An improvement of regularity result for pseudo Calabi flow

1. Problem Statement and Context

Background:
The search for canonical metrics in Kähler geometry, specifically Constant Scalar Curvature Kähler (cscK) metrics, is a central theme. The Calabi flow is a parabolic evolution equation proposed by Calabi to find such metrics. However, the Calabi flow is a fourth-order fully nonlinear parabolic equation, making its analytical study extremely difficult. Standard a priori estimates (like those developed by Chen and Cheng for the elliptic cscK equation) do not directly apply to the flow.

The Pseudo-Calabi Flow (PCF):
To circumvent the difficulties of the fourth-order flow, Chen and Zheng introduced the Pseudo-Calabi Flow (PCF), a formally second-order equation:
$$\partial_t \phi = \Delta_\phi^{-1}(R - R(\omega_\phi))$$
where $\omega_\phi = \omega_0 + i\partial\bar{\partial}\phi$. This can be rewritten as:
$$\partial_t \phi = F + P, \quad \text{where } F = \log \frac{\omega_\phi^n}{\omega_0^n} \text{ and } \Delta_\phi P = R - \text{tr}_{\omega_\phi}(\text{Ric}(\omega_0)).$$
While Chen and Zheng proved short-time existence and long-time existence for smooth initial data close to cscK in the $C^{2,\alpha}$ topology, the question of long-time existence for general smooth initial data remained open.

The Gap:
The primary difficulty in proving unconditional long-time existence is the lack of $C^{0,\alpha}$ estimates for the linearized complex Monge-Ampère equation in the parabolic setting (unlike the real case proved by Caffarelli-Gutiérrez). The authors aim to improve the regularity theory of the PCF by relaxing the initial data requirements from smoothness to conditions on the volume form.

2. Methodology

The authors employ a combination of geometric analysis techniques, including:

• Energy Methods: Utilizing the K-energy and J-functionals to derive bounds on the flow.
• Maximum Principle: Applying parabolic and elliptic versions of the Aleksandrov-Bakelman-Pucci (ABP) maximum principle to bound functions like $P$ and $F$.
• Mollification and Approximation: Using geodesic mollification to approximate discontinuous functions (like $P$) by smooth ones to derive gradient estimates.
• Bootstrap Regularity: Establishing a hierarchy of estimates where $C^0$ bounds on the volume form lead to $C^{0,\alpha}$ bounds on the potential, which in turn yield higher-order derivative bounds.
• Implicit Function Theorem: Used in the Fano case to handle perturbations of the Kähler class.

3. Key Contributions and Results

The paper establishes two main theorems regarding the long-time existence and convergence of the PCF under weaker initial data assumptions than previously known.

A. Regularity Improvement (Proposition 1.1)

The authors prove that if the initial volume form is $C^0$-close to a smooth reference volume form, the flow becomes immediately smooth for $t > 0$.

• Condition: $\| \log \frac{\omega_{\phi_0}^n}{\omega_0^n} \|_{C^0} \leq \epsilon_0$.
• Result: The flow exists on $M \times [0, T_0]$ and is $C^\infty$ smooth for any $t > 0$. All derivatives are bounded on $M \times [\epsilon, T_0]$.

B. Global Existence near cscK (Theorem 1.2)

This is the primary application of the regularity improvement.

• Setup: Let $(M, \omega_0, J)$ be a cscK manifold.
• Condition: The initial metric $\omega$ is in the same cohomology class as $\omega_0$, and its volume form is $C^0$-close to that of $\omega_0$:
  $$1 - \epsilon_0 \leq \frac{\omega^n}{\omega_0^n} \leq 1 + \epsilon_0$$
• Result: The PCF exists globally ($t \in [0, +\infty)$) and converges exponentially fast to a cscK metric (which may differ from $\omega_0$).
• Significance: This removes the requirement for the initial data to be close in the $C^{2,\alpha}$ topology; closeness in the $C^0$ topology of the volume form suffices.

C. Global Existence on Fano Manifolds (Theorem 1.3)

The authors extend these results to Fano manifolds (where $c_1(M) > 0$) admitting a Kähler-Einstein (KE) metric.

• Setup: $(M, J, \omega_0)$ is a compact KE manifold with $\text{Aut}_0(M, J) = 0$.
• Condition: The initial Kähler class is close to the canonical class $c_1(M)$, and the volume form is bounded (not necessarily close to 1, but within a factor $\Gamma$):
  $$\frac{1}{\Gamma} \leq \frac{\omega^n}{\omega_0^n} \leq \Gamma$$
• Result: The PCF exists globally and converges to a cscK metric.
• Mechanism: In this case, the closeness of the volume form is not required because the closeness of the Kähler class to $c_1(M)$ ensures that the function $P$ is close to a continuous function (specifically, a multiple of the potential), which is sufficient for the regularity estimates.

4. Technical Outline of Proofs

1. $C^0$ Estimates (Section 3):

   • The authors establish that if $P$ is bounded, then $F$ (and thus the volume form) is bounded.
   • Using the K-energy monotonicity and Gronwall's inequality, they derive exponential bounds on $\|F\|_{C^0}$ and $\|\phi\|_{C^0}$ in terms of initial data.
   • Crucially, they prove that a bound on $\|P\|_{C^0}$ implies a $C^{0,\alpha}$ bound on $\phi$ via Theorem 3.1 (Yau's $C^0$ estimate for Monge-Ampère).

2. Higher Order Estimates (Section 4):

   • The core difficulty is bridging the gap between $C^0$ bounds and $C^{2,\alpha}$ bounds.
   • Case 1 (P close to continuous): If $P$ is $\delta^*$-close to a continuous function, the authors use geodesic mollification to approximate $P$ by a smooth function $\tilde{P}$. They derive differential inequalities for $|\nabla_\phi F|^2$ and $(n+\Delta_\phi)$, utilizing the smallness of the error term $(P-\tilde{P})$ to absorb negative terms. This yields $W^{1,p}$ bounds on $F$, which by Chen-He's work implies $C^{2,\alpha}$ bounds on $\phi$.
   • Case 2 (Small Volume Form): If $\|F\|_{C^0}$ is small, similar estimates hold without needing $P$ to be close to a continuous function, as the nonlinearity is controlled by the smallness of $F$.

3. Long-Time Existence (Sections 5 & 6):

   • For cscK (Thm 1.2): They show that if the initial volume form is close to 1, the flow stays within a "small volume form" regime for a short time $t_0$. By the regularity improvement, the solution becomes smooth at $t_0$. Since the solution is now smooth and close to the cscK metric in $C^{2,\alpha}$, the local existence theorem (Chen-Zheng) can be applied iteratively to extend the flow to infinity.
   • For KE (Thm 1.3): They utilize the fact that on KE manifolds, the PCF coincides with the normalized Kähler-Ricci flow (Proposition 6.1). They use the Implicit Function Theorem to show that for classes close to $c_1(M)$, the flow behaves like the KE flow, which is known to converge.

5. Significance

• Lowering Regularity Requirements: The paper significantly lowers the bar for initial data in the study of the PCF. Previously, initial data needed to be close to cscK in a high-order Sobolev or Hölder space ($C^{2,\alpha}$). This work shows that $C^0$ closeness of the volume form is sufficient.
• Bridging Elliptic and Parabolic: The techniques developed, particularly the adaptation of elliptic estimates (Chen-Cheng) to the parabolic setting via the control of the volume form, provide a new toolkit for studying fourth-order geometric flows.
• Robustness of Convergence: The results demonstrate the stability of the cscK condition under perturbations of the volume form, reinforcing the connection between the existence of canonical metrics and the behavior of geometric flows.
• Fano Case: The extension to Fano manifolds with bounded volume forms provides a new perspective on the convergence of the Kähler-Ricci flow in the context of the PCF framework.

In summary, Cheng and Tian provide a rigorous proof that the Pseudo-Calabi Flow is robust: even if the initial metric is rough (only $C^0$ control on volume), the flow instantly regularizes and, under appropriate geometric conditions (closeness to cscK or KE), converges globally to a canonical metric.