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Session S19 - Geometric and Analytic Aspects of General Relativity

Monday, July 12, 16:00 ~ 17:00 UTC-3

Bakry-\'Emery Ricci curvature and general relativity

Eric Woolgar

University of Alberta, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The famous Penrose singularity theorem of 1965 depends on an energy condition which asserts the nonnegativity of null components of the Ricci tensor. Indeed, many important theorems in Riemannian geometry and relativity depend on a Ricci curvature bound. It was therefore surprising (to me) that often such theorems can be proved when the bounds only hold on the Ricci tensor modulo the hessian of some function or, more generally, the Lie derivative of the metric along some vector field. This is the Bakry-\'Emery Ricci tensor. The surprise is lessened only somewhat by the realization that Bakry-\'Emery Ricci curvature arises naturally in static spacetimes, near horizon geometries, Kaluza-Klein compactifications (warped products), and other natural applications. I will discuss a selection of these applications, including an application to the topology of closed universes (joint work with M Khuri and GJ Galloway) which yields topological constraints even when closure density is not quite achieved. Time permitting, I will also discuss synthetic Bakry-\'Emery Ricci tensor bounds in metric-measure spaces. Recent work by McCann and independently by Mondino and Suhr construct synthetic Ricci curvature bounds (energy conditions) in Lorentzian geodesic spaces. Cavalletti and Mondino have given a proof of a generalized Hawking cosmological singularity theorem for synthetic energy conditions. Burtscher, Ketterer, McCann, and I have given a Riemannian analogue.

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