Letizia Branca: Rigidity of Einstein manifolds with positive Yamabe invariant

In this talk I will present some optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. I will also present an improvement of the known bounds of the Yamabe invariant via the \(L^{n/2}\)-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions 5 and 6. This is a joint work with Giovanni Catino, Davide Dameno and Paolo Mastrolia.