Let us consider a projective manifold and a smooth volume form on . We define the gradient flow associated to the problem of -balanced metrics in the quantum formalism, the -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the -balancing flow converges towards a natural flow in Kähler geometry, the -Kähler flow. We also prove the long time existence of the -Kähler flow and its convergence towards Yau's solution to the Calabi conjecture of prescribing the volume form in a given Kähler class (see Theorem 2). We derive some natural geometric consequences of our study.
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H.-D. Cao, J. Keller, About the Calabi problem: a finite-dimensional approach. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 1033–1065DOI 10.4171/JEMS/385