Abstract
Comm. Math. Phys., 384 (2021), no. 3, 2067-2101 The almost splitting theorem of Cheeger-Colding is established in the setting
of almost nonnegative generalized $m$-Bakry-\'{E}mery 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-\'{E}mery 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.
