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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.1. Notations and results. Let M be a two-dimensional, orientable C '-manifold. A domain D C M is an open, connected subset with compact closure D and such that the boundary aD is a finite union of piece-wise smooth curves. Let x: M->R3 be a minimal immersion into the Euclidean space R 3. It is well known that D is a critical point of the area of the induced metric, for all variations of D which keep aD fixed. When this critical point is a minimum for all such variations, wesay that D is stable. The goal of this paper is to estimate the "size" of a stable minimal immersion and the main theorem is as follows. Set S = { (x, y,z) e R 3; X2 + y2 + z2 = 1} and denote by g: M-> S'2, the Gauss map of the immersion x. THEOREM 1.2. Let the area of the spherical image g(D) c S' of a domain D c M be smaller than 27. Then D is stable.This estimate is sharp, as can be shown, for instance, by considering pieces of the catenoid bounded by circles C1 and C2 parallel to and in opposite sides of the waist circle CO. By choosing C1 close to CO and C2 far from CO, we may obtain examples of unstable domains whose spherical image has area larger than 2v and as close to 27 as we wish. Further details will be given in Section 2.Since g may cover g(D) more than once, Theorem 1.2 implies (but it is stronger than) that if the total curvature is smaller than 27, then D is stable.Let N be a unit normal field along x(M). Let A = div grad and K denote the Laplacian and the Gaussian curvature of M, respectively, in the induced metric. Given a piece-wise smooth function u:D--R, with u_0 on aD, the second derivative of the area function for a variation whose deformation vector field is given by V= uN is (Cf. 3.2.3 of [9]). I(V,V)= U(-/Au+2uK)dM. (1.3)

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of pseudo-Riemannian manifolds Mn (K) of index s, with K K, flat normal bundle and such that the normal principal curvatures are different from K-K. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R" which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of M, by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n -1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n -1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of M which have constant curvatures.

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