Existence of twisted Calabi flow and deformation from the $J$-flow to Calabi flow
Jie He, Haozhao Li
TL;DR
The paper studies a twisted Calabi flow that interpolates between the $J$-flow and Calabi flow on a compact Kähler manifold with a cscK metric. The authors develop a two-pronged approach: first, near the $J$-flow, they prove finite-time existence and convergence to a twisted cscK metric by constructing high-order approximate solutions and applying a contraction mapping to the linearized problem; second, near the cscK metric they establish stability for small twisting parameter $s$ using weighted Hölder spaces and invertible linearized operators to obtain global existence and exponential convergence. They then combine these results to show openness of Chen's continuity method for twisted Calabi flow and demonstrate that, starting from a cscK background, small twisting parameters yield global existence and convergence to the cscK metric, with a key step showing exponential decay of the $J$-flow. The work provides a robust framework for studying long-time existence and convergence of twisted Calabi flows and shows how openness of the continuity method can be established in this geometric-analytic setting.
Abstract
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.