About the Calabi problem: a finite dimensional approach
Let us consider a projective manifold and $Ω$ a volume form. We define the gradient flow associated to the problem of $Ω$-balanced metrics in the quantum formalism, the Ω$-balacing flow. At the limit of the quantization, we prove that the $Ω$-balacing flow converges towards a natural flow in Kähler geometry, the $Ω$-Kähler flow. We study the existence of the $Ω$-Kähler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given Kähler class. We derive some natural geometric consequences of our study.
math.DG↗