Abstract:

Eugenio Calabi had a dream: that one could find in each Kähler class of a Kähler manifold a metric that could be called "canonical". He chose to try and find these metrics among those which minimise a certain functional. This functional is tightly linked to the scalar curvature of the manifold; then, it turns out that metrics with costant scalar curvature are indeed canonical. We'll make an overview of the constant scalar curvature problem and of the recent breakthroughs (most notably, the one of Chen and Cheng linking existence of cscK metrics to geodesic stability).