Bounds on harmonic radius and limits of manifolds with bounded Bakry-\'Emery Ricci curvature

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^{\alpha} W^{1, q}$ harmonic radius for manifolds with bounded Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^\alpha W^{1,q}$, where $q>2n$ and $n$ is the dimension of the manifolds. This is almost 1 order lower than that in the classical $C^{1,\alpha} W^{2, p}$ harmonic coordinates under bounded Ricci curvature condition [And]. The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $W^{2, p}$ convergence in the classical case. Based on this lower bound and the techniques in [ChNa2] and [WZ], we extend Cheeger-Naber's Codimension 4 Theorem in [ChNa2] to the case where the manifolds have bounded Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green's function argument and adopt a linear algebra argument in [Bam]. A new ingradient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.