Markus Khuri (Stony Brook): A Bakry-Emery Almost Splitting Result With Applications to the Topology of Black Holes and Cosmology

The almost splitting theorem of Cheeger-Colding is generalized to the setting of almost nonnegative -Bakry-Emery Ricci curvature, in which 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 -Bakry-Emery Ricci curvature. This analysis is then 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. Lastly, applications to cosmology are also described. Here restrictions are obtained on the list of possible topologies of the universe. This is joint work with Greg Galloway and Eric Woolgar.