About the Calabi problem: a finite-dimensional approach

  • H.-D. Cao

    Lehigh University, Bethlehem, USA
  • J. Keller

    Université de Provence, Marseille, France


Let us consider a projective manifold MnM^n and a smooth volume form Ω\Omega on MM. We define the gradient flow associated to the problem of Ω\Omega-balanced metrics in the quantum formalism, the Ω\Omega-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the Ω\Omega-balancing flow converges towards a natural flow in Kähler geometry, the Ω\Omega-Kähler flow. We also prove the long time existence of the Ω\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