CURVATURE 5 0. ABSTRACT. Clored minimal surfaces in a unit m-sphere SnL with Gauss curvature K p 0 are considered. 1. Introduction. Recently, S. S. Chern ...
PROCEEDINGS O F THE AiMERICAN MATHEMATICAL SOCIETY Volume 31, No. 1, January 1972
MINIMAL SURFACES IN S n W I T H GAUSS
CURVATURE 5 0
ABSTRACT. Clored minimal surfaces in a unit m-sphere SnL with Gauss curvature K p 0 are considered.
1. Introduction. Recently, S. S. Chern, M. do Carmo, and S. Kobayashi [2] studied the n-dimensional submanifolds of a unit rn-sphere S m with scalar curvature 2 n(n - 1) - n(m - n)/(2m - 2n - 1). In particular, they proved that the only closed minimal surfaces of S m with Gauss curvature K 2 (2m - 6)/(2m - 5) are the following surfaces: (i) equatorial sphere of S3, (ii) Clifford torus in S3,and (iii) Veronese surface in S4. The main purpose of this paper is to study the closed minimal surfaces of S m with Gauss curvature K 0. 2. Preliminarie~.~Let M be a surface in a unit m-sphere Sm. We choose a local field of orthonormal frames el, . . . , em in S m such that, restricted to M, the vectors el, e, are tangent to M (and, consequently, e 3 , . . . , e, are normal to M). With respect to the frame field of Sn" chosen above, let w,, . . , w , be the field of dual frames. Then the structure equations of S m are given by
We restrict these forms to M. Then
Since 0 = dm, = wl (4)
cot, = h;ul
+ o,A o,,,by Cartan's lemma we may write + hc2w2, hij = h:%, i, j = I , 2.
w,,
Received by the editors February 8, 1971.
ALMS 1970 subject c / a s s ~ ~ c a t i o tPrimary ~ s . 53A10, 53A05; Secondary 53C40. Key words and phrases. Minimal surfaces, Gauss curvature, flat surfacer, Clifford torus, minimal direction, Lipschitz-Killing curvature. This work was supported in part by NSF Grant GU-2648 Manifolds, mappings, functions, and other geometric objects are assumed to be differentiable and of class C". @Amellcan Mathematical Soclety 1972
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BANG-YEN CHEN
From these we obtain
(7) Put
dmiT =
Z:
Wij
A
WjT
+2
Wi,
A
W,,.
Then H is a well-defined normal vector field over M , and is called the mean curvature vector of M in Sn: If H = 0 identically on M , then M is called a minimal surface of S m .The Gauss curvature K of M is given by
Let e = C;L3 cos OreTbe a unit normal vector at p ; then the LipschitzKilling curvature G ( p , e) with respect to e is given by (10)
G(P, e) =
( F cos OTh:L)(4cos %,h:,) - (f cos O,h:,)l
Let V' be the covariant differentiation on S m ,and rj be a normal vector field over M in Sm.If the covariant differentiation V'rj has no normal component, then rj is said to be parallel in the normal bundle. A unit normal vector field P over M is called a minimal direction if the Lipschitz-Killing curvature with respect to e is minimal at every pointp E M, i.e. G ( p , 6) = min { G ( p ,e ) : e unit normal vector at p ) , for all p E M. THEOREM 1. Let M be a closed minimal surface of a unit m-sphere S m with Gauss curvature K 5 0. If there exists a unit normal vector jield 6 otler M such that d is parallel in the normal bundle and the LipschitzKilling curvature with respect to 2, G ( p , e ) , is nowhere zero, then M is a Clzfford torus in a unit 3-sphere S 3 c Sm. THEOREM 2. Let M be a closed minimal surface of a unit m-sphere 0. If there exists a minimal direction which is with Gauss curvature K parallel in the normal bundle, then M is a Clifford torus in a unit 3-sphere S 3 C sm. From Theorem 2 and the result of Chern-doCarmo-Kobayashi, we obtain
19721
MINIMAL SURFACES IN
sm
WITH GAUSS CURVATURE 5
o
237
COROLLARY( [ I ] ,[4]). Let M be a closed minimal surface of S3. If the Gauss curvature of M does not change its sign, then M is either an equatorial sphere or a Clzfford torus. 3. Proof of Theorem 1. Suppose that M is a closed minimal surface of a unit m-sphere S m with Gauss curvature K 5 0. If there exists a unit normal vector field 2 over M such that d is parallel in the normal bundle and the Lipschitz-Killing curvature with respect to E is nowhere zero. We consider only the orthonormal frames (p, el, e,, e,, . . . , em) in B such that em = e and el, e2 are in the principal directions of em. Since M is minimal in S m , the principal curvatures k,, k 2 in the direction of e,, are given in the forms: (11) k , = h , and k , = - h .
Since the Lipschitz-Killing curvature G ( p , em) = -h2 # 0 is defined
globally on M, we see that h is defined globally on M. Without loss of generality, we may assume that h > 0 on M . Then we have
(12) wlm = hol and o,, = -ho,.
By taking exterior derivatives of (12) and applying (5) and (7), we obtain
2h do), dh A ol = 2 olTA or,, (13) 2h doj2 dh A w, = - 2 co,, A co,,. Since em = E is parallel in the normal bundle, we have or, = 0. Thus ( 1 3) reduces to 2h do), dh A w, = 0 , 2h do), + dh A o, = 0. (14) From (14) we can consider local coordinates ( u , v) in an open neighborhood U of a point p E M such that ds2 = E du2 G du2, w1 = ElJ2du, co2 = G1J2d v, (15) where ds2 is the first fundamental form and E and G are local positive functions on U. From (15), equation (14) becomes d(hE) A du = 0 , d(hG) A dv = 0 , (16) which shows that (hE) is a function of u, and (hG) is a function of v. By making the following coordinates transformation:
+ +
+
+
(17)
S
U' = ( h ~ ) ~du, "
U'
S
= ( h ~ ) "dv, ~
we see, from (15), that there exists a neighborhood Vof each pointp in M such that there exist isothermal coordinates ( u , v) in V such that
238
BANG-YEN CHEN
where f = f (u, v) is a positive function defined on V. It is well known that the Gauss curvature K is given by
with respect to the isothermal coordinates (u, v). Hence, the condition K 5 0 with hf = 1 implies A log (h) = -A log (f) 0. From Hopf's lemma, we see that log (h) is a constant on M. Hence, the Gauss curvature K satisfies K = (- 1/2f)A log (f) = (h/2)A log ( h ) = 0, identically on M. This implies that M is a closed flat minimal surface in Sn
