Bounds on Harmonic Radius and Limits of Manifolds with Bounded Bakry–Émery 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^α∩ W^1,q harmonic radius for manifolds with bounded Bakry–Émery 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^α∩ W^1,q, where q> 2 n and n is the dimension of the manifolds. This is almost 1-order lower than that in the classical C^1,α∩ W^2,p harmonic coordinates under bounded Ricci curvature condition (Anderson in Invent Math 102:429–445, 1990). 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 Cheeger and Naber (Ann Math 182:1093–1165, 2015) and Wang and Zhu (Crelle’s J, http://arxiv.org/abs/1304.4490), we extend Cheeger–Naber’s Codimension 4 Theorem in Cheeger and Naber (2015) to the case where the manifolds have bounded Bakry–Émery 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 Bamler (J Funct Anal 272(6):2504–2627, 2017). A new ingredient 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.