一个Bakry-Emery几乎分裂结果及其在黑洞拓扑中的应用
A Bakry-Émery Almost Splitting Result With Applications to the Topology of Black Holes
摘要：
在几乎非负的广义$m$ -Bakry-Émery里奇曲率的背景下，建立了Cheeger-Colding的几乎分裂定理，其中$m$为正，且相应的向量场不一定是函数的梯度.在这种情况下，它表明，与直径上界和体积下界的基本群，这样的流形几乎是阿贝尔。进一步，推广了关于广义$m$ -Bakry-Émery里奇曲率的Ricci曲率下界的已知结果.其中包括：Gromov和Gallot的第一贝蒂数界，Anderson的基本群同构类型的有限性，体积比较，Abresch-Gromoll不等式，Cheng-Yau梯度估计.最后，这个分析被应用到更高维度的静态真空黑洞，发现低温视界必须有有限的拓扑结构，类似于零温度（极端）视界所表现出的限制。
英文摘要：
The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-Émery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-Émery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.
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