Ph.D Thesis
Ph.D Student Eliezer Appleboim Quasi Normality of Least Area Incompressible Surfaces in 3-Manifolds Department of Mathematics Full Professor Moriyah Yoav

Abstract

This work addresses the following question:

Can every least area surface in the smooth sense, embedded in a 3-manifold, be presented as a limit surface of a sequence of pl-minimal surfaces,appropriately constructed?

Along the line, a thorough study of the intersections of a least area surface with tetrahedra of a given fat triangulation is given.

It is shown that these intersections are relatively simple and controlled. In particular, the notion of quasi normal position of a surface with respect to  a given triangulation of the ambient manifold is defined. It is proved that every least area incompressible surface is quasi normal with respect to a fat triangulation,  which has a mesh that is smaller than a constant which depends only on the metric of the manifold and on the fatness of the triangulation.

Based on this analysis, it will be shown that there exist obstructions which prevent an approximating of a least area surface by pl-minimal surfaces as defined by Jaco and Rubinstein.  A modified version of pl-minimal surfaces is defined, and it is proved that this family of surfaces yields a good approximation of least area surfaces,  i.e., a least area surface is a limit surface, in an appropriate sense, of a sequence of simple approximating surfaces, and the area of the given least area surface  is also the limit of the areas of the approximating surfaces.

A similar study for geodesic curves on surfaces is also given.