Bounds on harmonic radius and limits of manifolds with bounded Bakry-Émery Ricci curvature
Bounds on harmonic radius and limits of manifolds with bounded Bakry-Émery Ricci curvature
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调和半径的界限以及具有有界 Bakry-ämery Ricci 曲率的流形的极限
DOI：
10.1007/s12220-018-0072-9
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通讯作者：
Meng Zhu
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作者：
Qi S. Zhang;Meng Zhu
在测地线球的体积接近于欧氏体积或内射半径有界的一般条件下，证明了当位势梯度有界时，具有有界Bakry-Émery Ricci曲率的流形的调和半径的一个下界.在这些条件下，调和坐标下的度规的正则性仅为，其中为流形的维数。在有界Ricci曲率条件下，这比经典调和坐标下的结果（安德森in Invent Math 102：429-445，1990）低近一阶。正则性的损失导致了证明方法的一些差异，这也可以用来解决经典情况下收敛的细节。基于该下限以及Cheeger和Naber（Ann Math 182：1093-1165，2015）以及Wang和Zhu（Crelle's J， http://arxiv.org/abs/1304.4490 ），我们将Cheeger和Naber（2015）中的Cheeger-Naber余维4定理推广到当势的梯度有界时流形具有有界Bakry-Émery Ricci曲率的情况。当势的梯度有界时，这一结果涵盖了Ricci孤子。在证明过程中，我们将使用一个绿色的函数参数，并采用Bamler中的线性代数参数（J Funct Anal 272（6）：2504-2627，2017）。一个新的组成部分是证明变换定理中矩阵的对角元素有界远离0。这些似乎简化了余维4定理的证明，即使在里奇曲率有界的情况下。
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 theharmonic 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, whereandnis the dimension of the manifolds. This is almost 1-order lower than that in the classicalharmonic 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 ofconvergence 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.