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Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and ametric $ds^2$. Suppose that $F\to Y$ is a principal bundle with structure group$U(2) = SU(2)\times_{\pm1}S^1$ such that $F/S^1$ is the principal SO(3) bundleof orthonormal frames for $TY$. A unitary connection $A_0$ on the Hermitianline bundle $F\times_{\det U(2)}\mathbb{C}$ determines a self-adjoint Diracoperator $D_0$ on the $\mathbb{C}^2$-bundle $F\times_{U(2)}\mathbb{C}^2$.  The contact form $a$ can be used to perturb the connection $A_0$ by$A_0-ira$. This associates a one parameter family of Dirac operators $D_r$ for$r\geq0$. When $r>>1$, we establish a sharp sup-norm estimate on theeigensections of $D_r$ with small eigenvalues. The sup-norm estimate can beapplied to study the asymptotic behavior of the spectral flow from $D_0$ to$D_r$. In particular, it implies that the subleading order term of the spectralflow is strictly smaller than the order of $r^{\frac{3}{2}}$. We also relatethe $\eta$-invariant of $D_r$ to certain spectral asymmetry function involvingonly the small eigenvalues of $D_r$.

Dirac spectral flow on contact three manifolds I: eigensection estimates and spectral asymmetry