This work presents an approximation method by simple pieces of least area surfaces in 3-manifolds.

In [JR] a piecewise linear (*PL*) version of minimal and least area surfaces is presented. This *PL *version is based on normal surfaces. A normal surface in a 3-manifold is a surface that intersects a given triangulation of the manifold in a special simple manner. It was shown in [JR] that *PL*-minimal and least area surfaces share many of the properties of classical differential geometric least area surfaces. However, the extent at which this analogy holds if far from being fully understood. For instance, the following question is very natural in this context, yet it is still open:

**Question: **Can every minimal surface in the smooth sense, be presented as a limit surface of a sequence of *PL*-minimal surfaces, appropriately constructed?

Addressing the above question is the main theme of this work. Specifically, a method of approximating a least area surface by simple pieces will be given. It will be shown that there exist obstructions that prevent the approximating simple pieces from being the *PL- *minimal surfaces as defined in [JR]. A modified version of *PL*-minimal surfaces will be defined, and it will be shown 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.

This study is of major interest of its own sake, since it gives a new topological insight and understanding of known geometric results about the global behavior of least area surfaces. Potential applications of this study in the areas of Image Processing and Computer Graphics will also be discussed. Examples are the problem of removing noise from an image, and finding the shortest path on a triangulated surface. Some preliminary experimental results will be shown.

**Reference: **[JR] W. Jaco and H. Rubinstein, *PL-Minimal Surfaces in 3-Manifolds*, J. Differntial Geometry, 27, 1988.