On the double curves of least area tori

Abstract

Abstract. The double curves of least area immersions of the torus into closed, orientable, irreducible 3-manifolds are simple in the torus. A related result for other least area surfaces is given.If f : F → M is a general position immersion of a surface into a 3-dimensional manifold, then the set of points where f (F ) fails to be an embedding consists of a finite collection of double curves and triple points. The preimages of these curves in F , also known as double curves, may not be simple or disjoint (they meet at the preimages of the triple points). A double curve in f (F ) is simple in M if and only if the corresponding double curves in F (2 if F and M are orientable) are simple and disjoint. Figure 1 shows the double curves of 3 homotopic immersions of the torus into some 3-manifold and illustrates how much they can change with the homotopy.Of special interest are immersions which are incompressible (π 1 -injective) and have least area in its free homotopy class for some Riemannian metric on the manifold. In this case one can hope that the singularities of the immersion are minimal and are related to the topology of M . The immersions in Figure 1 are incompressible, and the last 2 have least area for some Riemannian metrics.Freedman, Hass and Scott [2] proved that if M is orientable and irreducible (i.e. every sphere embedded in M bounds a 3-ball), and F is orientable and not S 2