On a Family of Polyhedral Singular Vertices

Abstract

We provide a local description of the curves with minimal length based at singularities in a family of polyhedral surfaces. These singularities are accumulation points of vertices with conical angles equal to π and 4π (or 3π, in a variation). While a part of the minimizing curves behaves quite like the ones reaching conical vertices, the singularities present features such as being connected to points arbitrarily close to them by exactly two minimizing curves. The spaces containing such singularities are constructed as metric quotients of an euclidean half-disk by certain identification patterns along its edge. These patterns are examples of what is known as paper-folding schemes, and we provide the foundational aspects about them which are necessary for our analysis. The arguments are based on elementary metric geometry and calculus.