Thurston’s asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

Abstract

Consider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface by a Euclidean rect-angle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston’s asymmetric metric on the Teichm¨uller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichm¨uller space to be geodesic and we prove that any two points of the space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm at the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the sub-space topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.