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

In this paper, we study a family of twisted Calabi flows connecting the $J$-flow and Calabi flow on a compact Kähler 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.