%0 Journal Article
%T About the Calabi problem: a finite-dimensional approach
%+ Department of Mathematics (Lehigh University)
%+ Institut de Mathématiques de Marseille (I2M)
%A Cao, H.-D.
%A Keller, Julien
%< avec comité de lecture
%@ 1435-9855
%J Journal of the European Mathematical Society
%I European Mathematical Society
%V 15
%N 3
%8 2013
%D 2013
%Z 1102.1097
%R 10.4171/JEMS/385
%K moment map
%K convergence
%K Calabi-Yau
%K Monge-Ampère
%K manifold
%K projective
%K quantization
%K flow
%K PDE
%Z 32Q20, 53C44, 53D20, 53D50
%Z Mathematics [math]/Differential Geometry [math.DG]
%Z Mathematics [math]/Complex Variables [math.CV]Journal articles
%X Let us consider a projective manifold and Ω a volume form. We define the gradient flow associated to the problem of Ω-balanced metrics in the quantum formalism, the \Omega−balacing flow.At the limit of the quantization,we prove that the \Omega$-balancing flow converges towards a natural flow in K\"ahler geometry, the$\Omega$-K\"ahler flow. We study the existence of the$\Omega$-K\"ahler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given K\"ahler class. We derive some natural geometric consequences of our study.
%G English
%L hal-01282429
%U https://hal.science/hal-01282429
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ INSMI
%~ I2M
%~ I2M-2014-
%~ AMIDEX
%~ ANR