Project B3: Geometry of scalar curvature
SummaryCurvatures have local and global influences on various properties of the underlying space. In the case of sectional curvature studying the impact of curvature constraints makes up whole branches in mathematics and reaches from topology, dynamics up to group theory. On the other hand, in physics, for example in General Relativity, scalar and Ricci curvature are more relevant than sectional curvature.
However scalar curvature which measures the local volume growth of distance balls around a point within some Riemannian manifold (M, g) gauged relative to the Euclidean case (Rn, gEucl) (where this quantity is set to zero) has quite less local but a subtle not yet well-understood global impact. And the results demanded from physics are mostly such global ones.
As we will discuss later on many structural results are related to obstructions for the existence of positive scalar curvature metrics. One important (and historically this was also the first) tool that is able to detect or name such obstructions uses extensions of the Lichnerowicz formula (and the Atiyah-Singer Index Theorem). However these methods exclusively apply to spin manifolds while the generic case (in dimension ≥ 5) is that of non-spin manifolds.
For the general case (i.e. without spin assumption) there is presently only one alternative: an inductive analysis using minimal hypersurfaces. Such hypersurfaces can be found within any given integral homology class of the manifold whose scalar curvature one wants to understand. They are capable to gather or capture positive scalar curvature from their ambience (into their own one dimension smaller world) and thus via induction they can carry some information of the topology and scalar curvature geometry of the original space down to a lower dimensional space where scalar curvature is already better understood.
Compared to the approach via Index theory (in spin geometry) the challenges are now very different in nature: understanding the analysis of such minimal hypersurfaces can be a delicate task and also they are less adapted to produce topological invariants (but different from the first approach they are also sensitive to the local geometry). As for the analysis we only mention the classical stumbling block: beyond dimension 8 these hypersurface may contain singularities complicated enough to inhibit the mentioned induction scheme. This problem was resolved only very recently.
The major aim of this long term project is to develop a theory for the geometry of scalar curvature: compared to the advanced state of development for other curvature notions the study of geometry of scalar curvature (in particular when we drop the spin assumption) is still in its infancy.
The starting point will be a deepening of our understanding of the analysis of/on minimal hypersurfaces and their relation to topology. In particular we aim at an obstruction theory, new and more general structure- and existence results based on these and certainly also other techniques.