Geodesic
About the Calabi problem: a finite-dimensional approach
Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1033-1065

Voir la notice de l'article provenant de la source EMS Press

Let us consider a projective manifold Mn and a smooth volume form Ω on M. We define the gradient flow associated to the problem of Ω-balanced metrics in the quantum formalism, the Ω-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the Ω-balancing flow converges towards a natural flow in Kähler geometry, the Ω-Kähler flow. We also prove the long time existence of the Ω-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.
DOI : 10.4171/jems/385
Classification : 53-XX, 32-XX, 00-XX
Keywords: Calabi problem, Balanced metrics, canonical flow, Kähler geometry, moment map, Bergman kernel, asymptotics, quantization 
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     title = {About the {Calabi} problem: a finite-dimensional approach},
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H.-D. Cao; J. Keller. About the Calabi problem: a finite-dimensional approach. Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1033-1065. doi: 10.4171/jems/385

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