Vortrag 'Positive versus non-negative scalar curvature'

 Tee um 10.30 im runden Zimmer, Vortrag um 11.00

Title: Positive versus non-negative scalar curvature.

Abstract:  Geometers, analysts, and topologists have intensely studied the question, whether a closed smooth manifold M does admit a Riemannian metric of positive scalar curvature or not  (as one example of the question how much the topology does determine the possible geometries), and also how rich the space of all such metrics is.

Most prominently, using the spectral theory of the Dirac operator, if the A-hat genus is non-zero and M is spin, no such metric exists. This (and generalizations) definitely use the positivity of the scalar curvature function. What about the borderline case: can we also rule out non-negative scalar curvature? What is the relation between the space of metrics with non-negative scalar curvature and its subspace of strictly positive scalar curvature? The Dirac operator gives information about the latter space well beyond the question whether it is empty or not.

We can give a fairly complete answer to the last question: except for some hypothetical mystery metrics which might be Ricci flat without special holonomy, the inclusion is a homotopy equivalence; and the Dirac operator methods are not really effected by the hypothetical special metrics. This is joint work with
David Wraith.

In the talk, we want to explain the basics of the Dirac operator method and also sketch the geometric input (from Ricci flow to special holonomy considerations) we borrow to arrive at the desired comparison.