06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Thomas Schick (Göttingen), Bernhard Hanke (Augsburg)
Thomas Schick (Göttingen): The topology of positive scalar curvature
Given your favorite smooth manifold M (which should be compact and without boundary), e.g the seven-dimensional sphere, you'd like to figure out what kind of Riemannian metric one can equip M with. Seen as a whole, the space of all such metrics is boring, it is just an infinite dimensional cone.
We will concentrate on the subset of metrics of positive scalar curvature. As discussed in Bernhard Hanke's colloquium talk, sometimes this space is empty. If it is not empty, it is an infinite-dimensional manifold. But what are its topological properties? Does it have different path components (from the point of view of the sphere: are there "exotic" metrics of positive scalar curvature beyond the standard metric)? Are the path-components contractible? Even more interesting are these questions for the moduli space of isometry classes.
In the talk, we will discuss a number of old and recent results whose combination says that essentially in all cases the structure of the space of metrics and of the moduli space is very rich and interesting: infinitely many path components, which themselves have infinitely many non-trivial homotopy groups,...
This requires the construction of metrics of positive scalar curvature, which uses tools from algebraic topology. To detect the difference of such metrics, we rely on index theory of Dirac operators.
Bernhard Hanke (Augsburg): Obstructions to positive scalar curvature from submanifolds
Scalar curvature measures the asymptotic volume growth of small balls in Riemannian manifolds. In the case of positive scalar curvature the growth rate is less than in Euclidean space. Spheres of dimension at least two are a model example for this.
We will explain how the geometry of submanifolds of low codimension can obstruct the existence of Riemannian metrics of positive scalar curvature on the ambient manifold.