FUJIMORI, S.

KIM, Y. W.

KOH, S. -E.

ROSSMAN, W.

SHIN, H.

UMEHARA, M.

YAMADA, K.

YANG, S. -D.

Fujimori, S

Kim, Y. W.

Koh, S.-E.

Rossman, W.

Shin, H.

Umehara, M.

Yamada, K.

Yang, S.-D.

Osaka University Knowledge Archive

ZERO MEAN CURVATURE SURFACES IN LORENTZ–MINKOWSKI 3-SPACE WHICH CHANGE TYPE ACROSS A LIGHT-LIKE LINE

Osaka City University Repository

Fujimori, S., Kim, Y.W., Koh, S.-E., Rossman, W., Shin, H., Umehara, M., Yamada, K. and Yang, S.-D.
Osaka J. Math.
52 (2015), 285–297
ZERO MEAN CURVATURE SURFACES IN
LORENTZ–MINKOWSKI 3-SPACE WHICH
CHANGE TYPE ACROSS A LIGHT-LIKE LINE
S. FUJIMORI, Y.W. KIM, S.-E. KOH, W. ROSSMAN, H. SHIN, M. UMEHARA,
K. YAMADA and S.-D. YANG
(Received October 2, 2013)
Abstract
It is well-known that space-like maximal surfaces and time-like minimal surfaces
in Lorentz–Minkowski 3-space R31 have singularities in general. They are both char-
acterized as zero mean curvature surfaces. We are interested in the case where the
singular set consists of a light-like line, since this case has not been analyzed before.
As a continuation of a previous work by the authors, we give the first example of
a family of such surfaces which change type across a light-like line. As a corollary,
we also obtain a family of zero mean curvature hypersurfaces in RnC11 that change
type across an (n   1)-dimensional light-like plane.
Introduction
Many examples of space-like maximal surfaces containing singular curves in the
Lorentz–Minkowski 3-space (R31I t , x , y) of signature ( CC) have been constructed in
[11], [1], [12], [8], [4] and [5].
In this paper, we are interested in the zero mean curvature surfaces in R31 changing
their causal type: Klyachin [10] showed under a sufficiently weak regularity assumption
that a zero mean 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 light-like) which
are non-degenerate (i.e., their projections into the xy-plane are locally convex plane
curves), or
• light-like lines, which are degenerate everywhere.
Given a non-degenerate null curve  in R31, there exists a zero mean curvature surface
which changes its causal type across this curve from a space-like maximal surface to
2010 Mathematics Subject Classification. Primary 53A10; Secondary 53B30, 35M10.
Kim was supported by NRF 2009-0086794, Koh by NRF 2009-0086794 and NRF 2011-0001565,
Shin by NRF-2014R1A2A2A01007324 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-in-Aid for Scientific Research (B) No. 20340012, Umehara by (A) No. 22244006 and Yamada
by (B) No. 21340016 from Japan Society for the Promotion of Science.
286 S. FUJIMORI ET AL.
a time-like minimal surface (cf. [6], [10], [9] and [7]). This construction can be ac-
complished using the Björling formula for the Weierstrass-type representation formula
of maximal surfaces. (The reference [3] is an expository article on this subject.) How-
ever, if  is a light-like line, the aforementioned construction fails, since the isothermal
coordinates break down at the light-like singular points. Locally, such a surface is the
graph of a function t D f (x , y) satisfying
(1   f 2y ) fxx C 2 fx fy fxy C (1   f 2x ) fyy D 0,()
where fx D  f =x , fxy D 2 f = yx , etc. We call this and its graph the zero mean
curvature equation and a zero mean curvature surface, respectively. Until now, zero
mean curvature surfaces which actually change type across a light-like line were un-
known. As announced in [2], the main purpose of this paper is to construct such an
example. In Section 1, we give a formal power series solution of the zero mean curva-
ture equation describing all zero mean curvature surfaces which contain a light-like
line. Using this, we give the precise statement of our main result and show how the
statement can be reduced to a proposition (cf. Proposition 1.3). In Section 2, we then
prove it. As a consequence, we obtain the first example of (a family of) zero mean
curvature surfaces which change type across a light-like line.
1. The main theorem
We discuss solutions of the zero mean curvature equation () which have the
following form
(1.1) f (x , y) D b0(y)C
1
X
kD1
bk(y)
k
xk ,
where bk(y) (k D 1, 2, : : :) are C1-functions. When f contains a singular light-like
line, we may assume without loss of generality that (cf. [2])
(1.2) b0(y) D y, b1(y) D 0.
As was pointed out in [2], there exists a real constant  called the characteristic of f
such that b2(y) satisfies the following equation
(1.3) b02(y)C b2(y)2 C  D 0 ( 0 D d=dy).
Now we derive the differential equations satisfied by bk(y) for k  3 assuming
(1.2). If we set
Y WD fy   1 D
1
X
kD2
b0k(y)
k
xk
ZERO MEAN CURVATURE SURFACES 287
and
P WD 2(Y fxx   fx fxy), Q WD Y 2 fxx   2 fx fxyY , R WD f 2x fyy ,
then, by straightforward calculations, we see that
P D  b2b02x2  
4
3
b2b03x3  
1
X
kD4

Pk C
2(k   1)
k
b2b0k C (3   k)b02bk

xk ,
Q D  
1
X
kD4
Qk xk , R D
1
X
kD4
Rk xk ,
where
(1.4)
Pk WD
k 1
X
mD3
2(k   2m C 3)
k   m C 2
bmb0k mC2,
Qk WD
k 2
X
mD2
k m
X
nD2
3n   k C m   1
mn
b0mb0nbk m nC2,
Rk WD
k 2
X
mD2
k m
X
nD2
bmbnb00k m nC2
k   m   n C 2
for k  4, and that the zero mean curvature equation () reduces to
1
X
kD2
b00k
k
xk D fyy D P C Q C R.
It is now immediate, by comparing the coefficients of xk from both sides, to see that
each bk (k  3) satisfies the following ordinary differential equation
(1.5) b00k (y)C 2(k   1)b2(y)b0k(y)C k(3   k)b02(y)bk(y) D  k(Pk C Qk   Rk),
where P3 D Q3 D R3 D 0 and Pk , Qk and Rk are as in (1.4) for k  4. Note that Pk ,
Qk and Rk are written in terms of b j ( j D 1, : : : , k   1) and their derivatives.
Now, we consider the case that 1   f 2x   f 2y changes sign across the light-like
line {t D y, x D 0}. This case occurs only when the characteristic  as in (1.3) of f
vanishes [2]. If we set
b2(y) D 0 (y 2 R),
then (1.3) holds for  D 0. So we assume
(1.6) b0(y) D y, b1(y) D 0, b2(y) D 0, b3(y) D 3cy,
288 S. FUJIMORI ET AL.
where c is a non-zero constant. Then f (x , y) in (1.1) can be rewritten as
(1.7) f (x , y) D y C cyx3 C
1
X
kD4
bk(y)
k
xk .
In this situation, we will find a solution satisfying
(1.8) bk(0) D b0k(0) D 0 (k  4).
Then (1.5) reduces to
b00k (y) D  k(Pk C Qk   Rk), bk(0) D b0k(0) D 0, (k  4),(1.9)
Pk D
k 1
X
mD3
2(k   2m C 3)
k   m C 2
bm(y)b0k mC2(y) (k  4),(1.10)
Qk D
k 4
X
mD3
k m 1
X
nD3
3n   k C m   1
mn
b0m(y)b0n(y)bk m nC2(y) (k  7),(1.11)
Rk D
k 4
X
mD3
k m 1
X
nD3
bm(y)bn(y)b00k m nC2(y)
k   m   n C 2
(k  7),(1.12)
and Qk D Rk D 0 for 4  k  6, where the fact that b2(y) D 0 has been extensively
used. For example,
b0 D y, b1 D b2 D 0, b3 D 3cy, b4 D  4c2 y3, b5 D 9c3 y5,
b6 D  24c2 y7, b7 D 70c5 y9   14c3 y3, : : : .
In this article, we show the following assertion:
Theorem 1.1. For each positive number c, the formal power series solution f (x , y)
uniquely determined by (1.9), (1.10), (1.11) and (1.12) gives a real analytic zero mean
curvature surface on a neighborhood of (x , y) D (0, 0). In particular, there exists a non-
trivial 1-parameter family of real analytic zero mean curvature surfaces each of which
changes type across a light-like line (see Fig. 1).
As a consequence, we get the following:
Corollary 1.2. There exists a family of zero-mean curvature hypersurfaces in
Lorentz–Minkowski space RnC11 each of which changes type across an (n   1)-
dimensional light-like plane.
Proof. Let f be as in the theorem. The graph of the function defined by
Rn 3 (x1, : : : , xn) 7! f (x1, x2) 2 R
ZERO MEAN CURVATURE SURFACES 289
Fig. 1. The graph of t D f (x , y) for c D 1=2 and jx j, jyj < 0.8
(The range of the graph is wider than the range used in our math-
ematical estimation. However, this figure still has a sufficiently
small numerical error term in the Taylor expansion.)
gives the desired hypersurface. In this case, the zero mean curvature equation
0
1  
n
X
jD1
f 2x j
1
A
n
X
iD1
fxi xi C
n
X
i, jD1
fxi x j fxi fx j D 0 ( fxi WD  f =xi , fxi x j WD 2 f =x jxi )
reduces to () in the introduction.
To prove Theorem 1.1, it is sufficient to show that for arbitrary positive constants
c > 0 and Æ > 0 there exist positive constants n0, 0, and C such that
(1.13) jbk(y)j  0Ck (jyj  Æ)
holds for k  n0. In fact, if (1.13) holds, then the series (1.7) converges uniformly over
the rectangle [ C 1, C 1]  [ Æ, Æ].
The key assertion to prove (1.13) is the following
290 S. FUJIMORI ET AL.
Proposition 1.3. For each c > 0 and Æ > 0, we set
(1.14) M WD 3 max{144c jÆj3=2, 4
p
192c2 },
where  is the positive constant given by (A.3) in the appendix, such that
(1.15) t
Z 1 t
t
du
u2(1   u)2   (0 < t < 1=2).
Then the function {bl(y)}l3 formally determined by the recursive formulas (1.9)–
(1.12) satisfies the inequalities
jb00l (y)j  cjyjl

M l 3,(1.16)
jb0l(y)j 
3cjyjlC1
l C 2
M l 3,(1.17)
jbl(y)j  3cjyj
lC2
(l C 2)2 M
l 3(1.18)
for any
(1.19) y 2 [ Æ, Æ],
where
(1.20) l WD 1
2
(l   1)   2 (l  3).
Once this proposition is proven, (1.13) follows immediately. In fact, if we set
0 D
3
c
(ÆM)3, C WD ÆM
and n0  7, then 1  l C 2 < l   3 and (1.13) follows from
3cjyjlC2
(l C 2)2 M
l 3
 0C l .
2. Proof of Proposition 1.3
We prove the proposition using induction on the number l  3. If l D 3, then
jb003(y)j D 0 
c
jyj
D cjyj3

M0,
jb03(y)j D 3c D
3cjyj3C1
3 C 2
M0,
jb3(y)j D 3cjyj D 3cjyj
3C2
(3 C 2)2 M
0
ZERO MEAN CURVATURE SURFACES 291
hold, using that b3(y) D 3cy, M0 D 1 and 3 D  1. So we prove the assertion for
l  4. Since (1.17), (1.18) follow from (1.16) by integration, it is sufficient to show
that (1.16) holds for each l  4. (In fact, the most delicate case is l D 4. In this case
l D  1=2 and we can use the fact that
R y0
0 1=
py dy for y0 > 0 converges.)
The inequality (1.16) follows if one shows that, for each k  4
(2.1) jk Pk(y)j, jk Qk(y)j, jk Rk(y)j  c3 jyj
k Mk 3 (jyj  Æ)
under the assumption that (1.16), (1.17) and (1.18) hold for all 3  l  k   1. In fact,
if (2.1) holds, (1.16) for l D k follows immediately. Then by the initial condition (1.9)
(cf. (1.8)), we have (1.17) and (1.18) for l D k by integration.
The estimation of jkPkj for k  4. By (1.10) and using the fact that (1.17),
(1.18) hold for l  k   1, we have for each jyj < Æ that
jk Pk j 
k 1
X
mD3
2kjk   2m C 3j
k   m C 2
jbm(y)j jb0k mC2(y)j

k 1
X
mD3
2kjk   2m C 3j
k   m C 2

3cMm 3jyjmC2
(m C 2)2

3cMk mC2 3jyj(k mC2)C1
(k   m C 2) C 2

D cMk 3jyjk

144cjyj3=2
M
k 1
X
mD3
kjk   2m C 3j
(m   1)2(k   m C 1)(k   m C 2)
 cMk 3jyjk

144cjÆj3=2
M
k 1
X
mD3
kjk   2m C 3j
(m   1)2(k   m C 1)(k   m C 2)

c
3
Mk 3jyjk

k 1
X
mD3
kjk   2m C 3j
(m   1)2(k   m C 1)2 .
Here, we used (1.14). Since
max
mD3,:::,k 1
jk   2m C 3j D max
mD3,k 1
jk   2m C 3j D max{jk   3j, j k C 5j},
by setting q D m   1, we have that
jk Pk j 
c
3
Mk 3jyjk

k 1
X
mD3
k2
(m   1)2(k   m C 1)2 D
c
3
Mk 3jyjk

1
k
k 2
X
qD2
k3
q2(k   q)2

c
3
Mk 3jyjk

1
k
Z 1 1=k
1=k
du
u2(1   u)2 
c
3
Mk 3jyjk

,
where we applied Lemma A.1 and (1.15) at the last step of the estimations. Hence,
we get (2.1) for k Pk .
292 S. FUJIMORI ET AL.
The estimation of jkQkj for k  7. By (1.11) and the induction assumption, we
have that
jk Qk j 
k 4
X
mD3
k m 1
X
nD3
kj3n   k C m   1j
mn
jb0m(y)j jb0n(y)j jbk m nC2(y)j

k 4
X
mD3
k m 1
X
nD3
kj3n   k C m   1j
mn

3cMm 3jyjmC1
m C 2



3cMn 3jyjnC1
n C 2

3cMk m nC2 3jyj(k m nC2)C2
((k   m   n C 2) C 2)2

D cMk 3jyjk

432c2
M4
k 4
X
mD3
k m 1
X
nD3
kj3n   k C m   1j
(m   1)2(n   1)2(k   m   n C 2)2 .
Now we apply the inequalities
max
3mk 4
3nk m 1
j3n   k C m   1j D max
(m,n)D(3,3),(3,k 4),(k 4,3)
j3n   k C m   1j
D max{j k C 11j, 4, j2k   10j}  2k,
and also
432c2
M4

1
36
,
which follows from (1.14). Setting p WD m   1, q D n   1, we have that
jk Qk j 
c
36
Mk 3jyjk

k 4
X
mD3
k m 1
X
nD3
2k2
(m   1)2(n   1)2(k   m   n C 2)2
D
c
18
Mk 3jyjk

k 5
X
pD2
k p 2
X
qD2
k2
p2q2(k   p   q)2 .
Now applying Lemma A.2, we have that
jk Qk j 
c
18
Mk 3jyjk

 6 
c
3
Mk 3jyjk

,
which proves (2.1) for k Qk .
ZERO MEAN CURVATURE SURFACES 293
The estimation of jkRkj for k  7. As in the case of jk Qk j, we have that
jk Rk j 
k 4
X
mD3
k m 1
X
nD3
kjbm(y)j jbn(y)j jb00k m nC2(y)j
k   m   n C 2

k 4
X
mD3
k m 1
X
nD3
k
k   m   n C 2

3cMm 3jyjmC2
(m C 2)2



3cMn 3jyjnC2
(n C 2)2

(cMk m nC2 3jyj(k m nC2))
D 144c3 Mk 7jyjk
k 4
X
mD3
k m 1
X
nD3
k
(k   m   n C 2)(m   1)2(n   1)2
D cMk 3jyjk

144c2
M4
k 4
X
mD3
k m 1
X
nD3
k2
(k   m   n C 2)2(m   1)2(n   1)2 .
Now we set p D m   1, q D n   1, and using the inequality
34  144c2  34  192c2 < M4,
we have that
jk Rk j 
c
34
Mk 3jyjk

k 5
X
pD2
k p 2
X
qD2
k2
p2q2(k   p   q)2 .
By applying Lemma A.2, we have that
jk Rk j 
c
34
Mk 3jyjk

 6 <
c
3
Mk 3jyjk

,
which proves (2.1) for k Rk . This completes the proof of Proposition 1.3.
Appendix A. Inequalities used in the proof of Theorem 1.1
For a > 0, it holds that
(A.1) 1
u2(a   u)2 D
1
a3

a
u2
C
2
u
C
a
(a   u)2 C
2
a   u

.
Therefore,
(A.2)
Z a t
t
du
u2(a   u)2 D
2
a3

a(a   2t)
t(a   t) C 2 log
a   t
t

(0 < t < a=2).
294 S. FUJIMORI ET AL.
In particular, one can show that there exists a positive constant  such that
(A.3) t
Z 1 t
t
du
u2(1   u)2   (0 < t < 1=2).
The following assertion is needed to prove (2.1) for k Pk(y):
Lemma A.1. Let p be a non-negative integer and k an integer satisfying k 
p C 4. Then the inequality
k p 2
X
qD2
k3
q2(k   p   q)2 
Z a 1=k
1=k
du
u2(a   u)2 (a WD 1   p=k)
holds.
Proof. In fact, if we set a WD 1   p=k, then (A.1) yields that
k3
q2(k   p   q)2 D
1
k
1
(q=k)2(a   q=k)2
D
1
a3

1
k

a
(q=k)2 C
2
q=k

C
1
k

a
(a   q=k)2 C
2
a   q=k

.
Since x 7! (a C 2x)=x2 is a monotone decreasing function and the function x 7! (a C
2(a   x))=(a   x)2 is monotone increasing on the interval (0, a=2), we have that
k3
q2(k   p   q)2 
1
a3

Z q=k
(q 1)=k

a
u2
C
2
u

du C
Z (qC1)=k
q=k

a
(a   u)2 C
2
a   u

du

,
which yields that
k p 2
X
qD2
a3k3
q2(k   p   q)2 
Z a 2=k
1=k

a
u2
C
2
u

du C
Z a 1=k
2=k

a
(a   u)2 C
2
a   u

du

Z a 1=k
1=k

a
u2
C
2
u

du C
Z a 1=k
1=k

a
(a   u)2 C
2
a   u

du

Z a 1=k
1=k

a
u2
C
2
u
C
a
(a   u)2 C
2
a   u

du
D
Z a 1=k
1=k
du
u2(a   u)2 .
This proves the assertion.
The following assertion is needed to prove (2.1) for k Qk(y) and k Rk(y):
ZERO MEAN CURVATURE SURFACES 295
Lemma A.2. For any integer k  7, the following inequalities holds:
k 5
X
pD2
k p 2
X
qD2
k2
p2q2(k   p   q)2 
6
k
Z 1 1=k
1=k
du
u2(1   u)2  6 ,
where  is a constant satisfying (A.3).
Proof. We set a D a(p) WD 1   (p=k). Applying Lemma A.1 and the identity
(A.2), we have that
k 5
X
pD2
k p 2
X
qD2
k2
p2q2(k   p   q)2 D
k 5
X
pD2
"
1
kp2
k p 2
X
qD2
k3
q2(k   p   q)2
#

k 5
X
pD2

1
kp2
Z a 1=k
1=k
du
u2(a   u)2

D
k 5
X
pD2

1
p2
2
a2

a   2=k
a   1=k
C 2
log(ka   1)
ka


k 5
X
pD2

2
p2a2

1C 2
log ka
ka


k 5
X
pD2
6
p2a2
,
where we used the fact that (log ka)=(ka) < 1. By applying Lemma A.1 and by using
the property (A.3) of the constant  , it holds that
k 5
X
pD2
k p 2
X
qD2
k2
p2q2(k   p   q)2  6
k 5
X
pD2
1
p2(1   p=k)2 D
6
k
k 5
X
pD2
k3
p2(k   p)2

6
k
k 2
X
pD2
k3
p2(k   p)2 
6
k
Z 1 1=k
1=k
du
u2(1   u)2 < 6 ,
which proves the assertion.
References
[1] F.J.M. Estudillo and A. Romero: Generalized maximal surfaces in Lorentz–Minkowski space
L3, Math. Proc. Cambridge Philos. Soc. 111 (1992), 515–524.
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K. Yamada and S.-D. Yang: Zero mean curvature surfaces in L3 containing a light-like line,
C.R. Math. Acad. Sci. Paris 350 (2012), 975–978.
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Zero mean curvature surfaces in Lorentz–Minkowski 3-space and 2-dimensional fluid mechanics,
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296 S. FUJIMORI ET AL.
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ZERO MEAN CURVATURE SURFACES 297
Shoichi Fujimori
Department of Mathematics, Faculty of Science
Okayama University
Okayama 700-8530
Japan
e-mail: fujimori@math.okayama-u.ac.jp
Young Wook Kim
Department of Mathematics
Korea University
Seoul 136-701
Korea
e-mail: ywkim@korea.ac.kr
Sung-Eun Koh
Department of Mathematics
Konkuk University
Seoul 143-701
Korea
e-mail: sekoh@konkuk.ac.kr
Wayne Rossman
Department of Mathematics, Faculty of Science
Kobe University
Kobe 657-8501
Japan
e-mail: wayne@math.kobe-u.ac.jp
Heayong Shin
Department of Mathematics
Chung-Ang University
Seoul 156-756
Korea
e-mail: hshin@cau.ac.kr
Masaaki Umehara
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Tokyo 152-8552
Japan
e-mail: umehara@is.titech.ac.jp
Kotaro Yamada
Department of Mathematics
Tokyo Institute of Technology
Tokyo 152-8551
Japan
e-mail: kotaro@math.titech.ac.jp
Seong-Deog Yang
Department of Mathematics
Korea University
Seoul 136-701
Korea
e-mail: sdyang@korea.ac.kr


ZERO MEAN CURVATURE SURFACES IN LORENTZ-MINKOWSKI 3-SPACE WHICH CHANGE TYPE ACROSS A LIGHT-LIKE LINE

http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/57676/1/ojm52_01_285.pdf

ZERO MEAN CURVATURE SURFACES IN LORENTZ-MINKOWSKI 3-SPACE WHICH CHANGE TYPE ACROSS A LIGHT-LIKE LINE

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