### About the Calabi problem: a finite-dimensional approach

Let us consider a projective manifold ${M}^{n}$ and a smooth volume form $\Omega $ on $M$. We define the gradient flow associated to the problem of $\Omega $-balanced metrics in the quantum formalism, the $\Omega $-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the $\Omega $-balancing flow converges towards a natural flow in Kähler geometry, the $\Omega $-Kähler flow. We also prove the long time existence of the $\Omega $-Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...