Arch. Math. Vol. 60, 568-578 (1993) 0003-889X/93/6006-0568 $ 3.70/0 9 1993 Birkh/iuser Verlag, Basel Some pinching and classification theorems for minimal submanifolds By BANG-YEN CHEN 1. Introduction. Let x: M -* E" be an immersion from an n-dimensional (n > 0) mani- fold into a Euclidean re, space. Denote by A the Laplacian operator of M with respect to the induced metric. Then we have the following formula of Beltrami: (1.1) Ax = -- nil, where H is the mean curvature vector of M in Em. From (1.1), it follows that an immersed submanifold of positive dimension in a Euclide- an space is a minimal submanifold if and only if all the coordinate functions are harmonic functions relative to the induced metric. In particular, one has the following necessary condition for the immersion to be minimal: Condition 1: If x : M --r E m is a minimal immersion from a manifold of positive dimension into a Euclidean m,space, then M is non-compact. Beside Condition t, the well-known equation of Gauss yields the second necessary condition: Condition 2: if x : M ~ E m is a minimal immersion from a manifold of positive dimension into a Euclidean m-space, then the Rieei tensor of M is negative semi-definite. In page 13 of [4] S. S. Chern asked to search for further necessary conditions on the Riemannian metric of M for M to admit an isometric minimal immersion into a Euclide- an space. In this article, we mention a third necessary condition for the existence of minimal isometric immersion from a given Riemannian manifold M into Euclidean space; namely, the sectional curvatures and the scalar curvature of M must satisfy the third condition: K(n) > 89 T(p) for any plane section n c TpM, p ~ M. We then construct examples of minimal submanifolds of any dimension _> 2 in E" which satisfy the condi- 1 In the last section, we obtain some classification theorems for minimal tion: infK - gz. 1 submanifolds in Euclidean space which satisfy the third condition: infK -= gz. In order to prove the above results, we establish in Section 3 several lemmas for submanifolds in Riemannian manifolds of constant sectional curvature.