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In this paper the author proves some pinching theorems of Simons type for complete minimal submanifolds in the sphere, which generalize the relative results by Simons and Yau. In this paper we first prove a pinching theorem for complete minimal submanifolds in the sphere, which has been proved by J. Simons [4] in the compact case. Namely, if Mn is a complete submanifold in the sphere Sn 'P, and if the square of the length of the second fundamental form of Mn is not greater than n/(2 -lp), then Mn is totally geodesic. When p = 1, this theorem has been proved by T. Hasanis [2]. Furthermore, it turns out that many pinching theorems in Yau's paper [5] can be proved to be still true for the complete case in the same way. 1. Local formulas and basic lemmas. Let Mn be an n-dimensional manifold in the Euclidian unit sphere Snlp. We choose a local field of orthonormal frames e1,. . . ,e,... en+p in Sn+p such that, restricted to Mn, the vectors e ,..., en are tangent to Mn. Let w1,. . . , wn be the field of dual frames. We shall make use of the following convention on the ranges of indices: 1 < i, j, k,... < n, n + 1 < a, /3, y,... < n + p; we shall agree that repeated indices are summed over the respective ranges. We denote the second fundamental form of Mn by Ea jhwlwjea. The square of the length of the second fundamental form is defined by S = Za i j(h(' )2. We call H = (1/n)(Za ih )ea the mean curvature vector. Mn is said to be a minimal submanifold if H = 0 and a submanifold with parallel mean curvature if the normalized mean curvature vector is parallel in the normal bundle of Mn. The following equality is well known: (1) Rijkl = ( ik J/Si/S k) + Z(hakha hahak), a where Rijkl are the components of the curvature tensor of Mn. The following two lemmas are basic for our aim. Received by the editors June 26, 1984. 1980 Mathematics Subject Classification. Primary 53C40; Secondary 53C20. ' The author was supported by the Max-Planck-Institut fur Mathematik, Sonderforschungsbereich 40 of the University of Bonn. (?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Pinching theorems of Simons type for complete minimal submanifolds in the sphere