Given a Riemannian manifold (M, g), the Riemann curvature tensor encodes all the information about the curvature. It is well known that in dimension n ≥ 3, it splits as a sum of three different parts. Many of the so-called canonical metrics arise as critical points of suitable Lp-norms of the components of the curvature tensor. Throughout the talk, we will provide some background on Riemannian manifolds and discuss some known results and open problems concerning Riemannian functionals and critical metrics in dimension 4 and beyond.
The last part of the talk is based on a joint work with G. Catino, D. Dameno and P. Mastrolia.