Besides the existence question, it has become more and more popular to study the homotopy type of the space of all psc-metrics R+(M).
In the first part of this talk, I will give the construction of a cubical set
that naturally contains all psc-metrics as zero-cubes and present some of its properties.
In the second part I will outline how this space might help to tackle the "isotopy-versus-concordance" question.
This is part of my ongoing PhD project and work in progress.
Thorsten Hertl, Göttingen
Cubical Approximations for Positive Scalar Curvature Metrics
Scalar curvature is a local invariant of a Riemannian manifold. Roughly said, it measures the volume growth of geodesic balls asymptotically.
Although scalar curvature is a fairly weak local invariant of a Riemannian metric, there are still topological obstructions for a manifold to
have a metric with positive scalar curvature (psc-metrics).How does total mass affect spatial geometry?
In mathematical general relativity, the ADM mass of an isolated gravitational
system is a geometric invariant measuring the total mass due to matter and
other fields present in spacetime.
The celebrated Positive Mass Theorem (of Schoen-Yau and Witten)
states that this invariant is non-negative and vanishes only for flat spacetime. In recent work, we showed how to compute ADM mass in 3 spatial dimensions by studying harmonic functions.
I will explain this how that works,
then use the resulting formula to consider the following
question: How flat is an "asymptotically flat"
space with very little total mass? The existence of wormholes and
gravity wells make this question subtle.
We make progress on this problem and partially
confirm conjectures made by Huisken-Ilmanen and Lee-Sormani.
Coarse embeddings and isometric actions of discrete groups
The aim of the talk is to show how the notion of coarse embedding between metric spaces, which was
introduced by M. Gromov in the 80's, allows to understand better actions of discrete groups preserving rigid geometric structures.
We will emphasize the case of the isometry group of Lorentzian manifolds, proving a Tits alternative for those groups.
Back to the plan of the ArbeitsgruppenseminarAbstracts of the Seminar on Advanced topics in Geometric Analysis
Prof. Bernd Ammann und Mitarbeiter, Zimmer 119