Existence of least area planes in hyperbolic 3-space with co-compact metric

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...

Geodesic planes in hyperbolic 3-manifolds

In this talk we discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a very strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. Manifolds with infinite volume, however, are far less understood and are the main subject of th...

Compact Non-orientable Hyperbolic Surfaces with an Extremal Metric Disc

The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus g ≥ 3. We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case g = 3, we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.

Tchebycheff Approximation in a Compact Metric Space

Manifolds With Many Hyperbolic Planes

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.