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

Gregory J. Galloway, Marcus A. Khuri, Eric Woolgar

Research output: Contribution to journalArticlepeer-review


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, as well as control on the Bakry–Émery vector field, 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.

Original languageEnglish (US)
Pages (from-to)2067-2101
Number of pages35
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - Jun 2021

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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