Dirac spectral flow on contact three manifolds II: Thurston-Winkelnkemper contact forms

Given an open book decomposition (Σ,τ) of a three manifold Y, Thurston and Winkelnkemper [16] construct a specific contact form a on Y. Given a spin-c Dirac operator D on Y, the contact form naturally associates a one parameter family of Dirac operators Dr=D−ir2cl(a) for r≥0. When r>>1, we prove that the spectrum of Dr=D0−ir2cl(a) within [−12r12,12r12] is almost uniformly distributed. With the result in Part I [18], it implies that the subleading order term of the spectral flow from D0 to Dr is of order r(log⁡r)92. Besides the interests of the spectral flow, the method of this paper provides a tool to analyze the Dirac operator on an open book decomposition.