About the Calabi problem: a finite dimensional approach

Abstract

Let us consider a projective manifold and $\Omega$ a volume form. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the Ω$-balacingflow.Atthelimitofthequantization,weprovethatthe$Ω$-balacingflowconvergestowardsanaturalflowinK\ddot{a}hlergeometry,the$Ω$-K\ddot{a}hlerflow.Westudytheexistenceofthe$Ω$-Kähler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given Kähler class. We derive some natural geometric consequences of our study.

38 pages. Revised version with improved exposition

Cited by in corpus (1)

• On the geometric flows solving Kählerian inverse ${\sigma }_k$ equations