# About the Calabi problem: a finite-dimensional approach

### H.-D. Cao

Lehigh University, Bethlehem, USA### J. Keller

Université de Provence, Marseille, France

## Abstract

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 volume form in a given Kähler class (see Theorem 2). We derive some natural geometric consequences of our study.

## Cite this article

H.-D. Cao, J. Keller, About the Calabi problem: a finite-dimensional approach. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 1033–1065

DOI 10.4171/JEMS/385