Topics in Geometric Analysis

04.07. bis 04.07.2019, 16.15  –  University Potsdam, Room 2.22
Seminar "Topics in Geometric Analysis"

Armando Cabrera Pacheco, Bruno Premoselli

16:15Armando Cabrera Pacheco Extensions of minimal Bartnik data and the Bartnik mass

The Bartnik mass is an important notion of quasi-local mass in mathematical relativity, although it has the drawback of being difficult to compute. Mantoulidis and Schoen constructed asymptotically flat extensions of a certain type of minimal Bartnik data, which allowed them to compute their Bartnik mass. In this talk, we will describe this type of extensions and discuss how they can be used to address some questions related to some conjectures regarding the Bartnik mass. This talk is based on a joint project with C. Cederbaum.
17:45 Bruno Premoselli Examples of Compact Einstein four-manifolds with negative curvature

We construct new examples of closed, negatively curved Einstein four-manifolds. More precisely, we construct Einstein metrics of negative sectional curvature on ramified covers of compact hyperbolic four-manifolds with symmetries, initially considered by Gromov and Thurston. These metrics are obtained through a deformation procedure. Our candidate approximate Einstein metric is an interpolation between a black-hole Riemannian Einstein metric near the branch locus and the pulled-back hyperbolic metric. We then deform it into a genuine solution of Einstein’s equations, and the deformation relies on an involved bootstrap procedure. Our construction yields the first example of compact Einstein manifolds with negative sectional curvature which are not locally homogeneous. This is a joint work with J. Fine (ULB, Brussels).
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