几何分析讨论班— Bounds on harmonic radius and limits of manifolds with bounded Bakry Emery Ricci curvature
主 题: 几何分析讨论班— Bounds on harmonic radius and limits of manifolds with bounded Bakry Emery Ricci curvature
报告人: Qi Zhang (UC Riverside and Fudan University)
时 间: 2017-06-21 10:10-12:00
地 点: 理科1号楼1303
Abstract: Under the usual condition that the volume of a geodesic ball is close to the Euclidean one, we prove a lower bound of the $C^{\alpha} \cap W^{1,q}$ harmonic radius for manifolds with bounded Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded. This is almost 1 order lower than that in the classical $C^{1,\a} \cap W^{2, p}$ harmonic coordinates under bounded Ricci curvature condition. The method of proof can also be used to address the detail of $W^{2, p}$ convergence in the classical case, which seems not in the literature.
Based on this lower bound and the techniques in Cheeger and Naber and F. Wang and X.H. Zhu, we extend Cheeger-Naber's Codimension 4 Theorem 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. Some short cuts and additional information in the original case are also obtained.
