About the Calabi problem: a finite-dimensional approach

Cao, H.-D., and Keller, J.. "About the Calabi problem: a finite-dimensional approach." Journal of the European Mathematical Society 015.3 (2013): 1033-1065. <http://eudml.org/doc/277573>.

@article{Cao2013,
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.},
author = {Cao, H.-D., Keller, J.},
journal = {Journal of the European Mathematical Society},
keywords = {Calabi problem; Balanced metrics; canonical flow; Kähler geometry; moment map; Bergman kernel; asymptotics; quantization; Calabi problem; balanced metric; Bergman kernel; moment map},
language = {eng},
number = {3},
pages = {1033-1065},
publisher = {European Mathematical Society Publishing House},
title = {About the Calabi problem: a finite-dimensional approach},
url = {http://eudml.org/doc/277573},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Cao, H.-D.
AU - Keller, J.
TI - About the Calabi problem: a finite-dimensional approach
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 3
SP - 1033
EP - 1065
AB - 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.
LA - eng
KW - Calabi problem; Balanced metrics; canonical flow; Kähler geometry; moment map; Bergman kernel; asymptotics; quantization; Calabi problem; balanced metric; Bergman kernel; moment map
UR - http://eudml.org/doc/277573
ER -