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, Andersons 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.
A universal geometric inequality for bodies relating energy, size, angular momentum, and charge is naturally implied by Bekensteins entropy bounds. We establi
We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbo
lic setting from counterparts in the asymptotically flat realm, whenever a geometrically motivated system of elliptic equations admits a solution. The inequalities treated here relate mass, angular momentum, charge, and horizon area.
We establish the conjectured area-angular momentum-charge inequality for stable apparent horizons in the presence of a positive cosmological constant, and show that it is saturated precisely for extreme Kerr-Newman-de Sitter horizons. As with previou
s inequalities of this type, the proof is reduced to minimizing an `area functional related to a harmonic map energy; in this case maps are from the 2-sphere to the complex hyperbolic plane. The proof here is simplified compared to previous results for less embellished inequalities, due to the observation that the functional is convex along geodesic deformations in the target.
A universal inequality that bounds the charge of a body by its size is presented, and is proven as a consequence of the Einstein equations in the context of initial data sets which satisfy an appropriate energy condition. We also present a general su
fficient condition for the formation of black holes due to concentration of charge, and discuss the physical relevance of these results.
We present a general sufficient condition for the formation of black holes due to concentration of angular momentum. This is expressed in the form of a universal inequality, relating the size and angular momentum of bodies, and is proven in the conte
xt of axisymmetric initial data sets for the Einstein equations which satisfy an appropriate energy condition. A brief comparison is also made with more traditional black hole existence criteria based on concentration of mass.
