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Twisted Calabi functional and twisted Calabi flow

Jie He, Haozhao Li

TL;DR

This work develops a comprehensive analytic framework for the twisted Calabi flow on compact Kähler manifolds by introducing the twisted Calabi functional ${\mathcal C}^s$ and showing it has favorable variational properties near twisted cscK metrics. It establishes a first-variation formula linking critical points to twisted cscK metrics, proves short-time existence via a quasi-linear parabolic fixed-point approach, and demonstrates stability in a neighborhood of twisted cscK metrics with exponential convergence. The results highlight a robust gradient-flow structure for the twisted K-energy ${\mathcal M}^s$, including strict convexity along geodesics and isolation of twisted cscK metrics, and provide a bootstrapped path to global existence and convergence under small perturbations. Overall, the paper lays a solid analytic foundation for studying the twisted Calabi flow and its local dynamics, with implications for uniqueness and rigidity of twisted cscK metrics in Kähler geometry.

Abstract

This paper investigates the twisted Calabi functional and the associated twisted Calabi flow on compact Kähler 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 Kähler metrics. These results provide an analytic foundation for studying the twisted Calabi flow and resolve questions about its local behavior.

Twisted Calabi functional and twisted Calabi flow

TL;DR

This work develops a comprehensive analytic framework for the twisted Calabi flow on compact Kähler manifolds by introducing the twisted Calabi functional and showing it has favorable variational properties near twisted cscK metrics. It establishes a first-variation formula linking critical points to twisted cscK metrics, proves short-time existence via a quasi-linear parabolic fixed-point approach, and demonstrates stability in a neighborhood of twisted cscK metrics with exponential convergence. The results highlight a robust gradient-flow structure for the twisted K-energy , including strict convexity along geodesics and isolation of twisted cscK metrics, and provide a bootstrapped path to global existence and convergence under small perturbations. Overall, the paper lays a solid analytic foundation for studying the twisted Calabi flow and its local dynamics, with implications for uniqueness and rigidity of twisted cscK metrics in Kähler geometry.

Abstract

This paper investigates the twisted Calabi functional and the associated twisted Calabi flow on compact Kähler 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 Kähler metrics. These results provide an analytic foundation for studying the twisted Calabi flow and resolve questions about its local behavior.

Paper Structure

This paper contains 26 sections, 26 theorems, 220 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some notations
  3. Main results
  4. Preliminaries
  5. Notations and basic conventions
  6. Twisted canonical metrics
  7. Twisted functionals and flows
  8. The twisted Lichnerowicz operator
  9. Variational theory of the twisted Calabi functional
  10. The twisted Calabi functional and its first variation
  11. Geometric properties of the twisted Calabi flow
  12. Second variation and convexity analysis
  13. Geometric consequences and the twisted $K$-energy
  14. Short-time existence of the twisted Calabi flow
  15. Main existence theorem
  16. ...and 11 more sections

Key Result

Theorem 1.2

Assume $s \in (0,1)$. The first variation of the twisted Calabi functional is given by where ${\mathbb L}^s$ is the twisted Lichnerowicz operator. Consequently, $\omega$ is a critical point of ${\mathcal{C}}^s$ if and only if it is a twisted cscK metric.

Figures (1)

  • Figure 1: Estimate \ref{['bootstrap-bound']} guarantees that the twisted Calabi flow $\varphi(t)$ starting from $B_{\delta_0}^{2,\alpha}$ remains in $\mathcal{V}^{3,\alpha}_{\delta,\lambda}$ for all $t \ge I$. The dashed curve illustrates that a twisted Calabi flow starting from $\mathcal{V}^{2,\alpha}_{\delta,\lambda}$ stays within $\mathcal{V}^{2,\alpha}_{\epsilon,\lambda}$ for $t \in [0,2I]$, but may leave $\mathcal{V}^{2,\alpha}_{\epsilon,\lambda}$ after $2I$.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1.2: \ref{['thm:first-variation']}
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 58 more