Thomas Tony
Llarull proved in the late '90s that the round n-sphere is area-extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized Llarull's rigidity statement to certain area-non-increasing spin maps \(f: M\to N\) of non-zero \(\hat{A}\)-degree.
In this talk, I give a brief introduction to scalar curvature comparison geometry and explain how higher index theory can be used in this context. More specifically, I present a recent generalization of Goette and Semmelmann’s theorem, in which the topological condition on the \(\hat{A}\)-degree is replaced by a weaker condition involving the so-called higher mapping degree. A key challenge in the proof is that a non-vanishing higher index does not necessarily give rise to a non-trivial kernel of the corresponding Dirac operator. I will present a new method that extracts geometrically useful information even in this more general setting.
