An unknotting result for complete minimal surfaces R3

Abstract

In this note we prove: Theorem. If S is a complete properly embedded minimal surface in R 3, with the Euclidean metric, and if A is a closed connected complementary domain of Z, then either A is metrically R 2  [0, hi or, up to diffeomorphism, A is made from a countable or finite collection of building blocks by attaching a proper collection of closed 1-handles. Each building blocks is a closed 3-ball or an open annulus x [0, 1). [] The first theorem of this kind was B. Lawson's I-L] unknottedness theorem for minimal surfaces in S 3. That result was obtained by considering the second variation of arc length. Because the flat Euclidean metric of R 3 does not contribute a favorable curvature term as found in S 3 a dual approach employing stable minimal surfaces (dual to Lawson's spanning arcs) is required. The key technical result, proved by an analysis of the second variation operator on surfaces, is that geometric planes are the only stable minimal surfaces in R 3 [C, P] and [F-C, S]. From this starting point a more topological theory has been developed by Frohman, Hoffman, Meeks, Yau, and others. Our argument is based, to a large degree, on the methods used by C. Frohman and W. Meeks [F, M] to classify 1-ended, proper, complete, minimal, imbeddings in R 3. We present four succinct topological assertions about any 3-manifold with boundary (A, S) which is a connected, closed complementary region of a complete proper minimal surface in R 3. These statements should be thought of as "axioms" for this emerging theory. Deduced from geometry, they help to mark the boundary with topology. The theorem follows quickly from the axioms. Axiom 1 occurs explicitly as corollary 3.2 I-F, M]. Axiom 2 is an immediate consequence of the methods of [F, M] and Axiom 3 is proved by an extension of these methods. Axiom 4 is related to an idea on the ordering of ends [M, Y] due to W. Meeks and S.-T. Yau. These are the axioms. Their verification actually only requires that the mean curvature vector of 2: be everywhere zero or pointing towards A.