The structure of complete stable minimal surfaces in 3‐manifolds of non‐negative scalar curvature

Abstract

The purpose of this paper is to study minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian three-manifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+(AI* be positive. Here Ric (v) is the Ricci curvature of N in the normal direction to M and (A)’ is the square of the length of the second fundamental form of M. In the case that N is the flat R3, we prove that any complcte stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true provided the image of the Gauss map of M omits an open set on the sphere. The relationship of the stable regions on M to the area of their Gaussian image has been studied by Barbosa and do Carmo [l] (cf. Remark 5 ) . The methods of Schoen-Simon-Yau [ 113 give a proof of this result provided the area growth of a geodesic ball of radius r in M is not larger than r6. An interesting feature of our theorem is that it does not assume that M is of finite type topologically, or that the area growth of M is suitably small. The theorem for R3 is a special case of a classification theorem which we prove for stable surfaces in three-dimensional manifolds N having scalar curvature SZO. We use an observation of Schoen-Yau [8] to rearrange the stability operator so that S comes into play (see formula (12)). Using this, and the study of certain differential operators on the disc (Theorem 2), we are