Equivariant minimax and minimal surfaces in geometric three-manifolds

Abstract

Minimal surfaces in Riemannian three-dimensional manifolds have played a major role in recent studies of the geometry and topology of 3-manifolds. In particular, stable and least area minimal surfaces have been used extensively [SY, MSY, HS]. On the other hand, explicit examples of unstable minimal surfaces have rarely been given. For the 3-sphere S with the standard metric of constant sectional curvature, there is the classical paper of Lawson [LH], showing that closed orientable surfaces of every genus occur as (unstable) minimal surfaces embedded in 5 3 . More recently, Karcher, Pinkall, and Sterling [KPS] have constructed several new examples in S. We have discovered new infinite families of minimal surfaces in S. More generally, we announce a number of new finite and infinite families of embedded minimal surfaces in geometric 3-manifolds, using the minimax procedure described below. Geometric structures on 3-manifolds were introduced by Thurston [TW]. (See also the excellent survey by Scott [SP1].) There are eight geometries: R , S , S x R, Nil, H 2 x R, S L ^ R ) , Solv, and H . A geometric structure on a 3-manifold S is a representation of E as a quotient of one of the above eight spaces divided out by a covering transformation group acting isometrically. Equivalently, E is locally isometric to one of these spaces, with its natural homogeneous space structure. The first six of these geometries give Seifert fiber spaces and we are mainly interested in such examples. The SO(2)-isometry actions associated with most Seifert fiber spaces yield infinite classes of embedded minimal surfaces. Note that interesting examples can also be obtained in the other two geometries; e.g., in hyperbolic geometry H 3 [PR3, §2]. The minimax procedure has proved to be a versatile and powerful means of constructing unstable minimal surfaces in 3-manifolds. For basic details of this technique, see [PJ, SS, P R l , and PR2]. Suppose that G is a finite group of isometries acting on a closed oriented Riemannian 3-manifold E. Assume that A is a Heegaard surface in E; i.e., the closures of the components of E ~ A are handlebodies K and K'. Assume furthermore that A is G-equivariant; i.e., gA = A for all g E G. We consider one-parameter smooth families At, t € [0,1], sweeping out E and having the following properties: Ao and Ai are graphs; At is isotopic to A for all 0 < t < 1; At is G-equivariant for all t; the handlebody Kt is chosen so that the orientation on At is induced from that on