Every Three-sphere of Positive Ricci Curvature Contains a Minimal Embedded Torus

One of the most celebrated theorems of differential geometry is the 1929 theorem of Lusternik and Schnirelmann, which states that for every riemannian metric on the 2-sphere there exist at least three simple closed geodesies. Jurgen Jost [J] (following important work of Pitts [P] and Simon and Smith [SS]) has recently generalized this result by showing that for every riemannian metric on S, there exist at least 4 minimal embedded 2-spheres. This is optimal in that there are metrics for which the number of embedded minimal 2-spheres is exactly four [W2,4.5]. However, one can also ask about surfaces of higher genus, and here our knowledge is very incomplete. On the one hand, Lawson [L] showed that S with its standard metric contains embedded minimal surfaces of every orientable toplogical type, and recently Pitts and Rubinstein [PR] have discovered many new infinite families of examples. But for general metrics on S, no known theorem asserts the existence of any minimal surface other than a sphere. The present paper takes a first step in this direction by proving: