27 nov 2019 -- 15:30
Aula Tricerri, DiMaI, Firenze
Arguably, the HyperK\"ahler with torsion metrics, HKT for short, constitute the right analogue of K\"ahler metrics for the hypercomplex manifolds. Building on the previous work of Verbitsky, on HKT manifolds, and Alesker, on a quaternionic Monge-Amp\`ere equation, a conjecture about the existence of HKT metrics with a prescribed volume form has been posted by them. It reduces to solving a certain second order PDE of an interesting nature coming from considering not one but two complex structures. Recent advances towards proving the conjecture and its consequences for the existence of extremal HKT metrics will be presented.