Counting Curves on surfaces

Viveka Erlandsson, University of Bristol. Part of geometry seminar series.

Let S be a closed surface and consider all curves on S which self-intersect a bounded number of times. Due to Mirzakhani we know the asymptotic growth of the number of such curves whose hyperbolic length is bounded by L, as L grows. More specifically, if S has genus g and is equipped with a hyperbolic structure, she showed that the number of such curves on S (in each mapping class group orbit) is asymptotic to a constant times L^{6g-6}.

In this talk I will explain, through the use of geodesic currents, why the same asymptotics hold for other metrics on the underlying topological surface, in particular for any Riemannian metric.