Least area incompressible surfaces in 3-manifolds

Abstract

Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function. The existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group [18, 17]. In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly. One should note that evidence supporting this view had been found by Meeks-Yau in their embedding theorems for minimal disks and 2-spheres [13, 143 . Our main result asserts that if a least area immersion is homotopic to an embedding, then it has no self-intersections, which clearly exemplifies the above philosophy. The precise result is the following.