{"paper":{"title":"Dirac spectral flow on contact three manifolds II: Thurston--Winkelnkemper contact forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DG","authors_text":"Chung-Jun Tsai","submitted_at":"2013-07-17T13:01:14Z","abstract_excerpt":"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)^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4605","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}