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

TL;DR

This paper studies the spectral distribution of a family of Dirac operators on contact three-manifolds derived from open book decompositions, revealing almost uniform spectral distribution and detailed spectral flow behavior.

Contribution

It introduces a method to analyze Dirac operators on contact three-manifolds with open book decompositions, focusing on spectral distribution and spectral flow asymptotics.

Findings

Spectra of Dirac operators are almost uniformly distributed for large parameters.

The subleading order term of spectral flow is of order r (log r)^{9/2}.

Provides a new analytical tool for Dirac operators on open book decompositions.

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 $r\geq0$. 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.