# About the Calabi problem: a finite-dimensional approach

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 3, page 1033-1065
- ISSN: 1435-9855

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topCao, 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 -

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