A project in Lorentzian Geometry
speaker: Ettore Minguzzi (Università Degli Studi Di Firenze)speaker: Stefan Suhr (Ruhr-Universität Bochum)
abstract: In recent years various notions of Lorentzian manifolds of (very) weak regularity have been proposed. The underlying idea is to develop a theory similar to ideas in Riemannian geometry for weak solutions of geometrically motivated equations (e.g. Yamabe problem, Ricci flow, bounded Ricci curvature etc). The motivation in Lorentzian geometry is the problem of understanding the solution space of the Einstein equations (even in the simple case of Ricci flat spacetimes, i.e. vacuum Einstein), which still today is very elusive. With the development of a theory of optimal transportation in Lorentzain geometry the question of Gromov-Hausdorff convergence in Lorentzian geometry has been renewed (after previous work by Noldus in the 2000’s). Recent progress on curvature conditions for Lorentzian length spaces and their stability by Kunzinger&Sämann and Cavaletti&Mondino strengthen the need for a good notion of Gromov-Hausdorff convergence in Lorentzian geometry.
In our project we are currently developing such a theory of an intrinsic Gromov-Hausdroff convergence for spacetimes. An extrinsic definition has been given by Cavaletti&Mondino. Our study is in an advanced state. Among other we have already obtained pre compactness theorems for Lorentzian spaces under natural assumptions. The main subjects for the coming weeks is to study the curvature conditions on Lorentzian length spaces and their stability. We will also study the Bishop-Gromov inequality for Lorentzian spaces and its implications for the pre compactness problem (in analogy to Cheeger&Colding) for spacetimes. We are aiming to finish a first draft in September.
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