About the Calabi problem: a finite dimensional approach

TL;DR

This paper introduces a finite-dimensional approach to the Calabi problem by defining a gradient flow for alancing metrics, showing its convergence to a flow solving the volume form prescription in Ka4hler geometry.

Contribution

It establishes a new finite-dimensional gradient flow framework that converges to a natural Ka4hler flow solving the Calabi problem.

Findings

The alancing flow converges to the alancing flow in the limit of quantization.

The alancing flow exists long-term and converges to the solution of the Calabi problem.

The approach provides geometric insights into the Calabi problem via a finite-dimensional perspective.

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 \Omega$-balacing flow. At the limit of the quantization, we prove that the $\Omega$-balacing 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.