Sim Be a Minimal Immersion of an N-Dimensional, N > 2, C' Manifold Ml

MINIMAL IMMERSIONS OF SPHERES INTO SPHERES BY MANFREDO P. DO CARMO* AND NOLAN R. WALLACH

I.M.P.A., RIO DE JANEIRO, BRAZIL; AND UNIVERSITY OF CALIFORNIA, BERKELEY Communicated by S. S. Chern, April 28, 1969 Abstract.-In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the rn-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is known that for n = 2 each admissible r determines a unique element of the above class. The main result here is that for each n > 3 and each admissible r > V8 there exists a continuum of distinct such submanifolds.

Let Slm c Rm+1 be the unit sphere in the euclidean space Rm+l, and let x: Mn Sim be a minimal immersion of an n-dimensional, n > 2, C' manifold Ml into Sim. It is well known that the condition of minimality gives an elliptic differential system for x; it is then reasonable to expect that some classification of such immersions can be obtained under topological and metrical restrictions on M. We shall consider the simplest situation where the induced metric has constant sectional curvature and Mn is itself a topological sphere. For the general facts on minimal immersions into riemannian manifolds, we refer to references 2 and 5. It is convenient to express our results in terms of the group actions naturally involved. Let G be a compact Lie group and K c G be a closed subgroup. An irreducible G-module W is called a class-one representation of the pair (GK) if there exists w e W, w $ 0, with kw = w, for all k e K. From now on, let G = SO(n + 1), K = SO(n). It is possible to associate uniquely to each positive integer s a class-one representation Vs of (G,K) and to endow V' with a G- invariant inner product (,). Let v e V', (v,v) = 1, be such that Kv = v, and denote by S, c V8 the unit sphere of VP. The following is easy to prove. PROPOSITioN 1. The map yns: GIK - S C Vs given by y,,,n(gK) = gv GI is a minimal immersion of G/K into S, with induced metric of constant sectional curvature k(s), where k() = n s(s + n -1) The dimension m(s) of S, is given by

m(s) = (2s + n- 1) )s!n- -1)

The immersions Yn,8 = y described in Proposition 1 will be called the standard minimal immersions. We say that an immersion x: M -- Sm c R"+1 is full if x(M) is not contained in a hyperplane of Rm+" and that two immersions are equivalent if they differ by an isometry of Sim. In what follows, x: (G/K)k 640 Downloaded by guest on September 26, 2021 VOL. 63, 1969 MATHEMATICS: DO CARMO AND WALLACH 641

SI' c RI+' is a minimal immersion of the n-sphere SO(n + 1)/SO(n) into the unit sphere Si, such that the induced metric has constant curvature k. PROPosiTION 2. Let x: (G/K)k Si' be as above. Then k = k(s), for some s, and the maximum of the dimensions l for which the above immersions are full is mr(s). Furthermore, there exists a symmetric positive semidefinite linear map A: V8 V- , such that x is equivalent to Aye,,. The above proposition gives some hope of obtaining a sort of "rigidity theorem," i.e., that a given x, as above, be equivalent to some standard minimal immersion. This has actually been proved for n = 2 (refs. 1 and 3). Here we announce that for an arbitrary n and small s (s = 1,2,3) this still holds, and that for n > 3 and s > 4, this is false. More specifically: THEOREM 1. Let x: (G/K)k _ 0SI be as above. By Proposition 2, k = k(s). Assume that x is full and s = 1, 2, or 3. Then x is equivalent toy,,,,; in particular, = m(s). THEOREM 2. Let x: (G/K)k -1 S1i be as above. By Proposition 2, k = k(s). Assume that x is full, s > 4, and n > 3. Then the set of equivalence classes of such immersions can be smoothly parametrized by a compact convex body L C W2 in a vector space W2, with dim W2 = N(n;s) > 18. The interior points of L correspond to those minimal immersions x for which 1 = m(s); in particular, any two such immersions can be deformed into each other by a homotopy of minimal immersions of the same type. The boundary points of L correspond to those minimal immersions for which 1 < m(s). Remark: Differentiation along the parameter gives Jacobi fields of the immersion x, which are orthogonal to the space of normal components of Killing fields in So. The crucial point in Theorem 2 is to obtain an estimate for dim W2 which is realized as a G-submodule of the symmetric square of V', S2(VS). This is done in several steps, the first one being to decompose the normal bundle of a standard immersion yn,,, (which is equivariant) in its osculating bundles of various orders. The interesting point is that, for the case of the sphere SO(n + 1)/SO(n), the corresponding decomposition of V8 at each point of y(G/K) can be described in terms of class one representations Vf,i = 1, .. ,s, of (SO(n),SO(n - 1)). This and the Frobenius reciprocity theorem are then used to show that W2 v W3, where W3 is the sum of all G-submodules in S2(V2), not containing Vo or V2. It turns out that dim W3 < N(n;s) is computable and increases with n and s; the number 18 in the statement is a lower bound for all n > 3 and s > 4. Complete proofs will appear elsewhere. From our analysis of the standard immersions, we found a minimal immersion (described below) of the unit sphere bundle of each sphere; for n = 2, this is equivalent to an example given by Hsiang4 of a minimal immersion of S3 into S4. PROPOSITION 3. Let y: G/K -- SI C V8 be a standard minimal immersion of the n-sphere G/K. Let S(G/K) be the unit sphere bundle of G/K, relative to the induced metric, and y*: S(G/K) - Si C Vs be the differential of y. Then y* is an immersion if and only if s > 2, and y* is a minimal immersion if and only ifs = 2. In the latter case, the metric induced by y* on S(G/K) has constant scalar curvature, and the square of the norm of the second fundamental form A is Downloaded by guest on September 26, 2021 642 MATHEMATICS: DO CARMO AND WALLACH PRoc. N. A. S.

J1AJ 2 2(t - 1) (n - 2) + (n - 1) {( where (1-k(s))2n n + 2 /= 1 n 1 * Guggenheim fellow; partially supported by NSF GP-8623 and C.N.Pq. 1 Calabi, E., "Minimal immersions of surfaces in Euclidean spheres," J. Differential Geom., 1, 111-125 (1967). 2 Chern, S.-S., "Minimal submanifolds in a Riemannian manifold," University of Kansas, Department of Mathematics, Technical Report 19 (New Series), 1968. 3 do Carmo, M., and N. Wallach, "Representations of compact groups and minimal immersions into spheres," J. Differential Geomn., to appear. 4Hsiang, W. Y., "Remarks on closed minimal submanifolds in the standard Riemannian r-spheres," J. Differential Geom., 1, 257-267 (1967). 6 Simons, J., "Minimal varieties in Riemannian manifolds," Ann. Math., 88, 62-105 (1968). Downloaded by guest on September 26, 2021