From Normal Surfaces to Normal Curves to Geodesics on Surfaces

Motivated by the topological theory of normal surface we give in this paper a complete study of the relations between geodesic curves and normal curves embedded in a triangulated Riemannian surface. Normal surface theory is a topological piecewise linear (p` for short) counterpart of the differential geometric theory of minimal surfaces. This theory studies the ways surfaces intersect with a given triangulation of a 3-manifold. A surface is normal if it intersects the tetrahedra of a triangulation in a fairly simple manner. It was proved by Haken in [1] that every incompressible (i.e., topologically non trivial) surface embedded in a triangulated 3-manifold can be continuously deformed to a normal surface with respect to any given triangulation of the manifold. Having the theory of normal surfaces, it is then desirable to interpret minimal surfaces in terms of normal surfaces. In this direction Jaco and Rubinstein presented in [2] a p` version, that is based on normal surfaces, of minimal and least area surfaces in a triangulated 3-manifold . It is shown in [2] that p` minimal and p` least area surfaces share many properties of classical minimal and least area surfaces. These results are the first to make precise the analogy between minimal and normal surfaces. However, the extent at which this analogy holds is far from being fully understood. For instance, the following two questions are very natural in this context, yet both of them are still open. (I) Is it possible to subdivide a given triangulation of a 3-manifold arbitrarily fine, to obtain sequences of p`-minimal surfaces that converge in some suitable way to classical minimal surfaces? An affirmative answer to the above question gives an alternative topological-combinatorial proof of many classical existence results of least area incompressible surfaces in 3-manifolds, such as those obtained for example in [3], using partial differential equations. Moreover, the use of existing algorithms for finding normal surfaces makes this topological-combinatorial result of computable algorithmic nature. The second question is the following: (II) Can every minimal surface in the smooth sense, be presented as a limit surface of a sequence of p`-minimal surfaces, appropriately constructed? Answering these questions, at least partially will be the focus of two followup papers. In this paper we will address these questions in one dimension lower. That is in the context of curves on surfaces. In particular the following theorems will be proved: