Rates of mixing for the Weil–Petersson geodesic flow: exponential mixing in exceptional moduli Spaces

We establish exponential mixing for the geodesic flow φ_t: T¹S→ T¹S of an incomplete, negatively curved surface S with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil–Petersson flows for the moduli spaces M_{1 , 1} and M_{0 , 4} are exponentially mixing, in sharp contrast to the flows for M_{g , n} with 3 g- 3 + n> 1 , which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space T¹S and rescaling the flow φ_t.