Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature

Abstract

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some canonical embedded surface. From the point of view of geometry, a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds. In [MY2], the theory of minimal surfaces was used to simplify and strengthen the classical Dehn's lemma, loop theorem and the sphere theorem. In the setting there, one minimizes area among all immersed surfaces and proves that the extremal object is embedded. In this paper, we minimize area among all embedded surfaces isotopic to a fixed embedded surface. In the category of these surfaces, we prove a general existence theorem (Theorem 1). A particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area. We also prove that there exists an embedded sphere of least area enclosing a fake cell, provided the complementary volume is not a standard ball, and provided there exists no embedded one-sided RP2. By making use of the last result, and a cutting and pasting argument, we are able to settle a well-known problem in the theory of three dimensional manifolds. We prove that the covering space of any irreducible orientable three dimensional manifold is irreducible. It is possible to exploit our existence theorem to study