On the convergence of the Calabi flow

TL;DR

This paper proves that on a compact Kähler manifold without holomorphic vector fields, the Calabi flow converges to a constant scalar curvature metric, assuming it exists indefinitely with bounded Ricci curvature.

Contribution

It establishes convergence of the Calabi flow under specific conditions, advancing understanding of geometric flows in Kähler geometry.

Findings

Calabi flow converges to constant scalar curvature metric under given assumptions

Bounded Ricci curvature is sufficient for convergence

Results apply to manifolds without holomorphic vector fields

Abstract

Suppose there is a constant scalar curvature metric on a compact Kahler manifold without holomorphic vector field. We prove that the Calabi flow, if it is assumed to exist for all time with bounded Ricci curvature, will converge to the constant scalar curvature metric.