About the Calabi problem: a finite dimensional approach

Let us consider a projective manifold and $\mathrm{\Omega }$ a volume form. We define the gradient flow associated to the problem of $\mathrm{\Omega }$-balanced metrics in the quantum formalism, the \Omega$-balacing flow. At the limit of the quantization, we prove that the $\mathrm{\Omega }$-balacing flow converges towards a natural flow in K"ahler geometry, the $\mathrm{\Omega }$-K"ahler flow. We study the existence of the $\mathrm{\Omega }$-K"ahler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given K"ahler class. We derive some natural geometric consequences of our study.