Vorträge in der Woche 25.04.2016 bis 01.05.2016

Donnerstag, 28.04.2016: Mass-like invariants for asymptotically hyperbolic manifolds

Dr. Julien Cortier (Université Joseph Fourier, Grenoble)

In this talk, we are interested in Riemannian manifolds which have an end whose geometry is asymptotic to the hyperbolic space geometry. Such situations occur in general relativity for some slices of asymptotically anti-de Sitter (adS) spacetimes. Similar to the Euclidean case, it is possible to define global quantities (mass and center of mass) under suitable decay rate assumptions. These quantities enjoy "asymptotic invariance" properties that we will review, and for which the group PO(n,1) of isometries of the hyperbolic space plays a central role. We will then see how to construct other such asymptotic invariants when we relax the assumption on the decay rate. They are attached to finite dimensional representations of PO(n,1). We shall finally see how every such invariant is naturally linked to a curvature operator (e.g. the scalar curvature for the classical mass). This is based on a joint work with Mattias Dahl and Romain Gicquaud.

Donnerstag, 28.04.2016: New perspectives on Mustafin Varieties

Marvin Anas Hahn (Universität Tübingen)

Donnerstag, 28.04.2016: On self gravitating solutions of the Einstein-Scalar field equations.

Dr. Martín Reiris (Universidad de la República del Uruguay)

We will discuss the existence of geodesically complete solutions of the Einstein-Scalar field equations in arbitrary dimensions depending on the form of the scalar field potential $V(\phi)$. As a main special case it will be shown that when $V(\phi)$ is the Klein-Gordon potential, i.e. $V(\phi)=m^{2}|\phi|^{2}$, geodesically complete solutions are necessarily Ricci-flat, have constant lapse and are vacuum, (that is $\phi=\phi_{0}$ with $\phi_{0}=0$ if $m\neq 0$). Therefore, if the spatial dimension is three, the only such solutions are either Minkowski or a quotient thereof. For $V(\phi)=m^{2}|\phi|^{2}+2\Lambda$, that is, including a vacuum energy or a cosmological constant, it will be shown that no geodesically complete solution exists when $\Lambda>0$, whereas when $\Lambda<0$ it is proved that no non-vacuum geodesically complete solution exists unless $m^{2}<-2\Lambda/(n-1)$, ($n$ is the spatial dimension) and the manifold is non-compact. The proofs are based on techniques in comparison geometry ?a la Backry-Emery.