Minimal surfaces of constant curvature in $S\sp n$

Abstract

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into S. 0. Introduction. In this paper, we classify the connected minimal surfaces of constant Gaussian curvature in the unit n-sphere Sn C Enll for all n. It is a classical fact (and follows easily from the structure equations) that the only examples up to rigid motion in S3 are the open subsets of the geodesic spheres (with K 1) and the Clifford torus (with K 0). In an early paper Boruvka constructed a linearly full immersion S2 C S2m for each m > 0, where the induced metric had K 2/m(m + 1). Later [Calabi] showed that, up to rigid motion, these Boruvka spheres were the only compact minimal surfaces with K Ko > 0 in S' for any n. [Wallach] proved that any connected piece of minimal surface with K a positive constant in Sn was a subset of a Boruvka sphere. [Kenmotsu, 1976] found all of the flat minimal surfaces (K 0) in S' and quite recently, [Kenmotsu, 1983] showed that K Ko < 0 is impossible for minimal surfaces in S4. The techniques used in the above proofs range from harmonic analysis to rather involved calculations with the moving frame. On the other hand, in this paper, we take a somewhat different point of view. If (M2, ds2) is a surface of constant Gaussian curvature K, then the minimal surfaces in S n C E n ? 1 of constant Gaussian curvature K are given locally by smooth maps f: M En' + lwhich satisfy three conditions: first, Kf, f) 1; second, Af =2f (A is the Laplace-Beltrami operator of ds2); and third, that f should be an isometry, Kdf, df) = ds2. We study the more general class of maps f: M En + 1 which only satisfy the first two conditions. Note that this set of equations is already overdetermined (by one equation). Received by the editors August 14, 1984. 1980 Mathematics Subject Classification. Primary 53A10; Secondary 53B25 53C40.