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Minimal surfaces with constant Gauss curvature

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Bang-Yen Chen at Michigan State University
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Minimal surfaces with constant Gauss curvature in real space forms are studied.
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume
34,
Number 2, August 1972
MINIMAL SURFACES WITH CONSTANT
GAUSS CURVATURE
ABSTRACT.
Minimal surfaces with constant Gauss curvature
in real space forms are studied.
1.
Introduction.
Let
RTn(c)
be an m-dimensional complete, simply-
connected Riemannian manifold of constant curvature
c
with covariant
differentiation
V,
and let
M
be a surface immersed in
Rm(c).
For a normal
vector field
q
of
M
in
Rm(c),
we denote by
V*q
the normal component
of
Vq.
Then
V*
defines a connection in the normal bundle.
A
normal
vector field
q
is said to be parallel in the normal bundle if
V*q=O
and
to be
constant
if
Vq
=O.
MAIN
THEOREM.
Let
M
be a minimal surface with constant Gauss
curvature in a space form Rm(c) of curvature c.
If
there exists a nonconstant
normal vector field which is parallel in the normal bundle, then c>O and
M
is an open piece of the Clzjford torus.
Since every unit normal vector field is parallel in the normal bundle for
m=
3
and is constant only if
M
is totally geodesic, we have the following
corollary.
COROLLARY
If
M
is a minimal surface with constant Gauss curvature
1.
in a 3-space form R3(c), then either
M
is totally geodesic or c>O and
M
is
an open piece of the Clzjford torus.
REMARK
1.
Corollary
1
was obtained by Lawson
[4]
for
c>O.
If
c=O,
then Corollary
1
also follows from the well-known Ricci condition.
REMARK
2.
Suppose
RnL(c)
is the euclidean m-space
Em.
Then by
combining a result of Osserman-Chern
[5]
for the generalized Gauss map
and a result of Calabi
[I]
for Riemann surfaces in the complex projective
spaces, we know that
every minimal surface in the euclidean m-space with
constant Gauss curvature must be totally geodesic.
Received by the editors October
1,
1971.
AMS
1970
subject classifications.
Primary 53A10, 53B25; Secondary 53C40.
Key words andphrases.
Minimal surface, Gauss curvature, scalar normal curvature,
Clifford torus.
This work was partially supported by the National Science Foundation under
grant GU-2648.
@
American Mathematical Society 1972
REMARK3. For the minimal surfaces with positive constant Gauss
curvature in the m-sphere, see do Carmo-Wallach
[Z].
The author would like to express his thanks to Professor
H.
B.
Lawson
for the valuable conversations.
2.
Preliminaries.
Let R1"(c) be a complete, simply-connected Rieman-
nian manifold of constant sectional curvature c, with scalar product
(
,
),
and let
M
be a surface immersed in Rin(c). We choose a local field
of orthonormal frames
el,
.
.
.
,
em
in RH1(c) such that, restricted to M,
the vectors
el, e,
are tangent to
M
(and, consequently,
e,,
.
. .
,
em
are
normal to M). With respect to the frame field chosen above, let
col,
. . .
,
wm
be the field of dual frame. Then the structure equations of R1"(c) are given
by
(1)
dcoA
=
-2
a;:
A
coU,
cog
+
=
0,
A,B,C;..=
I,...,
m.
(2)
dco;
=
-2
w$
A
w:
+
ccoA
A
coU,
We restrict these forms to M. Then
oT=O,
for
r,
s,
t,
.
. .
=3,
. . .
,
m.
Since
O=ctwT=co:~co1+co,"hw2,
by Cartan's lemma we may write
From these we obtain
(6)
dco;
=
-
2
coy
A
w:
-
2
coiA
wt.
The second fundamental form
h,
the mean curvature vector
H,
the
Gauss curvature
K
and the scalar normal curvature
K,.
are given by
respectively.
A
surface M is minimal if
H=O
identically on M.
506
BANG-YEN
CHEN
[AU~US~
3. Proof of the
Main
Theorem.
We first prove the following lemmas:
LEMMA
1.
Let
M
be a minimal surface of Rn'(c). If there exists a unit
normal vector jield
e
which is parallel in the normal bundle, then
M=
M1uM2,
where
M,
is an open subset of
M
with
Kay=O
and
M,=
{p
E
M: (h(X, Y), e)=0
for all tangent vectors
X,
Y
in
T,(M)).
PROOF.
Let
MI=
{p
E
M: 3X, Y
E
T,(M)
such that
(h(X,
Y),
e)
#O).
Then
Ml
is an open subset of
M.
We choose our frame field in such a way
that
on
M,.
Then we have
everywhere on
MI.
Since
e
is parallel in the normal bundle, we have
By taking exterior differentiation of
(13)
and applying
(4), (6)
and
(13),
we obtain
(14) h;,h?, 0,
for
r
=
4,
.
.
.
,
m.
=
From
(lo), (12)
and
(14),
we see that the scalar normal curvature
KAv=O
identically on
M,.
This proves Lemma
1.
LEMMA
2.
If
M
is a minimal surface of Rn"c) ~t'itli constant Gauss
curvature and zero scalar normal curvature, then
M
is either Jut and
c2_0
or totally geodesic.
PROOF.
Since the scalar normal curvature
KAV=O
identically on
M,
we have
IEl(h;ih",---hsihl,i)=~
for
r,
s=3,
. .
.
,
m.
Thus, we may choose
our local frame field in such a way that
Therefore we obtain
(16)
co;
=
hIlco1,
og
=
-hi1o2.
Taking exterior differentiation of the first equation of
(16),
we have
Multiplying
(17)
by
hi,
and summing up on
r,
we obtain
19721
MINIMAL
SURFACES
WITH
CONSTANT
GAUSS
CURVATURE
507
Since mi=
-a):,
this equation implies
On the other hand, the constancy of the Gauss curvature implies
Therefore, combining
(18)
and (19), we obtain
Similarly, by taking exterior differentiation of the second equation of
(16),
we obtain
Put V={p
E
M:dwl#O or dw2#0}. Then V is an open subset of
M.
If V=
Q;
,
then dw1=do2=0 identically on M. Hence, by
(4),
(5)
and (9),
we see that K=O on M. Now, suppose that V#
D,
and V, is a component
of V. Then we have
on Vl. This implies that V, is totally geodesic in Rn'(c). If c=O, then K=O
identically on
M.
Now assume that c#O. Then V, is of constant curvature
c. This implies that closure(V)=M. Thus
M
is totally geodesic. This
proves Lemma
2.
LEMMA
3. Let
M
be a surface in a 3-dimensional totally geodesic
submar~ifold R3(c) of Rm(c) n*ith unit normal vector
Z
in R3(c).
If
e is a
unit normal vector jeld of M in Rn"c) ~i~hiclzis parallel in tlze normal
bundle, then tlzere exist a constant unit normal vectorjeld a in Rn'(c) and
a
constant angle
K
suclz tlzat e=(cos ~)Z+(sin ~)a.
This lemma can be proved immediately.
Now, we return to the proof of the Main Theorem. Let M be a minimal
surface of
a
space form Rnl(c) with constant Gauss curvature
K.
If there
exists a nonconstant unit normal vector field e which is parallel in the
normal bundle, then by Lemma 1 we see that M= M, UM,, where MI
is open with scalar normal curvature KAY=O and Mz=M-MI=
{p
E
M:
(h(X,
Y),
e)=O for all tangent vectors
X,
Y
at p). Since e is non-
constant, M, is nonempty. For each
p
E
MI, there exist tangent vectors
X,
Y
at
p
such that (h(X,
Y),
e)#O. Hence by Lemma
2
we see that
M
is flat and not totally geodesic in RV'(c). Therefore by (9) we obtain c>O,
i.e., R1'"c) is an m-sphere Sn'(c). Now let
U
be a component of M. Then,
508
BANG-YEN CHEN
by
K=KN=O,
U
is an open piece of a Clifford torus in a great 3-sphere
S3(c)
[3].
Let
P
denote the unit normal vector field of
U
in
S3(c)
and
el, e,
be in the principal directions of
2.
Then we can find that the second
fundamental form in the direction of
Z
is given by
Therefore by Lemma 3 we see that the second fundamental form in the
direction of
e
is given by
((h(ei, ej),
e>)
=
-
COS
ci
where cos
ci
is a nonzero constant. Therefore, by the continuity of the
second fundamental form
h
on
M
and the definition of
M,,
we see that
M,=M.
This implies that
M
is an open piece of a Clifford torus in a
3-sphere
S3(c)c Sm(c).
This completes the proof of the theorem.
1.
E. Calabi,
Isometric imbedding of complex manifolds,
Ann. of Math. (2)
58
(1953),
1-23. MR
15,
160.
2.
M.
P.
do Carmo and
N.
R. Wallach,
Minimal immersions of spheres into spheres,
Ann. of Math. (2)
93
(1971), 43-62.
3.
K. Kenmotsu,
Some remarks on minimal submanifolds,
TBhoku Math.
J.
(2)
22
(1970), 240-248. MR
42
#3688.
4.
H.
B.
Lawson, Jr.,
Local rigidity theorems for minimal hypersurfaces,
Ann.
of
Math.
(2)
89
(1969), 187-197. MR
38
#6505.
5.
R. Osserman and S. S. Chern,
Complete minimal surfaces in euclidean n-space,
J.
Analyse Math.
19
(1967), 15-34. MR
37
#2103.
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