The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature

Abstract

In this paper we obtain a curvature estimate for embedded minimal surfaces in a three-dimensional manifold of positive Ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface. It should be mentioned that there are two main points in our result: One is the absence of a stability assumption and the other is the requirement of being embedded. Most known curvature estimates require the stability assumption, and once the stability assumption is dropped, many of these known results cease to be valid. (See [SS] and [An] for another example of a curvature estimate without the assumption of stability.) The embeddedness condition is rather subtle because of the way it enters in our proof. Our proof depends on the eigenvalue estimate and the area bound due to the first author and Wang [CW] which require embeddedness in an essential way. Present knowledge indicates that closed embedded minimal surfaces in S 3 are rare, while immersed surfaces are more plentiful. For example, only a finite number of minimal embeddings of a given genus are known. On the other hand, Otsuki [O] constructed infinitely many immersed minimal tori with arbitrarily large area. We obtain the curvature estimate indirectly by proving the smooth compactness theorem (Theorem 1). Theorem 1 has many interesting consequences. It shows that the set of conformal structures that can be realized on a minimal embedding in S 3 is a compact subset of the moduli space. This is in contrast with the result of Bryant [B] who showed that every Riemann surface is conformally and minimally immersed in S 4. In view of our compactness result and the scarcity of examples, it is very tempting to conjecture that there are only finitely many embedded minimal surfaces (up to rigid motion) in S 3 for each fixed genus. Throughout this paper, manifold means manifold without boundary unless explictly stated otherwise. When we say a sequence M i of surfaces converges to a