Recently, the paper “Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow,” coauthored by Dr. Jie He of our college and Professor Haozhao Li of the University of Science and Technology of China, was published in theJournal of Functional Analysis, a leading international mathematics journal (Vol. 291, 2026, Article 111634). Beijing University of Chemical Technology is the first-listed affiliation, and Jie He is the first author. The research was supported by the Tianyuan Fund for Mathematics of the National Natural Science Foundation of China and other funding programs.
Constant scalar curvature Kähler (cscK) metrics form an important class of canonical metrics in Kähler geometry. Introduced by the distinguished mathematician Eugenio Calabi, the Calabi flow is a geometric evolution equation used to seek such metrics. Its fourth-order and highly nonlinear nature makes the long-time existence and eventual convergence of solutions from arbitrary smooth initial data a major problem in complex differential geometry and geometric analysis. The related J-flow is a second-order geometric flow whose long-time behavior has been studied extensively. The twisted Calabi flow connects the J-flow and the Calabi flow through a deformation parameter, with the two flows corresponding to opposite endpoints of the parameter interval. This provides a natural framework for studying the Calabi flow using the continuity method.
In this work, the authors study the twisted Calabi flow on compact Kähler manifolds admitting cscK metrics. They first prove that, for any given initial Kähler potential, the twisted Calabi flow exists for all time and converges exponentially to a cscK metric when the deformation parameter is sufficiently close to the J-flow endpoint. They further show that if the twisted Calabi flow at a given parameter value exists for all time and converges smoothly, then flows with the same initial data at all sufficiently nearby parameter values also exist for all time and converge exponentially. Consequently, the set of parameters for which long-time existence and convergence hold contains a nontrivial interval starting at the J-flow endpoint and is open in the parameter space [0, 1].
To obtain these results, the authors develop a parabolic deformation theory for the twisted Calabi flow with respect to the deformation parameter. By constructing higher-order approximate solutions, studying the invertibility of the fourth-order linearized parabolic operator, and applying Schauder estimates and the contraction mapping principle, they establish stability under parameter variations on finite time intervals. They also introduce weighted Hölder spaces incorporating exponential decay to establish long-time stability near cscK metrics, uniformly for small values of the deformation parameter. Combining the long-time convergence of the J-flow with finite-time deformation and stability near critical metrics, they extend the conclusions to arbitrary initial Kähler potentials.
This work establishes a rigorous starting point at the J-flow endpoint and an openness result for studying the twisted Calabi flow and the Calabi flow. It provides a new analytical approach, through the continuity method, to Xiuxiong Chen’s conjecture on long-time existence and Donaldson’s conjecture on the asymptotic behavior of the Calabi flow. It also lays the groundwork for further study of the closedness of the parameter set and of long-time existence and convergence at the Calabi flow endpoint.
Paper link:https://doi.org/10.1016/j.jfa.2026.111634
About the author:Jie He received his Ph.D. in mathematics in 2018 and is a lecturer in the School of Mathematics and Physics at Beijing University of Chemical Technology. His research interests lie in two areas: quasilinear partial differential equations on manifolds, and moment maps and the associated geometric flows in complex geometry. His work has appeared in theJournal of Functional Analysis,Mathematische Zeitschrift,Calculus of Variations and Partial Differential Equations,Journal of Geometric Analysis, andJournal of Differential Equations, among other journals.