Thurston's Asymmetric Metric on the Space of Singular Flat Metrics with a Fixed Quadrangulatioṅ
2021
Consider a compact surface equipped with a fixed quadrangulation.
One may identify each quadrangle on the surface by a Euclidean rectangle 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 Teichmuller 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 Teichmuller 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 subspace 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.
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