Twisted Calabi functional and twisted Calabi flow

Abstract

This paper investigates the twisted Calabi functional and the associated twisted Calabi flow on compact K\"ahler manifolds. Our main contributions are threefold: first, we establish the convexity of the twisted Calabi functional at its critical points; second, we prove the short-time existence of the twisted Calabi flow; and third, we demonstrate the stability of this flow in the neighborhood of twisted constant scalar curvature K\"ahler metrics. These results provide an analytic foundation for studying the twisted Calabi flow and resolve questions about its local behavior.