ(z) = z0 ∂ϕ with respect to a base point z0 ∈ Σ2gives a holomorphic map defined on the universal cover of Σ2 whose realpart coincides with ϕ(z) − ϕ(z0 ). The condition that ϕ does not have any
branch point implies that Φ is an immersion. Moreover, since ϕ is conformal
on W , the map Φ satisfies−(dΦ0 )2 + (dΦ1 )2 + (dΦ2 )2 = 0
Φ = (Φ0 , Φ1 , Φ2 ) ,namely, Φ is a null immersion. So ϕ satisfies the definition of maxface as in[21] and [9]. We call Φ the holomorphic lift of the maxface ϕ.

Remark 2.9. By the above definition, maxfaces are orientable. However,there are non-orientable maximal surfaces, as shown in [6]. The definitionof non-orientable maxfaces is given in [6, Def. 2.1]. In this paper, we work
only with orientable maximal surfaces. It should be remarked that nonorientable maxfaces will be orientable when taking double coverings.
Let ϕ : Σ2 → R31 be a maxface with Weierstrass data (G, η) (see [21] forthe definition of Weierstrass data). Using the data (G, η), the maxface ϕhas the expression
Z z(−2G, 1 + G2 , i(1 − G2 ))η.
(2.6)
ϕ = Re(Φ), Φ =z0

The imaginary part(2.7)

ϕ∗ := Im(Φ) : Σ̃2 −→ R31

also gives a maxface called the conjugate surface of ϕ, which is defined on
the universal cover Σ̃2 of Σ2 . The following fact is known:
Fact 2.10 ([21, 9]). A point p of Σ2 is a singular point of ϕ if and only if|G(p)| = 1.
Definition 2.11. A singular point p of ϕ is called non-degenerate if dG
does not vanish at p.

Fact 2.12 ([21, 9]). If a singular point p of ϕ is non-degenerate, then there
exists a neighborhood U of p and a regular curve γ(t) in U so that γ(0) = p
and the singular set of ϕ in U coincides with the image of the curve γ.
This curve γ is called the singular curve at the non-degenerate singularpoint p.

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Definition 2.13. A regular curve γ on Σ2 is called a non-degenerate foldsingularity if it consists of non-degenerate singular points such that the realpart of the meromorphic function dG/(G2 η) vanishes identically along the
singular curve γ. Each point on the non-degenerate fold singularity is calleda fold singular point.
A singular point of a C ∞ -map ϕ : Σ2 → R3 has a fold singularity at
p if there exists a local coordinate system (u, v) centered at p such thatϕ(u, v) = ϕ(u, −v). Later, we show that a non-degenerate fold singularity
is actually a fold singularity (cf. Lemma 2.17).
Suppose that p is a non-degenerate fold singular point of ϕ. The following
duality between fold singularities and generalized cone-like singularities (cf.
[7, Definition 2.1]) holds:
Proposition 2.14 ([16]). Let ϕ : Σ2 → R31 be a maxface and ϕ∗ the conjugate maxface. Then p is a non-degenerate fold singular point of ϕ if andonly if it is a generalized cone-like singular point of ϕ∗ .
Proof. This assertion is immediate from comparison of the above definitionof non-degenerate fold singularities and the definition of generalized conelike singular points as in [7, Definition 2.1 and Lemma 2.3].

We now show the following assertion, which characterizes the non-degenerate
fold singularities on maxfaces.
Theorem 2.15. Let ϕ : Σ2 → R31 be a maxface which has non-degenerate
fold singularities along a singular curve γ : (a, b) → Σ2 . Then γ̂ := ϕ ◦ γ isa non-degenerate null curve, and the image of the mapγ̂(u + v) + γ̂(u − v)
2is real analytically connected to the image of ϕ along γ as a time-like minimal
immersion. Conversely, any real analytic zero mean curvature immersion
which changes type across a non-degenerate null curve is obtained as a real
analytic extension of non-degenerate fold singularities of a maxface.

(2.8)

ϕ̃(u, v) :=This assertion follows immediately from Fact 2.12 and the following Lemmas 2.16 and 2.17.
Lemma 2.16. Let γ : (a, b) → R31 be a real analytic non-degenerate nullcurve. Thenγ(u + iv) + γ(u − iv)
ϕ(u + iv) :=2
gives a maxface with non-degenerate fold singularities on the real axis.

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183

Proof. We set z = u + iv. Then it holds that11
ϕz = (ϕu − iϕv ) = γ ′ (u + iv),22′
where γ (t) := dγ(t)/dt. Since γ is a regular real analytic curve, the map(2.9)

Φ(u + iv) := γ(u + iv)

gives a null holomorphic immersion if v is sufficiently small. Thus ϕ = Re(Φ)
gives a maxface.
Since γ is a null curve, it holds that
(2.10)

γ0′ (t)2 = γ1′ (t)2 + γ2′ (t)2 ,where we set γ = (γ0 , γ1 , γ2 ). Moreover, since γ is a regular curve, (2.10)
implies(2.11)

γ0′ (t) 6= 0

(a < t < b).

It can be easily checked that the maxface ϕ has the Weierstrass data
(2.12)(2.13)

γ1′ (z) − iγ2′ (z)1dz,η := (dΦ1 − idΦ2 ) =22γ0′ (z)γ1′ (z) + iγ2′ (z)
dΦ0
=− ′
=
−
,
G := −2η
γ1 (z) − iγ2′ (z)γ0′ (z)where we set Φ = (Φ0 , Φ1 , Φ2 ) and use the identity(γ1′ − iγ2′ )(γ1′ + iγ2′ ) = (γ1′ )2 + (γ2′ )2 = (γ0′ )2 .

In particular, (2.13) implies that |G| = 1 holds on the u-axis, which implies
that the u-axis consists of singular points. By (2.13), dG vanishes if andonly if∆ := (γ1′ + iγ2′ )′ γ0′ − (γ1′ + iγ2′ )γ0′′ = (γ0′ γ1′′ − γ1′ γ0′′ ) + i(γ0′ γ2′′ − γ2′ γ0′′ )vanishes. In other words, ∆ = 0 if and only if (γ0′′ , γj′′ ) is proportional
to (γ0′ , γj′ ) for j = 1, 2, namely γ ′′ is proportional to γ ′ . Since γ is nondegenerate, this is impossible. So the image of the curve γ consists of nondegenerate singular points (cf. Definition 2.11).
By (2.11), (2.12) and (2.13), we have thatdz2(γ1′ − iγ2′ )=.
G2 η(γ0′ )2
Thus the u-axis consists of non-degenerate fold singular points if and only
if the real part of ∆1 := (γ1′ − iγ2′ )∆ vanishes. HereRe(∆1 ) = γ0′ (γ1′ γ1′′ + γ2′ γ2′′ ) − γ0′′ ((γ1′ )2 + (γ2′ )2 ) = γ0′ (γ0′ γ0′′ ) − γ0′′ (γ0′ )2 = 0,

where we used the identity (2.10) and its derivative. This implies that γ
consists of non-degenerate fold singularities.
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FUJIMORI ET. AL.

Finally, we prove the converse assertion:
Lemma 2.17. Let ϕ : Σ2 → R31 be a maxface which has non-degenerate
fold singularities along a singular curve γ : (a, b) → Σ2 . Then, the spacecurve γ̂(t) := ϕ ◦ γ(t) is a non-degenerate real analytic null curve such that1γ̂(u + iv) + γ̂(u − iv)ϕ̂(u, v) :=2
coincides with the original maxface ϕ. In particular, ϕ̂ satisfies the identityϕ̂(u, v) = ϕ̂(u, −v).
Proof. The singular set of ϕ can be characterized by the set |G| = 1, where(G, η) is the Weierstrass data as in (2.6). Let T be a Möbius transformation
on S 2 = C ∪ {∞} which maps the unit circle {ζ ∈ C ; |ζ| = 1} to the realaxis. Then T ◦ G maps the image of the singular curve γ to the real axis.
Since dG 6= 0 (cf. Definition 2.11), we can choose T ◦ G as a local complex
coordinate. We denote it byz = u + iv.
Then the image of γ coincides with the real axis {v = 0}. Let Φ be theholomorphic lift of ϕ. Since the real axis consists of non-degenerate foldsingularities, Proposition 2.14 implies that Im(Φ) is constant on the realaxis. Since Φ has an ambiguity of translations by pure imaginary vectors,
we may assume without loss of generality that
(2.14)Im(Φ) = 0on the real axis.

Thus, the curve γ̂ is expressed by (cf. (2.9))(2.15)

γ̂(u) = ϕ(u, 0) = Re Φ(u, 0) = Φ(u, 0),

namely, the two C 3 -valued holomorphic functions Φ(u + iv) and γ̂(u + iv)
take the same values on the real axis. Hence Φ(z) = γ̂(u + iv) and thusγ̂(u + iv) + γ̂(u − iv)= ϕ̂(u, v).
2
So it is sufficient to show that γ̂(t) is a non-degenerate null curve. Since|G| = 1 on the real axis, there exists a real-valued function t = t(u) suchthat
ϕ(z) =(2.16)

G(u) = eit(u)(u ∈ R).

Differentiating this along the real axis, we have Gu (u) = ieit (dt/du). Here,dt/du does not vanish because dG 6= 0 on the real axis. Since γ consists of
non-degenerate fold singularities,idG
e−it(u) dt=−
G2 ηw(u) duZERO MEAN CURVATURE SURFACES

185

must be real valued (cf. Definition 2.13), where η = w(z) dz. Since Φ is an
immersion, G must have a pole at z = u if w(u) = 0 (cf. (2.6)), but thiscontradicts the fact that |G| = 1 along γ. Thus we have w(u) 6= 0. It thenfollows that
ξ(u) := eit(u) w(u) = G(u)w(u)is a non-vanishing real valued analytic function. Now, if we write γ̂ =(γ̂0 , γ̂1 , γ̂2 ), (2.6) yields′γ̂0 (u) = Re −2G(u)w(u) = −2ξ(u),and(1 + G(u) )w(u) = 2ξ(u) cos t(u),′2γ̂2 (u) = Re i(1 − G(u) )w(u) = 2ξ(u) sin t(u).
γ̂1′ (u) = Re

2This implies that γ̂(u) is a regular real analytic null curve. Since dt/du 6= 0,
the acceleration vector



dt
′
γ̂ ′′ (u) = log ξ(u) γ̂ ′ (u) + 2ξ(u) 0, − sin t(u), cos t(u)duis not proportional to γ̂ ′ (u). Thus, γ̂(u) is non-degenerate.



Corollary 2.18. Let ϕ : Σ2 → R31 be a maxface. Then a singular pointp ∈ Σ2 lies on a non-degenerate fold singularity if and only if there exists a
local complex coordinate z = u + iv with p = (0, 0) satisfying the following
two properties:(1) ϕ(u, v) = ϕ(u, −v),(2) ϕu (0, 0) and ϕuu (0, 0) are linearly independent.
Proof. Suppose p is a non-degenerate fold singularity of ϕ. Then (1) follows from Lemma 2.17, and (2) follows from the fact that the null curveparameterizing fold singularities is non-degenerate.
Conversely, suppose there is a coordinate system around a singular point
p with p = (0, 0) which satisfies (1) and (2). Differentiating (1), we have thatϕv (u, 0) = 0. Let Φ be a holomorphic lift of ϕ. Since Φ is a null holomorphic
map, the relation ϕv (u, 0) = 0 implies that0 = hΦz (u, 0), Φz (u, 0)i = 4 hϕz (u, 0), ϕz (u, 0)i = hϕu (u, 0), ϕu (u, 0)i ,where h, i is the canonical inner product of R31 . This implies γ : u 7→ ϕ(u, 0)
is a null curve. By the condition (2), this null curve is non-degenerate. Then

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FUJIMORI ET. AL.

by Lemma 2.16,γ(u + iv) + γ(u − iv)
2
is a maxface such that γ parametrizes a non-degenerate fold singularity ofϕγ . Moreover, Φγ := γ(u + iv) gives a holomorphic lift of ϕγ (cf. (2.9)).
SinceΦz (u, 0) = 2ϕz (u, 0) = γ ′ (u) = (Φγ )z (u, 0),ϕγ (u + iv) :=the holomorphicity of Φ and Φγ yields that Φz (z) coincides with (Φγ )z (z).
Thus Φ coincides with Φγ up to a constant. Then ϕγ coincides with ϕ, andwe can conclude that γ parametrizes a non-degenerate fold singularity ofϕ.

So far, we have looked at the singular curves of maxfaces. Now we turn
our attention to the singular curves of zero mean curvature surfaces andprove the following assertion, which can be considered as the fundamental
theorem of type change for zero mean curvature surfaces:
Theorem 2.19 (Gu [10, 11, 12] and Klyachin [17]). Let γ : (a, b) → R31 bea non-degenerate real analytic null curve. We set
√
√

γ(u
+
i
v)
+
γ(u
−
i
v)


(v ≥ 0),
2ϕ̂γ (u, v) :=pp γ(u + |v|) + γ(u − |v|) (v < 0),2
for sufficiently small |v|. Then ϕ̂γ gives a real analytic zero mean curvature
immersion such that the image of γ consists of non-degenerate points of typechange with respect to ϕ̂γ .
Conversely, let f : Ω2 → R be a C ∞ -function satisfying the zero mean
curvature equation (1.1), and let p = (x0 , y0 ) be a non-degenerate pointof type change with respect to f , where Ω2 is a domain in the xy-plane.
Then there exists a real analytic non-degenerate null curve γ in R31 through(f (x0 , y0 ), x0 , y0 ) with ϕ̂γ coinciding with the graph of f in a small neighborhood of p.
This assertion was proved by Gu [10, 11, 12]. Later, Klyachin [17] analyzed type-changes of zero mean curvature surfaces not only at non-degenerate
points of type changes but also degenerate cases as mentioned in the introduction, and got the same assertion as a corollary. Note that the conclusion
for regularity of the converse statement is stronger than that of Proposition 2.6. We remark that Gu [12] gave a generalization of Theorem 2.19 for(n ≥ 2).
2-dimensional zero mean curvature surfaces in Rn+1
1

ZERO MEAN CURVATURE SURFACES

187

Proof. We have already proved the first assertion. (In fact, Proposition 2.5
implies that γ consists of non-degenerate points of type change with respect
to ϕ̂γ .) Then it is sufficient to show the converse assertion, which is provedin [10, 11, 12] and [17]. Here referring to [17], we give only a sketch of theproof: Let f : Ω2 → R be a C ∞ -function satisfying the zero mean curvatureequation (1.1), and let p = (x0 , y0 ) ∈ Ω2 be a non-degenerate point of typechange with respect to f . (As pointed out in Gu [11, 12] and Klyachin [17],one can prove the real analyticity of f at p, assuming only C 3 -regularity off , using the same argument as below.) By Proposition 2.6, there exists a
C ∞ -regular curve σ(u) (|u| < δ) such thatγ(u) := (f ◦ σ(u), σ(u))

is a non-degenerate null curve passing through (f (x0 , y0 ), x0 , y0 ), where δ isa sufficiently small positive number. We set B := 1 − fx2 − fy2 . Let Ω+ be asimply connected domain such that B > 0, and suppose that σ lies on theboundary of Ω+ . We set
Z q−fy dx + fx dy√
t = t(x, y) := f (x, y),s = s(x, y) :=,B
q0
where q0 ∈ Ω+ is a base point and q := (x,Zy). Since α := (−fy dx +q√
fx dy)/ B is a closed 1-form, its (line) integral
α does not depend on the
q0

choice of path. Let τ (v) = (a(v), b(v)) (0 ≤ v ≤ ǫ) be a path starting from p
and going into Ω+ which is transversal to the curve σ. Since B(p) = 0 and∇B(p) 6= 0, there exists a constant C > 0 such that B ◦ τ (v) = Cv + O(v 2 ),where O(v 2 ) denotes the higher order terms. Then there exists a constantm such thatdbda−fy ◦ τ (v) (v) + fx ◦ τ (v) (v)
m
dv
dv
p
for 0 < v ≤ ǫ,<√v
B ◦ τ (v)

hence

Z ǫ

m
√ dv < ∞,v
τ0which is just the case (1) of [17, Lemma 6], and (t, s) gives an isothermal
coordinate system of Ω+ with respect to the immersion
λ :=

Z

|α| <

ϕ : (t, s) 7→ (f (x(t, s), y(t, s)), x(t, s), y(t, s)) = (t, x(t, s), y(t, s))(s > 0).

Moreover, the function s(x, y) can be continuously extended to the imageof the curve σ. Since σ is an integral curve of ∇f , we may assume that σparametrizes the level set s = 0, where we have used the fact that s(x, y)
is constant along each integral curve of ∇f . In particular, ϕ satisfies ϕtt +188

FUJIMORI ET. AL.

ϕss = 0. Then ϕ(t, s) can be extended to a harmonic R3 -valued function
for s < 0 satisfying ϕ(t, s) = ϕ(t, −s) via the symmetry principle (see theproof of [17, Theorem 6]). In particular, f is a real analytic function whosegraph coincides with the image of ϕ on Ω+ near p. Moreover t 7→ ϕ(t, 0)
parametrizes the curve γ (cf. [17, Page 219]). By Corollary (2.18), γ canbe considered as a non-degenerate fold singularity of the maxface (t, s) 7→ϕ(t, s). (In fact, the condition (2) of Corollary (2.18) corresponds to thefact that γ is a non-degenerate curve near p.) Then Theorem 2.15 implies
that ϕ̂γ coincides with the graph of f on a sufficiently small neighborhoodof p.

As an application of Theorem 2.19, embedded triply periodic zero mean
curvature surfaces of mixed type in R31 with the same topology as the
Schwarz D surface in the Euclidean 3-space R3 have been constructed, in[8].
3. The conjugates of hyperbolic catenoids and Scherk-typesurfaces
The two entire graphs of n variables
f1 (x1 , . . . , xn ) := x1 tanh(x2 ),f2 (x1 , . . . , xn ) := (log cosh x1 ) − (log cosh x2 ),given by Osamu Kobayashi [18], are zero mean curvature hypersurfaces inwhich change type from space-like to time-like. When n = 2, the image
Rn+1
1
of f1 is congruent to C0 and the image of f2 is congruent to S0 . On the other
hand, the space-like catenoid C+ (resp. the space-like Scherk surface S+ ) andthe time-like catenoid C− (resp. the time-like Scherk surface S− ) are typicalexamples of zero mean curvature surfaces which contain singular light-likelines. Moreover, they are closely related to C0 (resp. S0 ) by taking theirconjugate surfaces as follows:
Fact 3.1 ([18], [14] and [4]). The conjugate space-like maximal surface ofthe space-like hyperbolic catenoid C+ and the conjugate time-like minimal
surface of the time-like hyperbolic catenoid C− are both congruent to subsetsof the entire graph C0 .
The space-like hyperbolic catenoid C+ was originally given by Kobayashi[18] as the catenoid of 2nd kind, and he also pointed out that the space-likepart C0+ of C0 is the conjugate surface of C+ (see [18]). C0+ is connected andis called the space-like hyperbolic helicoid. The time-like part C0− of C0 splits
into two connected components, each of which is congruent to the time-like
hyperbolic helicoid (see Figure 3, left). The entire assertion, including theZERO MEAN CURVATURE SURFACES

189

case of the time-like part, has been pointed out in [14, Lemma 2.11 (3) ] andthe caption of Figure 1 in [4] without proof.
Proof. We give a proof here as an application of the results in the previous section. A subset of the space-like hyperbolic catenoid C+ can beparametrized by(3.1)

ϕ1 (u, v) = (cosh u sin v, v, sinh u sin v) = − Re i(sinh z, z, cosh z),
ψ1 (u, v) = (− cosh u sin v, v, − sinh u sin v),

where z = u + iv. In fact, C+ is the union of the closure of the images of
ϕ1 and ψ1 . The surface ϕ1 has generalized conical singularities at (u, nπ)
for any u ∈ R and n ∈ Z, as pointed out in [4]. Using this, one can easily
compute that the conjugate of ϕ1 is congruent to the following surface(3.2)

ϕ∗1 (u, v) : = − Im i(sinh z, z, cosh z) = − Re(sinh z, z, cosh z)= −(sinh u cos v, u, cosh u cos v).

By Proposition 2.14, the conjugate surface has non-degenerate fold singularities. Then by Theorem 2.15 one can get an analytic continuation of ϕ∗1
as a zero mean curvature surface in R31 which changes type across the foldsingularities. We can get an explicit description of such an extension of ϕ∗1
as follows: We set(t, x, y) = ϕ∗1 (u, v) = −(sinh u cos v, u, cosh u cos v).
Then the surface has fold singularities at (u, nπ) for any u ∈ R and n ∈ Z.
Then it holds thatt
= tanh u = − tanh xyand the image of −ϕ∗1 is contained in the surface C0 .
On the other hand, a subset of the time-like hyperbolic catenoid C− hasa parametrization(3.3) ϕ2 (u, v) :=1α(u) + β(v)(sinh u + sinh v, u + v, cosh u − cosh v) =
,
2
2

where
(3.4)

α(u) := (sinh u, u, cosh u),β(v) := (sinh v, v, − cosh v).

Also1ψ2 (u, v) := (− sinh u − sinh v, u + v, − cosh u + cosh v)
2gives a parametrization of a subset of C− . More precisely, C− is the union of
the closure of the images of ϕ2 and ψ2 . We get the following parametrization
(3.5)

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FUJIMORI ET. AL.

of the conjugate surface ϕ∗2 of ϕ2(3.6) ϕ∗2 (u, v) := 11α(u) − β(v) = (sinh u − sinh v, u − v, cosh u + cosh v),
22

where α and β are as in (3.4). (See [13] for the definition of the conjugate surfaces of time-like minimal surfaces.) To find the implicit function
representation of the image of ϕ∗2 , take a new coordinate system (ξ, ζ) asu = ξ + ζ,v = ξ − ζ.

Thenϕ∗2 (ξ, ζ) = (cosh ξ sinh ζ, ζ, cosh ξ cosh ζ),
which implies that the image is a subset of C0 . The entire graph C0 changes
type across two disjoint real analytic null curves {y = ± cosh x}.
Next, we prove a similar assertion for the Scherk surfaces, which is also
briefly mentioned in the caption of Figure 1 in [4]:Theorem 3.2. The conjugate space-like maximal surface of the space-like
Scherk surface S+ and the conjugate time-like minimal surface of the timelike Scherk surface S− of the first kind are both congruent to subsets of theentire graph S0 .
Proof. Using the identities
cos arg z = Re(

z
),
|z|

sin arg z = Im(

z
),
|z|

one can prove that a subset of the space-like Scherk surface S+ is parametrized
by the complex variable z as


π
1−z
1 − iz
1 + z2+ (1, 1, 1)
(3.7)
, log
, log
ϕ1 (z) = − Re i log
2
1−z
1+z
1 + iz2
1 + z2π1−z
1 − iz
= arg
+
,
arg
,
arg
(1, 1, 1).
1 − z21+z
1 + iz2
In fact, ϕ1 (z) is a multi-valued R31 -valued function, but can be considered
as a single-valued function on the universal cover of C ∪ {∞} \ {±1, ±i}.
We now set

ϕ1 (z) = t(z), x(z), y(z)and
ψ1 (z) := π − t(z), x(z), π − y(z) .

ZERO MEAN CURVATURE SURFACES

191

Then S+ is the union of the closure of the images of ϕ1 and ψ1 . The
conjugate ψ1∗ of the space-like Scherk surface ψ1 is obtained by
1−z
1 − iz
1 + z2
∗
, log
, log
(3.8)
ψ1 (z) = Im i log
1 − z21+z
1 + iz
1 − iz
1−z
1 + z2
.
, log
, log
= log
1 − z21+z
1 + izSince ψ1 admits only generalized cone-like singularities (cf. Proposition 2.14),
ψ1∗ admits only fold singularities, and has a real analytical extension acrossthe fold singularities to a time-like minimal surface in R31 (cf. Theorem 2.15).
More precisely, the image of the conjugate ψ1∗ is contained in the graph S0 ,shown as follows: The singular sets of ψ1 and ψ1∗ are both parametrized as{z = eiu }. Then the image of a connected component of singular curve by
ψ1∗ as in (3.8) is parametrized as
(3.9)



1
1 − cos u
1 + sin uπγ(u) =
2 log cot u, log
, log
0<u<
.
2
1 + cos u
1 − sin u
2
By the singular Björling formula (2.8) in Section 2, we have the following
analytic extension of γ
(3.10)

ψ̂2 (u, v) :=γ(u) + γ(v).
2Now, we check that the conjugate ψ2 := ψ̂2∗ of ψ̂2 as in (3.10) coincides
with the time-like Scherk surface S− . By (3.10), the conjugate ψ2 of ψ̂2
is parametrized by (see [13] for the definition of the conjugate surfaces oftime-like minimal surfaces)(3.11)1
ψ2 (u, v) = γ(u) − γ(v)
2



 1
1
1 − cos v
1 − cos u=log(cot u) − log(cot v) ,
− log
log
,
2
4
1 + cos u
1 + cos v

1 + sin u
1 + sin v
1
log
.
− log
4
1 − sin u
1 − sin v
We set (t, x, y) = ψ2 (u, v), and will show that (t, x, y) lies in S− : In fact, by(3.11), we have
e2t =

cot u
cos u sin v
=
,
cot v
sin u cos v

192

FUJIMORI ET. AL.

which impliesr1
cosh t =2

cos u sin v+
sin u cos v

r

sin u cos v
cos u sin v!

sin(u + v)=√.
sin 2u sin 2v

Using
e4x =
we have thatuv 2
1 − cos u 1 + cos v ,×= tan cot
1 + cos u 1 − cos v
22
cosh x = √

1
u+v
.
sin
2
sin u sin v

Similarly,
1
u+v
cosh y = √cos
2cos u cos v
holds. Hence the analytic extension of the conjugate of S+ coincides with
S0 . As pointed out in [14], the space-like part of S0 is connected, and thetime-like part of S0 consists of four connected components, each of which iscongruent to the image of ψ1 (see Figure 3, right).
In the introduction, we saw that the conjugate surfaces of C+ and C− (resp.
S+ and S− ) are both subsets of the same zero mean curvature surface C0(resp. S0 ). As pointed out in [14], a similar phenomenon also holds between
elliptic catenoids and parabolic catenoids: The helicoid x sin t = y cos t iswell known as a ruled minimal surface in the Euclidean 3-space R3 , and also
gives a zero-mean curvature in R31 . The space-like elliptic catenoid
ϕE+ (u, v) := (v, cos u sinh v, sin u sinh v)
and the time-like elliptic catenoid
ϕE− (u, v) := (v, cosh u sinh v, sinh u sinh v)
induce their conjugate surfaces, both of which are subsets of the helicoids.
The space-like parabolic catenoid, on the other hand,


3
3
v
v
ϕP+ (u, v) := v −+ u2 v, v +− u2 v, 2uv
3
3
is given by Kobayashi [18] as an Enneper surface of the 2nd kind. Consider
the ruled zero-mean curvature surface, which we call the parabolic helicoid



u3
u3
2
P
,
ϕ0 (u, v) := γ(u) + v(u, −u, 1),γ(u) := −u − , −u + , −u33ZERO MEAN CURVATURE SURFACES

193

Figure 4. The completions of parabolic catenoids ϕP+ (left)and ϕP− (right).
where γ(u) is a non-degenerate null curve at which the surface changes
type. If v > 0, ϕP0 gives the conjugate surface of ϕP+ . If v < 0, ϕP0 gives the
conjugate surface of the time-like parabolic catenoid 2 given by


3
3
u
u
ϕP− (u, v) := −u −− uv 2 , −u ++ uv 2 , −u2 − v 2 .
3
3

The image of the parabolic catenoid ϕP+ (u, v) (resp. ϕP− (u, v)) is a subset of(cf. Figure 4)(3.12)
12(x2 − t2 ) = (x + t)4 − 12y 2

resp. 12(x2 − t2 ) = −(x + t)4 − 12y 2 .

We call it the completion of the space-like (resp. time-like) parabolic catenoid,
which contains a light-like line
L := {y = x + t = 0}.

The initial parametrization ϕP± does not include this line L. Kobayashi [18,
Example 2.3] noticed the line L in the surface and drew a hand-drawn figureof it that coincides with the left-hand side of Figure 4.
In [4], the zero mean curvature surfaces containing singular light-like lines
are categorized into the following six classes(3.13)

α+ ,

α0I ,

α0II ,

α−
I ,

α−
II ,

α−
III .

−
−
The surfaces belonging to α+ (resp. α−
I , αII , αIII ) are space-like (resp. timelike). On the other hand, the causalities of surfaces in α0I and α0II are not
2In [14, Examples 2.8 and 2.9], these are called spacelike (timelike) parabolic helicoids.

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FUJIMORI ET. AL.

unique. In fact, the light cone and the hyperbolic catenoids C± are examples
of surfaces3 of type α0II . We get the following assertion:
Proposition 3.3. The completion of space-like (resp. time-like) parabolic
catenoids gives an example of surfaces of type α0II at each point (−c, c, 0)(c 6= 0) on the light-like line L. In other words, both space-like surfaces
and time-like surfaces exist in the class α0II of zero mean curvature surfaces
containing light-like lines.
Proof. We set
F± := (x + t){12(x − t) ∓ (x + t)3 } + 12y 2 .

Then (3.12) is rewritten as F± = 0. We fix a point (−c, c, 0) (c 6= 0) on theset {F± = 0}. Since∂F(−c, c, 0) = −24t ∓ 4(x + t)3 (t,x)=(−c,c) = 24c(6= 0),∂t
the implicit function theorem yields that there exists a C ∞ -function t =t± (x, y) such that the set {F± = 0} is parametrized by the graph t = t± (x, y)around the point (−c, c, 0). By [4], we know that t = t± (x, y) has thefollowing expression
α(c) 2t± (c, y) = −c −y + β(c, y)y 3 ,2where α = α(x) and β = β(x, y) are C ∞ -functions. Differentiating theequation F± (t± (x, y), x, y) = 0 with respect to y, we have that
(3.14)−24t(x, y)ty (x, y) − 4(x + t(x, y))3 ty (x, y) + 24y = 0,

where ty = ∂t/∂y. Substituting (t, x, y) = (−c, c, 0), we getwhich implies that
(3.15)−cty (c, 0) = t(c, 0)ty (c, 0) = 0,ty (c, 0) = 0.

Differentiating (3.14) with respect to y again, we have−24t2y − 24t tyy ∓ 12(x + t)2 t2y ∓ 4(x + t)3 tyy + 24 = 0.

Substituting (t, x, y) = (−c, c, 0) and (3.15), we get−24ctyy + 24 = 0,

namely α(c) = 2/c. This implies that t = t± (x, y) is of type α0II at (c, 0).
3In [4], we wrote that the hyperbolic catenoids C0± are examples of surfaces of type αI ,but this is a typographical error. Also, in [4], we wrote that the time-like Scherk surface−of the first kind (resp. of the second kind) is of type α−
I (resp. of type αII ), however this−is again a typographical error, and it is, in fact, of type α−
II (resp. of type αI ).

ZERO MEAN CURVATURE SURFACES

195

In the authors’ previous work [5], surfaces of type α0I changing type acrossa light-like line have been constructed. The only other possibility for the
existence of surfaces changing type across a light-like line must be of typeα0II (cf. [4]). So the following question is of interest:
Problem. Do there exist zero-mean curvature surfaces of type α0II whichchange type across a light-like line?
Also, the existence of space-like maximal surfaces of type α0I is unknown(the time-like surfaces given in [4, Example 1] are of types α0I and α−
III ).
4. A relationship to fluid mechanics
As mentioned in the introduction, we give an application of Theorem 2.19
to fluid mechanics: Consider a two-dimensional flow on the xy-plane withvelocity vector field v = (u, v), and with density ρ and pressure p. We
assume the following:(i) The fluid is barotropic, that is, there exists a strictly increasing function p(s) (s > 0) such that the pressure p is expressed by p = p(ρ).
A positive function c defined by
(4.1)

c2 =

dp
= p′ (ρ)
dρ

is called the local speed of sound, cf. [1, pages 5–6].
(ii) The flow is steady, that is, v, p and ρ do not depend on time.
(iii) There are no external forces,(iv) and the flow is irrotational, that is, rot v(= vx − uy ) = 0.
By the assumption (ii), the equation of continuity is reduced todiv(ρv) = (ρu)x + (ρv)y = 0.
Hence there exists locally a smooth function ψ = ψ(x, y) such that(4.2)

ψx = −ρv,

ψy = ρu,which is called the stream function of the flow. The following assertion iswell-known:
Fact 4.1. The stream function ψ of a two-dimensional flow under the conditions (i)–(iv) satisfies(4.3)(ρ2 c2 − ψy2 )ψxx + 2ψx ψy ψxy + (ρ2 c2 − ψx2 )ψyy = 0.

Proof. By the assumptions (ii), (iii) and (iv), Euler’s equation of motion
∂v
∂v
∂v
1
+u
+v
+ grad p = 0
∂t
∂x
∂y
ρ

196

FUJIMORI ET. AL.

is reduced to(4.4)

uux + vvx +

px
= 0,
ρ

uuy + vvy +

py
= 0,
ρ

that is,(4.5)

dp + ρq dq = 0

p(q = |v| = u2 + v 2 ).

Here, by the barotropicity (i), we have
(4.6)

px =

∂ρ
∂
p(ρ) = p′ (ρ)= c2 ρx ,∂x∂xpy = c2 ρy .

Substituting these into the equation of motion (4.4), we have
(4.7)

ρx =

ρ
px
= − 2 (uux + vvx ),2ccρρy = − 2 (uuy + vvy ),cand henceρ(ρv)x = ρx v + ρvx = − 2 (uux + vvx )v + ρvx ,cρ(ρv)y = ρy v + ρvy = − 2 (uuy + vvy )v + ρvy ,cρ(ρu)x = ρx u + ρux = − 2 (uux + vvx )u + ρux ,cρ(ρu)y = ρy u + ρuy = − 2 (uuy + vvy )u + ρuychold. Thus, we have(ρ2 c2 − ψy2 )ψxx = (ρ2 c2 − ρ2 u2 )(−ρv)x
 ρ

2 2
2
= −ρ (c − u ) − 2 (uux + vvx )v + ρvx ,
c

 ρ
2
2
ψx ψy ψxy = −ρ uv(−ρv)y = ρ uv − 2 (uuy + vvy )v + ρvy , cρ
ψx ψy ψyx = −ρ2 uv(ρu)x = −ρ2 uv − 2 (uux + vvx )u + ρux ,
c
2 2
2
2 2
2 2
(ρ c − ψx )ψyy = (ρ c − ρ v )(ρu)y

 ρ
2 2
2
= ρ (c − v ) − 2 (uuy + vvy )u + ρuy .
cSumming these up, it holds that

(4.8) (ρ2 c2 −ψy2 )ψxx +2ψx ψy ψxy +(ρ2 c2 −ψx2 )ψyy = ρ3 (u2 +v 2 −c2 )(vx −uy ).
Here, by the assumption (iv), we have vx = uy . Then we have the conclusion.
ZERO MEAN CURVATURE SURFACES

197

When ρc = 1, the equation (4.3) coincides with the zero mean curvatureequation (1.1). In this case, (4.1) yields that1(4.9)
p = p0 − ,ρwhere p0 is a positive constant. This means that the zero mean curvatureequation induces a virtual gas. For actual gas, the pressure p is proportionalto ργ , where γ is a constant (> 1, γ ≈ 1.4 for air).
Differentiating (4.9), we have thatdρ
(4.10)
dp = 2 .
ρ
Substituting (4.10) into (4.5), we have that12d − 2 + q = 0,ρthat is, there exists a constant k such that1
(4.11)− 2 + q 2 = k.
ρ
Here, by (4.2), it holds that1
(4.12)(u, v) = (ψy , −ψx ).
ρ
Thus, (4.11) is equivalent to

1 − ψx2 − ψy2 = −kρ2 .

We suppose that 1 − ψx2 − ψy2 does not vanish identically. Then k 6= 0 andp
we may set |k| = 1/ρ0 , where ρ0 is a positive constant. Then we have
ρ = ρ0 |1 − ψx2 − ψy2 |1/2 .

Note that by (4.12) and the assumption cρ = 1, the speed |v| = q is greaterthan (resp. less than) the speed of sound c, that is, the flow is supersonic(resp. subsonic), if and only if 1 − ψx2 − ψy2 < 0 (resp. > 0).
Suppose now that the flow changes from being subsonic to supersonic at
a curve
σ(t) := x(t), y(t)(a ≤ t ≤ b).
Without loss of generality, we may assume that t is an arclength parameterof the curve σ. In particular,ρ = ρ0 |1 − ψx2 − ψy2 |1/2 = 0

holds on the curve σ. Since the local speed of sound is given by2
2 −1/2,c = ρ−1 = ρ−1
0 |1 − ψx − ψy |

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FUJIMORI ET. AL.

the curve σ is a singularity of the flow, although the stream function itself isreal analytic near σ. Moreover, we suppose that each point of the curve σ(t)is a non-degenerate point of type change with respect to ψ. Then, as seenin the proof of Proposition 2.6, we can take the parameter t of the curve σsuch that t 7→ (t, σ(t)) gives a non-degenerate null curve in R31 . Moreover,
it holds thatx′ (t) = ψx x(t), y(t) ,y ′ (t) = ψy x(t), y(t)and x′ (t)2 + y ′ (t)2 = 1. Since ∇B 6= 0 at a non-degenerate point of typechange, the proof of Proposition 2.5 yields that σ ′′ (t) 6= 0. Since x′ (t)2 +y ′ (t)2 = 1, this implies that σ(t) is a locally convex curve. Consequently, we
get the following assertion:

Theorem 4.2. Let σ(t) := x(t), y(t) be a locally convex curve on the xyplane with an arc-length parameter t. Then the graph t = ψ(x, y) of the zeromean curvature surface
ϕ̂σ̃ as in Theorem 2.19 associated to the null curveσ̃ := t, x(t), y(t) gives a real analytic stream function satisfying (4.3) with
(4.9) (i.e. cρ = 1) which changes from being subsonic to supersonic at the
curve σ. Moreover,1(u, v) := (ψy , −ψx )ρ
gives the velocity vector of the flow such that(1) u2 + v 2 diverges to ∞ as (x, y) approaches the image of σ.
(2) The flow changes from being subsonic to being supersonic across σ.
(3) The acceleration vector σ ′′ (t) points to the supersonic region.
Proof. Since σ is locally convex, its lift σ̃ is a non-degenerate null curve in
R31 . Then we can apply Theorem 2.19 for the curve σ̃, and get a graphψ(x, y) of a zero mean curvature surface which changes type at σ̃. Then ψ
can be considered as a stream function which changes from being subsonicto supersonic at the curve σ. We set
σ(t + s) + σ(t − s)σ ′′ (t) 2≈ σ(t) +s ,22which is the midpoint of the two points σ(t + s), σ(t − s) of the curve σ. By
Theorem 2.19, the flow is supersonic at the point P , which proves the lastassertion.
P :=Acknowledgements
The seventh author thanks Osamu Kobayashi for a fruitful discussion
at Osaka University, who suggested that our subject might be applied tothe theory of 2-dimensional compressible gas flow. The authors also thankZERO MEAN CURVATURE SURFACES

199

Miyuki Koiso for valuable comments. They would also like to thank thereferee for reading our manuscript carefully.
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Shoichi Fujimori
Department of Mathematics, Faculty of Science, Okayama University,
Okayama 700-8530, Japane-mail address: fujimori@math.okayama-u.ac.jp
Young Wook Kim
Department of Mathematics, Korea University, Seoul 136-701, Korea
e-mail address: ywkim@korea.ac.kr
Sung-Eun Koh
Department of Mathematics, Konkuk University, Seoul 143-701, Korea
e-mail address: sekoh@konkuk.ac.kr
Wayne Rossman
Department of Mathematics, Faculty of Science, Kobe University, Kobe
657-8501, Japane-mail address: wayne@math.kobe-u.ac.jp
Heayong Shin
Department of Mathematics, Chung-Ang University, Seoul 156-756, Korea
e-mail address: hshin@cau.ac.kr
Masaaki Umehara
Department of Mathematical and Computing Sciences, Tokyo Institute ofTechnology, Tokyo 152-8552, Japane-mail address: umehara@is.titech.ac.jp
Kotaro Yamada
Department of Mathematics, Tokyo Institute of Technology, Tokyo
152-8551, Japane-mail address: kotaro@math.titech.ac.jp
Seong-Deog Yang
Department of Mathematics, Korea University, Seoul 136-701, Korea
e-mail address: sdyang@korea.ac.kr(Received August 22, 2013 )(Revised March 27, 2014 )