Title: Bounds on harmonic radius and limits of manifolds with bounded Bakry-Émery Ricci curvature,
Abstract: 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-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 $C^\alpha W^{1,q}$, 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-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 [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.