A Bakry–Émery Almost Splitting Result With Applications to the Topology of Black Holes

The almost splitting theorem of Cheeger-Colding is established in the setting nonnegative generalized m-Bakry–Émery Ricci curvature, which m positive and associated vector field not necessarily required to be gradient a function. In this context it shown that with diameter upper bound volume lower bound, as well control on Bakry–Émery field, fundamental group such manifolds abelian. Furthermore, extensions well-known results concerning curvature bounds are given for curvature. These include: first Betti number Gromov Gallot, Anderson’s finiteness isomorphism types, comparison, Abresch–Gromoll inequality, Cheng–Yau estimate. Finally, analysis applied stationary vacuum black holes higher dimensions find low temperature horizons must have limited topology, similar restrictions exhibited by (extreme) zero temperature.