Existence of twisted Calabi flow and deformation from the $J$-flow to Calabi flow

TL;DR

This paper investigates the twisted Calabi flow on compact Kähler manifolds, demonstrating long-term existence and convergence to constant scalar curvature metrics, thus supporting the continuity method for Chen's conjecture.

Contribution

It establishes the existence and convergence of twisted Calabi flows near the $J$-flow, and shows the openness of the continuity method for studying Calabi flow on cscK manifolds.

Findings

Twisted Calabi flow exists long-term and converges to cscK metrics.

Long-term convergence is stable under small perturbations.

Supports the continuity method for Chen's conjecture.

Abstract

In this paper, we study a family of twisted Calabi flows connecting the $J$-flow and Calabi flow on a compact K\"ahler manifold with a constant scalar curvature (cscK) metric. We show that for any initial data the twisted Calabi flow near the $J$-flow has long time existence and converges smoothly to the cscK metric. Moreover, we show that if a twisted Calabi flow has long time existence and converges, then the nearby twisted Calabi flow with the same initial data also has long time existence and converges. These results imply the openness of the continuity method to study Chen's long time existence conjecture on (twisted) Calabi flow on cscK manifolds.