Closed geodesics on ideal polyhedra of dimension 2
Ideal polyhedra of dimension 2 are unions of ideal hyperbolic triangles which are glued together by isometries along their sides. The infimum of the lengths of broken geodesics defines a pseudodistance on such ideal polyhedra. In Theorem 1 the author proves that this is in fact a geodesic metric under certain finiteness assumptions. In Theorem 2 it is shown that for each closed curve there is a unique distance realizing closed geodesic in the same homotopy class. The only exception are curves homotopic to a cusp, meaning that the infimum of the length in this homotopy class is zero. However, this minimal closed geodesic is not necessarily simple.
scientific article; zbMATH DE number 944405