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

Abstract

Given an open book decomposition $(\Sigma ,\tau )$ of a three manifold $Y$, Thurston and Winkelnkemper [TW] 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 $D_r = D - \frac{ir}{2}\cl(a)$ for $r\ge 0$. When $r>>1$, we prove that the spectrum of $D_r = D_0 - \frac{ir}{2}\cl(a)$ within $[-(r^{1/2})/2,(r^{1/2})/2]$ are almost uniformly distributed. With the result in Part I, it implies that the subleading order term of the spectral flow from $D_0$ to $D_r$ is of order $r(\log r{)}^{\frac{9}{2}}$. Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.

arXiv comment48 pages. All comments welcome

References in corpus (2)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Asymptotic spectral flow for Dirac operators of disjoint Dehn twists