What is an Almost Normal Surface

Abstract

A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to characterize the 2-sphere among surfaces, and similar patterns characterize the 3-sphere among 3-manifolds. Analogous patterns of normal and almost normal surfaces led Rubinstein to an algorithm for recognizing the 3-sphere.