Area minimizing surfaces in mean convex 3-manifolds

Least Area Incompressible Surfaces in 3-Manifolds

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 ...

Area-minimizing Projective Planes in Three-manifolds

The Existence of Least Area Surfaces in 3-manifolds

This paper presents a new and unified approach to the existence theorems for least area surfaces in 3-manifolds. Introduction. A surface F smoothly embedded or immersed in a Riemannian manifold M is minimal if it has mean curvature zero at all points. It is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this cla...

A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-manifolds

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, and whose intersection is a separable subgroup, there are finite index subgrou...

Minimal Surfaces in Geometric 3-manifolds

In these notes, we study the existence and topology of closed minimal surfaces in 3-manifolds with geometric structures. In some cases, it is convenient to consider wider classes of metrics, as similar results hold for such classes. Also we briefly diverge to consider embedded minimal 3-manifolds in 4-manifolds with positive Ricci curvature, extending an argument of Lawson to this case. In the ...