Twisted Calabi Flow in Kähler Geometry

• Twisted Calabi flow is a fourth-order parabolic flow on the space of Kähler potentials that interpolates between the J-flow and classical Calabi flow.
• It defines a twisted Calabi functional whose gradient flow characterizes twisted cscK metrics through strict convexity and spectral gap analysis.
• Analytic techniques including parabolic theory, spectral gap estimates, and weak metric flows ensure long-time existence, stability, and convergence.

The twisted Calabi flow is a one-parameter family of fourth-order parabolic flows on the space of Kähler potentials over a compact Kähler manifold, interpolating between the J-flow and the classical Calabi flow. This construction links the analytic and geometric study of extremal metrics, including constant scalar curvature Kähler (cscK) and twisted cscK metrics, and provides a deformation theory for exploring long-time existence and convergence in complex geometry.

1. Definition and Fundamental Properties

Let $(M^n, \omega)$ be a compact Kähler manifold. The space of smooth Kähler potentials is defined by

$$\mathcal{H}_\omega := \{\, \varphi \in C^\infty(M) : \omega_\varphi = \omega + i\partial\bar\partial\varphi > 0 \, \},$$

often normalized by $\int_M \varphi \,\omega^n = 0$.

Given a parameter $s \in [0,1]$, the twisted Calabi flow is defined via the evolution equation for a time-dependent Kähler potential $\varphi(t)$: $$\frac{\partial \varphi}{\partial t} = s\bigl(R(\omega_\varphi) - \underline{R}\bigr) + (1-s)\bigl(n - \operatorname{tr}_{\omega_\varphi}\omega\bigr),$$ where $R(\omega_\varphi)$ is the scalar curvature of $\omega_\varphi$, $$\underline{R} = \frac{1}{V} \int_M R(\omega_\varphi)\, \omega_\varphi^n$$ is the average scalar curvature (fixed in time), and $\operatorname{tr}_{\omega_\varphi}\omega$ is the trace of the reference form with respect to $\omega_\varphi$.

Alternatively, the evolution equation is written as $L_s(\varphi) = 0$ with

$$L_s(\varphi) = \dot\varphi - s\,(R(\omega_\varphi) - \underline{R}) - (1-s)\,(n - \operatorname{tr}_{\omega_\varphi}\omega).$$

For $s=0$ the twisted Calabi flow reduces to the J-flow (Chen–Donaldson flow) and for $s=1$ it coincides with the classical Calabi flow. This interpolation is central to the continuity method for analyzing the Calabi flow on cscK manifolds (He et al., 4 Dec 2025).

2. Twisted Calabi Functional, Gradient Flow, and Variational Structure

The twisted Calabi functional, parameterized by $s \in [0,1]$, is defined as

$$C^s(\omega) = \int_M \left(R^s(\omega) - \underline{R}^s\right)^2 \omega^n,$$

where

$$R^s(\omega) = s\,R(\omega) - (1-s)\,\operatorname{tr}_\omega \chi, \quad \underline{R}^s = s\,\underline{R} - (1-s)\,\underline{\chi},$$

for a fixed Kähler form $\chi$.

The gradient flow of $C^s$ with respect to the $L^2$-metric on potentials yields the (alternative) twisted Calabi flow: $$\frac{\partial \varphi}{\partial t} = R^s(\omega_\varphi) - \underline{R}^s.$$ Critical points of $C^s$—twisted cscK metrics—are characterized by $R^s(\omega_\varphi) = \underline{R}^s$ (He et al., 2 Dec 2025).

The first variation and Hessian analysis show that $C^s$ is strictly convex at its critical points, with the Hessian given by the $L^2$-norm of the twisted Lichnerowicz operator: $$\mathcal{L}^s(f) = s(\Delta^2 f + \mathrm{Ric}^{\alpha\bar\beta} f_{\alpha\bar\beta}) - (1-s) i\bar\partial^*(\nabla^{1,0} f \lrcorner \chi),$$ guaranteeing isolated minima and stability properties near twisted cscK metrics (He et al., 2 Dec 2025).

3. Analytic Theory: Existence, Openness, and Stability

Assume $(M,\omega)$ admits a cscK metric. Given any initial Kähler potential $\psi_0$, there exists $s_0>0$ such that for $s \in (0,s_0)$, the twisted Calabi flow admits a global (for all $t \ge 0$) solution $\varphi_s(t)$ with exponential convergence to the cscK potential as $t \to \infty$ [(He et al., 4 Dec 2025), Theorem 1.2].

Moreover, an openness theorem holds: if for some $s_*$ the flow $L_{s_*}(\varphi_{s_*})=0$ converges for all time, then there is a neighborhood of $s_*$ where the same is true for nearby $s$ (Theorem 1.3). Thus, for each initial data, the set

$I_{\psi_0} = \{\, s \in [0,1] : \text{the %%%%31%%%%-flow exists globally and converges from %%%%32%%%%} \,\}$

is open (He et al., 4 Dec 2025).

Stability results for the flow in a neighborhood of a twisted cscK metric establish a uniform spectral gap for the linearized operator, exponential decay of the energy $$\mathcal{E}(t) = \int (R^s(t) - \underline{R}^s)^2 \omega^n_t$$, and a priori $C^{3,\alpha}$-estimates, providing dynamical stability and exponential convergence for sufficiently small perturbations (He et al., 2 Dec 2025).

4. Methodology: Continuity and Analytic Techniques

The continuity method involves deforming the parameter $s$ from $0$ to $1$, leveraging the well-understood J-flow at $s=0$ and attempting to transfer existence and convergence to Calabi flow at $s=1$.

• Finite-Time Existence: For small $s$, a formal Taylor expansion in $s$ produces approximate solutions, with residuals controlled by higher powers of $s$. Linear parabolic theory (Schauder estimates for equations of the form $\dot{u} + \Delta^2 u = \cdots$) yields invertibility and a true local solution via contraction mapping in, e.g., $C^{4,1}$-norms (He et al., 4 Dec 2025).
• Global Existence Past Proximity to cscK: Once the solution is sufficiently close to the cscK metric, the flow can be extended for $t \to \infty$ with exponential decay, using weighted Hölder norms and bootstrapping arguments.
• Openness in $s$: Replacing formal approximations with genuine solutions at $s_*$ and repeating the linear theory plus contraction-mapping argument guarantees openness of global existence with respect to $s$.
• Spectral Gap and Functional Analysis: A priori $L^2$-estimates, spectral gaps for linearizations, and interpolation inequalities as $s\to 0$ are central to the analysis. Weighted Hölder spaces and parabolic regularity are used to control solution norms in both finite and infinite time intervals (He et al., 4 Dec 2025, He et al., 2 Dec 2025).

5. Metric and Variational Aspects in Weak and Smooth Settings

In the metric completion $\mathcal{E}^p$ of the space of Kähler potentials with respect to $L^p$-path metrics, the (twisted) Mabuchi K-energy functional extends as a $d_p$-lower semicontinuous convex functional along finite-energy geodesics. The twisted Calabi flow admits a weak (minimizing-movement) formulation in the CAT(0) metric space $(\mathcal{E}^2, d_2)$, ensuring existence of unique continuous flows for all $t \ge 0$. When smooth solutions exist, they coincide with this weak flow (Berman et al., 2015).

For the weak flow, a long-time convergence dichotomy holds: either the trajectory diverges in $d_2$ or it $d_1$-converges to a minimizer of the K-energy, the latter confirming part of Donaldson’s conjectural picture for cscK metrics. In the case of positive twist ($\chi >0$), the twisted cscK potential is unique up to constants in the metric completion, and the behavior of diverging flows can be related to destabilizing geodesic rays in $\mathcal{E}^1$ (Berman et al., 2015).

6. Implications, Open Conjectures, and Future Directions

A key consequence of the openness results is that the existence and convergence to cscK metrics for the twisted Calabi flow is realized on an interval $[0, s_0)$ and forms an open subset of $[0,1]$. If this set is shown to be closed, one would conclude global existence and convergence for Calabi flow for all initial data on backgrounds admitting cscK metrics. Establishing closedness would likely require new compactness, blow-up analysis, or Gromov–Hausdorff-type arguments. This would resolve the major open conjecture attributed to X.X. Chen on the long-time existence of the Calabi flow (He et al., 4 Dec 2025).

The analytic techniques underlying the twisted Calabi flow also extend to higher-order deformation theories, such as flows associated with extremal or conical metrics. A strong local and semi-global analytic framework has now been established, including short-time existence, convexity at minimizers, uniform spectral gaps, stability, and energetic contraction properties (He et al., 4 Dec 2025, He et al., 2 Dec 2025).

A plausible implication is that as more quantitative curvature or energy-functional bounds become available, they will combine with continuity arguments to establish the full continuity path and convergence for broader classes of initial potentials and geometric backgrounds.

7. References to Key Results and Tools

Topic	Main Source (arXiv)	Essential Results/Methods
Existence, continuity, deformation	(He et al., 4 Dec 2025)	Existence and openness theorems, continuity method, analytic framework
Twisted Calabi functional, stability	(He et al., 2 Dec 2025)	Convexity, spectral gap, stability, short-time existence
Metric-variational theory (weak flow)	(Berman et al., 2015)	Metric structures, minimizing flow, asymptotic convergence

Key analytic ingredients include parabolic Schauder theory for fourth-order operators, precise interpolation and functional inequalities in parabolic Hölder and weighted spaces, and careful analysis of the twisted Lichnerowicz operator. Proofs and further technical details are consolidated in the cited papers, and auxiliary results—such as interpolation and metric equivalence—are collected in the cited appendices (He et al., 4 Dec 2025, He et al., 2 Dec 2025, Berman et al., 2015).