Generic Regularity of Unoriented Two-Dimensional Area Minimizing Surfaces

Abstract

Every smooth curve in R' bounds a surface (flat chain mod 2) of least possible area among all surfaces, orientable or non-orientable, having that boundary. Examples show that such a surface can have singularities; that is, it must be a smoothly immersed surface, but it can have finitely many points of self-intersection. In this paper we show that for almost every curve y, every unoriented area minimizing surface bounded by y is completely regular (i.e., has no self intersections). The idea of the proof is as follows. Consider a self-intersecting surface that is area minimizing. At each point of self-intersection, it is known that the sheets of the surface must cross each other absolutely orthogonally. (When they do not, it is possible to construct a comparison surface with higher genus, fewer self-intersections, and less area [M3].) What we show is that for generic perturbations of the boundary curve, nearby minimal surfaces of the same topological type no longer cross absolutely orthogonally, and hence they fail to be area minimizing. In other words, for a generic curve y, every self-intersecting surface bounded by y fails to be area minimizing. Equivalently, every area minimizing surface it bounds is embedded. The results are essentially the same when Rn is replaced by a riemannian manifold (see Section 3), but the situation is quite different for surfaces of more than two dimensions. For example if the boundary is an embedding of RP2, then (for purely topological reasons) it cannot bound any compact manifold (immersed or embedded). (This is because the Euler characteristic of a 3-manifold with boundary is half the Euler characteristic of the boundary; since the Euler characteristic of Rp2 is odd, it cannot bound a manifold.) Thus the area minimizing surfaces it bounds are all singular. In other settings there may be other obstacles to perturbing away singularities: recent work of F. Morgan [M4] suggests that there is an open set of embeddings F of S2 U S2 into R6 such that F bounds a singular area minimizing integral current. Open problems in this vein include proving or disproving generic regularity of oriented area minimizing hypersurfaces and of oriented two-dimensional area minimizing surfaces