sion one in the unit sphere and calculate their principal curvatures. In. Section 5, we ..... regular R-space N(H) is a minimal hypersurface in the unit sphere S in p.
Tdhoku Math. Journ. 36 (1984), 253-267.
HOMOGENEOUS MINIMAL HYPERSURFACES IN THE UNIT SPHERES AND THE FIRST EIGENVALUES OF THEIR LAPLACIAN HIDEO MUTO, YOSHIHIRO OHNITA AND HAJIME URAKAWA (Received June 11, 1983)
1. Introduction. The geometry of minimal closed submanifolds of the unit sphere is closely related to the eigenvalue problem of the Laplacian. In this paper, we study the first eigenvalue of the embedded minimal hypeasurfaces in the unit sphere. Let (ikf, g) be an w-dimensional compact connected Riemannian manifold without boundary and A its (non-negative) Laplacian acting on C°°functions on ikf. Let {0 = Xo < Xx M is a Covariation ft whose variation vector field Vx = (d/dt)ft(x) \t=0, xeM, belongs to F(TML). It is known (cf. [11]) that for every normal variation ft of the minimal immersion f:M-*M with the variation vector field V e r(TML), we have (d2/^2)(vol(M,/*ft))|t=0 - \ h(JV, V)dvg where dvg is the volume element of (M, g). Here the operator J of F(TML) into itself, called the Jacobi operator, is a self-ad joint strongly elliptic differential operator. So it has a discrete spectrum: The index of the minimal immersion f:M^M is the sum of the dimensions of the eigenspaces corresponding to the negative eigenvalues. The nullity of f:M—>M is the dimension of the O-eigenspace. The Jacobi operator J of F(TML) is determined as follows: Define the operators J\ R and A of F(TML) into itself by
and
=
±B(Avei,ei),
for evey Fe/XTM 1 ), respectively. Here {eJJU is an orthonormal local
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frame field of M, and R is the curvature tensor of (M, h). Then we have (cf. [11]) J = -A1 + R - A . In the following, we always assume that (M, h) is the (n + ^-dimensional unit sphere Sn+\1) with constant sectional curvature one. Let (M, g) be an ^-dimensional compact orientable Riemannian manifold isometrically and minimally immersed into Sn+1(l). Then we have R(V) = ± (h(eif V)ei - h(eif e%)VY = -nV , i
for every V e F{TML). Let N be the unit normal vector field on M. Then each element in F^TM1) can be written as a multiple of N by a C°°-function on M. Let C°°(M) be the space of all real valued C00functions on M. Then, since F^N = 0 , i = 1, • • •, n, we have - A\(j>N) = (A, (p, V>) 6 D(K, L)} , is a complete orthogonal system of the space C?(K/L) of all C°°-complex valued functions of K/L with respect to the following inner product: (5.1)
((&, &)) = (
h(xL)$&L)dv,
, ft, & e C%(K/L) .
JK/L
Here dvg is the volume element of (K/L, g) and (5.2)
pij(x) = ((p(x)vj,vi)) , xeK,
l^j^dimVpL,
1 ^ i ^ dimVp
are regarded as functions on K/L since p(V)Vj = vjf le L, implies pi5(xl) = piS(x) for all xeK, leL. We identify piS with 7c*pi:}. Now since the Laplacian A of (.K/L, g) is expressed in terms of the Lie algebra ! (cf. Proposition 3.1), we have (APij)(x) = ((p(x)p(D)vjf vt)) , 1 ^ j ^ dim VpL , 1 ^ i £ dim F ' . Here p(D) is the endomorphism of Vp given by (5.2) where {-3T;,4; X.e ^+, i = 1, • • •, m(X)} is as in Section 3 and p(X), Xe I, is the infinitesimal representation of p on Vp. We note that p(D) leaves VPL invariant, since {^(H^Xx^; Xe 2+, i = 1, • • •, m(X)} is an orthonormal basis of m with respect to the Ad (L)-invariant inner product < , > on m induced by the invariant Riemannian metric g on K/L. Therefore if we find all the eigenvalues of the endomorphism p(D) on the finite dimensional space VPL for each (p, Vp) e D(K, L), then these eigenvalues exhaust all the eigenvalues of the Laplacian A of (K/L, g). Except in the case SO (3)/Z2 x Z2f however, it is very difficult to find all the eigenvalues of p(D). In the remaining cases, we will determine only the first eigenvalue \ of the Laplacian A of (K/L, g) by estimating all the eigenvalues of p(D). S^^jOase of SO(3)/Z2 x Z2. In this case, let (u,!) = (8u(3), 8o(3)), p = {V^lX\ X 3 x 3-real matrices, X - lX = 0, tr X = 0}, and (U,K) = (SU(3), SO (3)). Here lX (resp. t r X ) is the transpose (resp. the trace) of X. Then the Ad ( i n a c t i o n on p is given by SO (3) x p 9 (k, X) ^ kXk-'ep. Pat a = {V-\ diag (yly y2, y,); yte R, SJ-i Vt = °1- H e r e w e d e n o t e b y d i a g (yu y2, y3) t h e d i a g o n a l 3 x 3 m a t r i x w h o s e d i a g o n a l e n t r i e s
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are y19 y2 and y3. Put /0 - 1 0\ 0 0 , \0 0 0/ and H = j / ^ 1 diag ( - 1 , 1, 0) e a. Let ( , ) be the Ad (?7)-invariant inner product on 8u(3) defined by (X, F) = - 2 " 1 t r ( X r ) , X, TG§U(3). Then (H, H) = 1. Let ^ be the linear form on a defined by a 3 V — 1 diag (ylf V2, Vz) -> 1/i- Then we have 2 = I7* = {±(fa - ft); 1 ^ i < j ^ 3}. Fix a lexicographic order > on I so that fa> ft> fa- Then 2^ = {fit — ft; 1 ^ i < j S 3} and p = 3. Put c = l/i/"3" and J?o = V 7 ^ ! diag ( — 2c, c, c) , i?2 = l/" 11 ! diag (c, -2c, c) . Let Xo = fi2 — ^ 3 , X1 = fi± — ftz, \ ,) = 0 ,
0 ^ i ^ 2 ,
ff, = V^l
= fa — fts.
and
diag (c, c, -2c) ,
Then we h a v e
0O = TT/6 ,
^ = TT/2 ,
52 = 57T/6 .
Thus by Proposition 4.1, the Ad(JST) orbit through H is minimal in the 4-dimensional unit sphere S = {Xe p; (X, X) = 1}. The stabilizer L of K at H under the Ad (K) action is a finite subgroup of SO (3) consisting of diag (1,1, 1) , diag ( - 1 , 1 , - 1 ) , diag ( - 1 , - 1 , 1 ) and diag(l, - 1 , - 1 ) . Thus the homogeneous space K/L is just SO(3)/Z2 x Z2, ! = tn and I = {0}. We note that {2~1X1, X2, XJ is an orthonormal basis of 8o(3) with respect to the inner product < , > given by
