The Collar Theorem

Abstract

A fundamental property of Riemann surfaces, know as the collar theorem, is that the small closed geodesics have large tubular neighborhoods which are topological cylinders. This was first observed in Keen [4], Chavel-Feldman [1], Halpern [1], Matelski [1], Randol[4], Seppälä-Sorvali [1,3] and others. The collar theorem is a basic tool in different parts of this book. The proof is given in Section 4.1. In Section 4.2 we apply the collar theorem to obtain a lower bound for the lengths of closed geodesic with transversal self-intersections. Another application is the triangulation of controlled size in Section 4.5. In Section 4.3 we extend the collar theorem to surfaces with variable curvature, and in Section 4.4 we extend the collar theorem to non-compact Riemann surfaces of finite area.