Publicacions
Ma emá iques,
Vol
36
(1992),
39-46
.
LINEARIZATION
AND
EXPLICIT
SOLUTIONS
OF
THE
MINIMAL
SURFACE
EQUATION
A
bs ac
ALEXANDER
G
.
REZNIKOV
We
show
ha
he
appa a us
o
suppo
unc ions,
usually
used
in
con ex
su aces
heo y,
leads
o
he
linea
equa ion
Ah+2h
=
0
de-
sc ibing
locally
ge ms
o
minimal
su aces
.
He e
A
is
he
Laplace-
Bel ami
ope a o
on he s anda d
wo-dimensional
sphe e
.
I
explains
he
exis en e
o
he
sum
ope a ion
o
minimal
su -
aces,
in oduced
ecen ly
.
In
4-dimensional
space
he
equa ion
Oh
+
2h
=
0
becomes
inequali y
whe e e
he
Gauss
cu a u e
o
a
minimal
hype su ace
is
nonze o
.
0
.
In oduc ion
Recen ly
g ea
p og ess
was
achie ed
in
he
in es iga ion
and
con-
s uc ion
o
examples
o
minimal
su aces
in
R3
[1]-[3]
.
The
Gauss
map
usually
plays
a
signi ican
ole
and
i s
singula i ies
in a
sense
con ol
opology
i
he
su ace
is
comple e
[4]
.
I
was
also
no iced
[5], [6]
ha
he e
exis s
a
"sum"
ope a ion
M
l
+
M2
o
wo
mini nal
su aces
Ml,
M2
.
I
may
seem
o
be
s ange,
o
he
usual
o m
o
he
minimal
su ace
equa ion
is
essen ially
nonlinea
.
T ue, gi en
a
con o mal
minimal
map
R2
D
U
x>
M
C
R3
we
ha e
a
linea
equa ion
áx
=
0
[7]
.
Howe e ,
he
condi ion
o
con o mali y
is
nonlinea
i sel
.
In his
pape ,
we
show
ha
appa a us
o
suppo
unc ions
usually
used
in
con ex
su ace
heo y
leads
o
he
linea
and
comple ely
in e-
g adle
equa ion
o
minimal
su aces
in
R3
.
We
a e
able
o
w i e
down
an
explici
o mula
desc ibing
locally
all
minimal
su aces
wi h
non anishing
cu a u e
which
is
qui e
di e en
om
he
Weie s ass
desc ip ion
.
We
hope
ou
me hod
will
be
use ul in
global
p oblems,
oo
.
I
au oma ically
implies
he
exis en e
o
he
sum
ope a ion
.
The
main
pa
o
his
wo k
was
done
du ing
he
au ho 's
isi
o
Li huania
in
1987
.
I
wish
o
hank
P o esso
F
.
Weiksa
o
ui ul dis-
cussions
.
I
also
wish
o
hank
he
e e ee
o
his
e y
aluable
ema ks,
in
pa icula ,
o
indica ing
o
me
ha
he
ela ion
o equa ion
(6)
o
minimal
su aces
was
independen ly
s a ed
in
[7]
.
40

A
.G
.
REZNIKOV
1
.
The
main
equa ion
Le
M
be
a
smoo h
o ien ed
hype su ace
in
RN
and
G
:
M
->
SN-1
be
i s
Gauss
map
.
Then
[7]
G
is
a
local
di eomo phism
whe e e
he
Gauss
cu a u e
o
M
is
nonze o
.
F om
now
en,
we
assume
ha
his
condi ion holds
a
e e y poin
o
M
.
Then
G
becomes a
co e ing
o e
i s
image
G(M)
.
Le
U
C
G(M)
be a
simply-connec ed
p ope
domain,
hen
G
-1
(U)
is
a
disjoin
union
o
open
Vi,
i
E
I,
and
GI
S
.:
Vi
-
U
is
a
di eomo phism
which we
call
Gi
.
We
supply
S
N-1
wi h
he canonical
me ic
o
cu a u e
1
.
De ini ion
.
By
suppo
unc ion
hi
:
U-
.
R
we
mean
(
1 )

hi(n)
=
(Gi
1(n),n)
.
Lemma
1
.

The
unc ion
hi(n)
de e mines
G7
1
(n) in he
ollowing
way
:
(2)

G~
1
(n)
=
h
i
(n)n
+
g ad
h
i
(n)
,
whe e
g ad
hi(n)
is
compu ed
in
e ms
o he
me ic o
SN
-1
and
looked
a
as
a
ec o
in
RN
.
(3)

hi
(Gi(y))
=
(y,Gi(y))
.
I
X
E
T~,
:SN-
1
hen,
di e en ia ing
(3)
along
X
we
ob ain
o
P oo
..
Le
G~
1
(n)
=
x
.
Fo
yE
Vi
we
ha e
by
(1)
(g ad
h
i
(G
i
(x))
,
Gi
.
X)
=
(X,
G
i
(x))
+
(x,
Gi
.
X
)
Bu
(X,
Gi(x))
=
0
by
he
de ini ion o
he
Gauss map,
so
(4)

((x
-
g ad
hi(Gi(x)))
,
Gi*X)
=
0
.
By
he
nondegene acy
condi ion,
Gi
.
maps
isomo phically
T
x
M
on o
TG
(
x
)S
N-1
.
The
la e
space
coincides
wi h
TM
as a
subspace
o
HN
so,
o
some
i
E
R,
we
ha e
x
-
g ad
hi
(Gi
(x))
=
pCi
(x)
,
G~
1
(n)
=
pn
+
g ad
hi(n)
Taking
scala
p oduc
wi h
n and
accoun ing
(1)
and
(g ad
hi(n), n)
=
0
we
ob ain
(2)
.
MINIMAL
SURFACE
EQUATION

4
1
Lemma
2
.
Le
A(x)
be
he
second
undamen al
ope a o
in
T,,M
.
Then
(5)

A(x)
=
(hi(n)E+Hesshi(n))
-1
.
He e
n
=
Gj(x),
E
is
he
iden i y
ope a o
in
T,,S
N-1
=T
x
M
and
Hessh
i
(n)
is
he
Hessian
ope a o
[8]
on
he
sphe e
S
N-1
.
P ooL
Deno e
o
a
momen
Fi
=
Gi
1
on
U
.
By
(2)
we
ha e
Fi(n)
_
hi(n)n+
g ad
hi(n)
.
Le
Z
E
T
n
SN
-1
.
By
he
de ini ion
o
he
Hessian
ope a o ,
Hess hi(n)
(Z)
=
(Oz
g ad
hi)
(n),
whe e
0
is
he
Le i-Ci i a
connec ion
on
he sphe e
.
Fo any
ec o
ield
on
he
sphe e
we
ha e
[8]
Vz
=
z
+
( ,
Z)n,
whe e
¿
deno es usual
di e en ia ion
in
RN
.
So by
(2),
(Hess hi(n))(Z)
=F
i ,
Z
-
(g ad
hi(n),
Z)n
-
hi(n)Z
+
(g ad
hi(n),
Z)n
=
Fi
.
Z
-
hi
(n)
Z
,
o
Fi,
Z
=
h
i
(n)
Z
+
(Hess
h
i
(n))
(Z)
.
Le
Y
=
Fi,
Z,
hen
Z=
G,
Y
.
By
de ini ion
o
A,
A(x)Y
=G
i
,Y
=
Z,
which
p o es
(5)
.
Theo em
1
.
Suppose
N
=
3
and
M
is
minimal
.
Then
o
any
p ope
simply-connec ed
domain
U
C
G(M)
andany
b anch h
i
we
ha e
(6)

áh
i
+
2hi
=
0
,
whe e
A
is
he
Laplace-Bel ami
ope a o
on
he
sphe e
52
.
Con e sely,
i
h
is
a
solu ion
o
(6)
in
an
open
U
CS
2
hen
he
o mula
x(n)
=
h(n)n+g ad
h(n)
de e mines
a
smoo h
map
om
U
o
R3,
which
is
ei he
a
cons an
o
a
con o mal
and
minimal
imme sion
ou side
a
locally
cni e
se
o isola ed
singula i ies
(b anch
poin s)
.
P oo
. .
M
is
minimal
i
A(x)
=
0
e e ywhe e
.

Fo
an
in e ible
ope a o
A
in
2-space
we
ha e
A
-1
=

1, AA
so
by
(5),
A(x)
=
0
de
is
equi alen
o 0
=
(h
¡
(n)
E
+
Hess
h¡
(n»
=
2hi
+
Ah
i
.
This
p o es
he
i s
s a emen
o
he
heo em
.
Now
Suppose
h
yields
(6)
.
Deno e
F(n)
=
h(n)n
+
g ad
h
.
F om
he
p oo
o
he
Lemma
2
weknow
ha
F,
(n)
=
h(n)E
+
Hessh(n)
.

In
pa icula ,
i
neans
ha F,
(n)
maps
T,
S
2
o
i sel
and
is
symme ic
in
T
n
S
2
.

Fu he ,
by
(6),
F,
(n)
=
0
.
No e ha
any
symme ic
ope a o wi h he
ze o
ace
in
2-space
is
ep esen ed
by
a
ma ix

b
ba
)
in
any
o hono mal
basis
and
is
hus
con o mal,
so
F
is
con o mal,
and
o
n
E
U
ei he
ank
F,
(n)
=
2
o
F,
(n)
=
0
.

Deno e
by
Z
he
se
o
poin s
whe e F,
=
0
.

As
(6)
is
4 2

A
.
G
.
REZNIKOV
ellip ic,
F(n)
is
analy ic
along
wi h
h(n),
so
i
F
is
non-cons an ,
hen
Z
is
nowhe e
dense
.
Ou side
Z,
F(n)
ís
a
con o mal
ímme sion
and
we
ha e
jus
shown
ha
T
F
(

,)F(U)
=T

,
S
2
,
so, o
he
Gauss
map
we
ha e
G(F(n))
=
n
.
Hence
he
suppo
unc ion
h
o
F(U
-
Z)
is
h(n)
=
(h(n)n
+
g ad
h(n),
n)
=
h(n)
.
F om
he
i s
pa
o
he
heo em
we
see
ha
Flu-z
is
minimal
.
I
ollows
ha
F
is
ha monic
in
U
-
Z,
bu
F
is
analy ic
in
U
and
Z
is
nowhe e
dense,
hence
F
is
ha monic
e e ywhe e
in
U
.
Locally
in
con o mal
coo dina es
(x,
y)
we
ha e
n
E
Z
<=>
á

=
áF
=
0,
hence Y~(n)
=
0
whe e
z
=
x+
iy
and
.F
is
holomo phic
and
Re
.F
=
F,
so
Z
is
locally
ini e
.
Theo em
2
.
Suppose
N
=
4
and
M
is
minimal
.
Then
o
any p ope
simply-connec ed
U
C
G
(M) and
any
b anch
hi,
Ahi(n)
+
2hi(n)
doesn'
change
sign in
U
.
P oo
..
Le
Ai
(x),
1<_
i
<_
3,
be
he
p incipal
cu a u es
o
M
in
x
.
Then
I 1
+
A2
+
3
=
0
by minimali y
condi ion
.

Suppose
Ah
i
(n)
+
2hi
(n)
=
0
.

somewhe e
in
U
and
le
x
=
Gz
1
(n)
.

As
A
-1 (x)
_
a,
a2+aLa3+- 2-3
d<,,
A

we
ob ain
by
(5)
ha
A1l 2
+
113
+
1 21 3
=
0
.

This
'(x)
implies
1
+
; 2
+
a3
=
(
,
1
+
1 2
+
A3)
2
-
2(A1
, 2
+
1A3
+
2
,
3)
=
0
in
x which
is
impossible
by
nondegené acy
condi ion
.
We
u n o
applica ions
o
ou
esul
.
Le
M1,
M2
be
wo'
minimal
su aces
in
R3 such ha
G(M1)
n
G(M2)
has
a
nonemp y
in e io
in
S
2
.
In
[5]
and
[6]
hei
sum
M1
+
M2
is
de ined
by
pa ame iza ion
x(n)
=
Gi
1
(n)
+
G2
1
(n)
.
Le
h
2
~
1)
and
h~
2
)
be wo
b anches
o
suppo
unc ions
o
M1
and
M2
espec i ely
.
Then
by
(2)
we
ha e
x(n)
_
h(n)n+
g ad
h(n)
.
whe e
h
=
hW
+
h~
2
)
.
Nex ,
bo h
h
~
1)
and
h~
2
)
yield
(6)
which
is
linea ,
hence
h
yields
(6),
oo
.
Theo em
1
implies
hus
he
minimali y
o
M1+M2
.
Mo eo e ,
gi en
a
minimal
M
and
any
Killing
ec o
ield
Z
in
S
2
we
can
de ine
he
de i a i e
su ace
MZ
by
(
7
)

h(M')
=
h'
,
which
is
also
minimal
.
2
.
Applica ions
MINIMAL
SURFACE
EQUATION

43
Example
.
Le
M
l
be
a
ca enoid
de ined
in
euclidean
coo dina es
(x,
y,
z)
by
he
equa ion
x
2
+
y
2
=
(ch
z)
2
.
Then
di ec
compu a ions
show
ha
(8)

hl
(n)
=
h(n
x
,
n
y
,
ni)
=
1
-
nZ
a c an
h
nZ
,
and
G(M1)
=
S
2
-
{ p}
whe e
p
=
(0,0,1)
.
Le
g
i
E
SO(3),
i
=
2,
. .
.
,
m,
gi
=
id
be
o a ions
such
ha
he
se s
{ g
i
p}
a e
pai wise
disjoin
.
Le
h
=

hl
og
i
,
and
V
=S
2
-
U{ gip}
.
Then
we
ha e
he
i
ollowing
P oposi ion
1
.

The
su ace
0
:
n
~--+
h(n)n
+
g ad
h
is
a
comple e
minimal
su ace
in
R3
wi h only
a
ini e
numbe
o
b anch
poin s,
and
i s
Gauss image
omi s
p ecisely
2m-poin
se
U{ gip}
(compa e
[5])
.
i
P oo
.:
Fi s
we
no e
ha
in
some
neighbou hood
o
p,
V)
canno
ha e
b anch
poin s
.
Indeed,
suppose
de
(h(n)E
+
Hess
h(n))
=
0,
hen
o
some
ec o
X
E
T

S?,
IIXII
=
1,
(hi(n)E+
Hesshi(n))X
=
-
E7
.>2
(hi(n)E+
Hess
hi(n))
X
:
We
know
ha
o
i >_
2
hi(n)
does
no
ha e
singula i ies
nea
p,
so in
some
neighbou hood
o
p
and
o
some
cons an
C
we
would
ha e
11
(hi(n)E+Hessh
i
(n))XII
<
CIIXII
.
Deno e
(hi
(n)E+
Hess
h
i
(n))
X=Y,
hen
11
(hi
(n)E+Hess
h
i
(n))
-1
Y11
>
C-1
IIYII
.
Ac ually
(hi
(n)E
+
Hess
h,
(n»
-
'
is
he
second
undamen al
ope a o
o
he
ca enoid
M
i ,
as
we
saw
in
Lemma
2,
he e o e,
i s
eigen-
alues a e
±
-K(n),
whe e
K(n)
is
he
Gaussian
cu a u e
a
G
-1
(n)
.
As
i is
well-known
(and easy
o
e i y)
ha
K
is
decaying
o
ze o
a
in ini y,
he
abo e
inequali y
is
impossible
.
O
cou se,
he
same
is
ue
abou
all
he
singula
poin s
gip,
hence,
being
locally
ini e,
he
se
o
b anch
poin s
should be
ini e
.
Nex ,
as
he
me ic
o
ca enoid
is
comple e,
we
ha e
7
~~
(hi(n)E
+
Hess
hi
(n))
y( )jj
=
co
o
any
cu e
y
:
[0,
oc)
-
S
2
such
lia

lim
y( )
=
p
.

Hence
he
same
a gumen s
ac
show
ha
his
is
ue
o
h
ins ead
o
h,
and
inally,
V) is
comple e
.
Now
conside
he
Ennepe
su ace
[7]
e
:
R2
_
R3
.
The
composi-
ion
G
o
e
wi h
he
Gauss
map
coincides
wi h
he
in e se
s e eog aphic
p ojec ion
7 -
1
:
R2
_
S
2
-
{p}, so
K
:~
0
and
he
suppo
unc-
ion
h
is
de ined
in
S
2
_
{p}
.
S aigh o wa d
compu a ions
show
ha
K
-
-0
on
E,
hence
we
can
apply
he
same
cons uc ion
o
ob ain

44

A
.G
.
R ZNIKOV
P oposi ion
2
.
Fo
any
gi en
ini e se
E
C
S
2 he e
exis s
a
com-
ple e
minimal
su ace
in
R3
wi h only
a,
ini e
numbe
o
b anch
poin s
whose Gauss image
omi s
p ecisely he
se
E
.
The
conjec u e
o
Meeks
[4]
s a es
ha
o
e e y
k
>
1
he e
exis s
an
embedded
minimal
su ace
homeomo phic
o
a
compac
mani old
punc-
u ed
in
k
poin s
.
The
p oblem
o
Osse man
[7],
sol ed
by
Fujimo o
[4],
asks
whe he
he
s a emen
o
ou
P oposi ion
2
holds
o
some
smoo h
comple e
minimal
su ace
(wi hou
b anch
poin s)
.
Concluding
Rema ks
.
1
.
Ou
main
equa ion
(6)
admi s
sepa a ion
o
a iables
.
Fix
xo
E
S
2
,
hen
in
sphe ical
pola
coo dina es
nea
( ,
cp)
nea xo
he
sphe ical
me ic
becomes
d
2
+
sin
2
dcp
2
and
he
Laplace-
Bel ami
ope a o
becomes
A
=
+
s°ñ
+
S
-,
1
~,
.
~
.

Sub
s i u ing
( , cp)
=

sin
g( ,
w)
we
ob ain
ha
A
+
2
=
0
is
equi alen
o
g'
.
+g
(4

+
2
)
+
5

=
0
.
By
Fou ie
me hods
one
inds
whe e
C,,( )
sa is y
Cm
( )e"I'
1
9
)
_
sin
2
+
4

0
.
2
.
A
a he
su p ising
phenomenon
ollows
om
ou
desc ip ion
.
Namely,
i
Oh
+
2h
=
0
in
an
open
U
C
S
2
,
hen
he
"Monge-
Ampe e"
0
=
de (hE+Hessh)
sa is ies
some
second
o de
PDE
.
Indeed,
we
know
om
Theo em
1
ha
he
su ace
M
pa ame ized
by F(n)
:
n
H
h(n)n,
+
g ad
h(n)
is
minimal
and
i s
cu a u e
a
F(n)
is V)
-1
(n)
.
Le 's pull
back
on
S2
he
minimal
su ace's
me -
ic
.
We
will
ob ain
g
=
e(n)go,
because
he
Gauss
nap
F
-1 (n)
is
con o mal
(he e go
is
he
sphe ical
me ic)
.
To
compu e
p(n),
,
, e
no e
ha
F*
ds
=
±O(n)
dso,
whe e
ds,
dso
a e
he
a ca
ele-
men s on
M,S
2
espec i ely
.
Hence,
g
=
10(n)Igo
.
The e o e, he
cu a u e
o
-V)(n)go
is
0
-1
(n)
(compa e
wi h
Rice¡-Cu bas o
Theo em,
[10])
.
This
is
equi alen
o
some
PDE
.
3
.
Suppose
M
is
a
comple e
minimal
su ace
o
ini e
o al
cu a u e
.
Then
by
he
heo em
o
Osse man
[7]
he
Gauss
mapG
_
:
M
S
2
ex ends
o
a
holomo phic
map
G
o
he
comple ion
M,
and
MINIMAL
SUR ACE
EQUATION

45
M
-
M
is
ini e,
say
M
-
M
=
{p
l
, . . .
p
z}
.

Le
G C
M
be
he
ini e se
_
o
b anch
poin s_

o
G,
say
G={q1
. . .
qk}
.

Then
N
=
M
-
((M
-
M)
U£)
(is
he
ini e
co e ing
o
S
2
-
G((M
-
M)
U
G)
.)
Nex ,
he
suppo
unc ion
h(n)
becomes
single- alued
on
N
and
we
see
ha
e e y
comple e
minimal
su ace
o
ini e
o al
cu a u e
de e mines
a
solu ion
o he
equa ion
Oh+
2h
=
0
in
a
ini e
co e ing
o he
s anda d
sphe e
punc u ed
in
a
ini e
numbe
o
poin s
.
4
.
Conside
he
ia
me ic
g
=
dx
2
+dy
2
-dz
2
in
R2,1
.
I
o
a
su ace
M
C
R
2,1
,
g
¡
M
is
posi i ely de ined,
hen
he e
exis s
a
co ec ly
de ined
Gauss
map
G
om
M
o
he
hype boloid
S
:
x2
+
y
2
-
z
2
=
-1
.
I
is
well-known
ha
g
1
S
is
he
s anda d
hype bolic
me ic
.
Jus
as
be o e
we can
de ine
a
suppo
unc ion
h(n)
.
Fo mula
(2)
in
his
case
eads
G~
1
(n)
=
-hi(n)
+
g adhi(n)
.
Fo mula
5
becomes
A
(x)
=
(-hi(n)E
+
Hess
hi(n))-1
and
(6)
becomes Ohi(n) -
2hi(n)
=
0 o
minimal
su aces
M
wi h
ime-
like
no mals
.
Re e ences
1
.

COSTA
C
.,
Bull
.
Soc
.
B as
.
Ma h
.
1
5
(1984),
47-54
.
2
.

Ho MAN
D
.,
MEEKS
W
.
III,
Bull
.
Ame
.
Ma h
.
Soc
.
1
2
(1985),
134-136
.
3
.
HO FMAN
D
.,
MEEKS
W
.
III,
J
.
Di e en ial
Geom
.
21
(1985),
109-127
.
4
.

FUnMOTo
H
.,
J
.
Ma h
.
Soc
.
Japan
40
(1988),
237-249
.
5
.
MEEKS
W
.
III,
"P os
.
In
.
Con,g
.
Ma h
.
Be keley
1986,"
1987,
pp
.
551--558
.
6
.
ROSENBERG
H
.
TOUBIANA
E
.,
J
.
Di e en ial
Geom
.
28
(1988),
115-132
.
7
.

LANGEVIN
R
.,
ROSENBERG
H
.,
Duke
Ma h
.
J
.
57
(1988),
819-828
.
8
.

LANGEVIN
R
.,
LEVITT
G
.,
ROSENBERG
H
.,
Banach
cen e
publ
.
20
(1987)
.
9
.

OSSERMAN
R
.,
"A
Su ey
o
Minimal
Su aces,"
Van
Nos and
Reinhold
Company,
New
Yo k,
Cincinna i,
To on o,
London,
Mel-
bou ne,
1969
.
46

A
.G
.
REZNIKOV
10
.
GROMOLL
D
.,
KIANGCNBGRG
W
.,
MGYGR
W
.,
"Riemannische
Ge-
ome ie
in'G o len,"
Lec u e
No es
in
Ma h,
55,
Sp inge ,
Be lin,
Heidelbe g,
New
Yo k,
1968
.
11
.
LAWSON,
H
.B
.,
JR
.,
"Lec u es
on
Minimal
Submani olds,"
Publish
o
Pe ish,
1980
.
Pe manen
add ess
:
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o
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Sciences
Raymond
and
Be e ly
Sackle
I'acul y
o
Bxac
Sciences
Tel
A i
Uni e si y
Ra na -A i ,
Tel
A i
69978
ISRAEL
Cu en
add ess
:
In e na ional
Cen e
o
Theo e ical
Physics
P
.O .B
.
586
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34100
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