The topological uniqueness of complete minimal surfaces of finite topological type

Abstract

IN 1970 Lawson [22] proved that two embedded closed diffeomorphic minimal surfaces in the unit three-dimensional sphere S3 in lRJ are ambiently isotopic in S3. Lawson proved this theorem by first proving that an embedded orientable closed minimal surface of genus y in a closed orientable Riemannian three-manifold M’ with positive Ricci curvature disconnects M-’ into two genus-8 handlebodics. A result of Frankel [7] was used to prove this. Lawson then applied a deep result of Waldhauscn [33] that states that decompositions of SJ into two genus-cl handlebodies arc unique up to ambient isotopy. More prcciscly. Waldhauscn’s uniqucncss thcorcm states that whenever a closed surface of genus y in S” separates SJ into handlcbodics, then the embedding of the surface is as simple as possible; in other words, the surface is obtained from a two-sphere Sz c S’ by adding handles in an unknotted manner. Mccks [23] gcncralizcd Lawson’s argument to the case of oricntablc closed minimal surfaces in a closed ,V3 with nonnegative Ricci curvature. Meek’s result and another topological uniqueness result of Waldhauscn [33] implies that two closed diffeomorphic minimal surfaces in S2 x S’ with the usual product metric arc ambiently isotopic. Finally, Meeks-Simon-Yau [25] proved that if X is a closed minimal surface in S’ equipped with a metric of non-negative scalar curvature, then X disconnects S3 into two handlebodies and hence dctcrmincs a unique ambient class in S’. In [23] Mceks considered the problem of the topological uniqueness of minimal surfaces in three-dimensional Euclidean space. In particular, Meeks proved that any two compact diffeomorphic minimal surfaces, with boundary a simple closed curve on the boundary of a smooth convex ball, are ambiently isotopic in the ball. Later Hall [I33 showed that there exist two simple closed curves on the unit sphere S* that bound a knotted minimal surface of genus one. In this paper we will use global properties of stable minimal surfaces in [w’ to prove: A properly embedded minimal surface of finite topological type in R’ is unknotted. In particular, if X, and IEz are two such diffeomorphic surfaces, then they are ambiently isotopic. This is the main result of our paper and it is restated in a more precise form in Theorem 5.1. Because of Hall’s counterexample to unknottedness for compact minimal