Parole chiave

• quaternionic plurisubharmonic functions
• quaternionic Monge-Ampère equation

Abstract Alesker initiated the study of quaternionic plurisubharmonic functions and used them as a tool to solve the Dirichlet problem for the quaternionic Monge-Ampère equation on quaternionic domains. In 2010, Alesker and Verbitsky extended this problem to certain hypercomplex manifolds, motivated by the search for special Riemannian metrics. The quaternionic Monge-Ampère equation belongs to a broader family of elliptic PDEs of quaternionic type, developed in the spirit of the seminal work of Caffarelli, Nirenberg, and Spruck. In this talk, we will discuss the conjecture of Alesker and Verbitsky, its geometric significance, and survey the current state of the art. Finally, I will present a solvability result for the aforementioned class of equations on compact manifolds under the assumption of the existence of a hyperkähler metric.

This is based on joint work with Luigi Vezzoni.