Chen's Long Time Existence Conjecture

• Chen’s Long Time Existence Conjecture is a framework studying maximal existence times for higher-order geometric flows in both intrinsic Calabi and extrinsic Chen settings.
• It predicts perpetual smooth evolution for the Calabi flow on compact Kähler manifolds and finite-time singularity for nonminimal biharmonic Chen flows in Euclidean spaces.
• Recent analyses using regularity theory, blow-up techniques, and concentration-compactness methods have yielded partial confirmations and highlighted key analytic challenges.

Chen's Long Time Existence Conjecture concerns the maximal existence time of certain geometrically motivated higher-order evolution equations, most notably the Calabi flow on compact Kähler manifolds in complex geometry and the extrinsic Chen flow of submanifolds in Euclidean space. At its core, the conjecture posits fundamentally different behaviors in the two settings: for the Calabi flow, perpetual regular evolution is expected for any smooth initial data; for the extrinsic Chen flow, finite-time singularity is expected except in the minimal (i.e., stationary) case. Both conjectures have motivated extensive work in higher-order geometric analysis, regularity theory, and flow-compactness methods, producing partial resolutions and advanced analytic frameworks.

1. Formulation of Chen’s Long Time Existence Conjecture

For the Calabi flow on compact Kähler manifolds $(X,[\omega_0])$, the conjecture is stated as follows: For any smooth initial potential $\varphi_0$, the solution $\varphi(t)$ of the Calabi flow,

$$\frac{\partial\varphi}{\partial t} = S(\varphi) - \overline{S}, \quad \varphi(0) = \varphi_0,$$

where $S(\varphi)$ is the scalar curvature of the metric $$\omega_\varphi = \omega_0 + i\partial\bar\partial\varphi$$ and $\overline{S}$ is the average scalar curvature, exists for all $t \geq 0$ and remains smooth (Feng et al., 2012, He et al., 4 Dec 2025). Explicitly, $T_{\max} = +\infty$ for any initial Kähler potential.

In contrast, for the extrinsic (biharmonic) Chen flow of submanifolds $f \colon M^n \to \mathbb{R}^{n+k}$,

$$\partial_t f = -\Delta^\perp \vec{H} + Q(A)\vec{H},$$

the conjecture asserts that any nonminimal initial closed immersion must become singular in finite time; there are no nonminimal closed smooth biharmonic stationary immersions (Bernard et al., 2017).

2. Definitions and Evolution Equations

Calabi Flow

The flow is a fourth-order degenerate parabolic equation with variational origin. On torus-invariant settings such as the Abelian variety $X = \mathbb{C}^n / (\mathbb{Z}^n + i \mathbb{Z}^n)$, the metric admits symplectic coordinates $x \in P = [-1/2,1/2]^n$,

$$\omega = \sum_{i,j} u_{ij}(x) dx_i \wedge d\theta_j, \quad u_{ij} = \frac{\partial^2 u}{\partial x_i \partial x_j},$$

where the Calabi flow becomes Abreu’s equation: $$\frac{\partial u}{\partial t} = \sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j}(u^{ij}), \quad u^{ij} = (u_{ij})^{-1}.$$

Chen Flow for Submanifolds

Let $A$ denote the second fundamental form, $\vec{H}$ the mean curvature, and $\Delta^\perp$ the normal Laplacian. The flow is explicitly

$$\partial_t f = -(\Delta^\perp \vec{H}) + Q(A)\vec{H} = -(\Delta^2 f)^\perp + P_3^0(A),$$

where $Q(A)\phi = A_{ij} \langle A^{ij}, \phi\rangle$ and $P_3^0(A)$ denotes cubic normal bundle terms.

3. Progress and Main Results in Specific Settings

Torus-Invariant Calabi Flow (Abelian Varieties)

Feng and Huang proved in (Feng et al., 2012) that for $X = \mathbb{C}^n / (\mathbb{Z}^n + i \mathbb{Z}^n)$, with torus-invariant initial data, the flow exhibits:

• Non-collapsing: Under an $L^\infty$ curvature bound, total-energy control, and Donaldson’s M-condition (a uniform $C^{0,1}$ bound on gradients of the symplectic potential $u$), the injectivity radius is bounded below at points of maximal curvature. Regularity results and two-sided Hessian bounds yield uniform (non-collapsed) geometry throughout the flow.
• Dimension 2: In complex dimension $n=2$, the flow exists for all time and maintains a uniform curvature bound; this confirms Chen's conjecture in this explicit scenario.
• Higher Dimensions: If uniform curvature bounds are available for all time, the flow converges exponentially to the flat Kähler metric, giving partial confirmation of Donaldson’s conjectural convergence in general dimensions.

Calabi Flow via Continuity Method (General cscK Setting)

A one-parameter family of twisted Calabi flows interpolates between the J-flow ($s=0$) and the Calabi flow ($s=1$): $$\frac{\partial\varphi}{\partial t} = s (R(\omega_\varphi) - \bar{R}) + (1-s)(n - \text{tr}_{\omega_\varphi}\!\omega).$$ Main results in (He et al., 4 Dec 2025):

• Long-Time Existence Near the J-flow: For any initial data, there exists $s_0>0$ such that for all $s < s_0$, the twisted Calabi flow exists for all time and converges exponentially to the cscK metric.
• Openness Along the Continuation Path: If a twisted Calabi flow exists for all time and converges at $s_*$, then the same holds for nearby $s$. This demonstrates openness, an essential step in the continuity method.
• Outstanding Issue: Closedness of the set $I_{\varphi_0}$ (parameters $s$ for which the flow exists/converges) remains open; resolving this would establish full confirmation of Chen’s conjecture in this context.

Extrinsic Chen Flow and Singularity Formation

In (Bernard et al., 2017), for the extrinsic Chen flow of submanifolds, Bernard, Wheeler, and Wheeler established:

• Finite-Time Extinction: Every closed, smooth, nonminimal initial immersion becomes extinct in finite time, i.e., exists only up to $T < \infty$. This is derived via sharp area monotonicity and mean curvature inequalities.
• Concentration-Compactness/Lifespan Theorem: For $n=2,4$, flow existence time is quantified by local $L^n$ curvature smallness. Singularities correspond to local blowup of curvature integrals.
• Small-Energy Regularity and Shrinking Spheres: In $n=2$, codimension 1, if the trace-free part of the curvature is initially small ($\leq \varepsilon_2 < 8\pi$), the flow shrinks smoothly to a round sphere. Thus, singularity formation and its structure are sharp and fully classified in this regime.

4. Analytic Methods and Technical Ingredients

Key analytic tools developed and employed include:

• Regularity Theory: Streets’ high-order regularity for fourth-order flows given a uniform curvature bound; role of Donaldson’s interior estimates for the Abreu equation and the M–condition to control Hessians of symplectic potentials (Feng et al., 2012).
• Blow-up Analysis: In dimension two, blow-ups and parabolic rescalings around points of maximal curvature yield limiting solutions; Bernstein-type rigidity excludes nontrivial eternal solutions on $\mathbb{R}^2$ with zero scalar curvature, ensuring boundedness (Feng et al., 2012).
• Function and Weighted Spaces: Weighted parabolic Hölder spaces $C^{4,1;\alpha}_\eta$, used in establishing exponential decay, are fundamental in the continuity-path approach to long-time existence (He et al., 4 Dec 2025).
• Contraction Mapping and Approximate Solutions: Establishment of contractive maps around approximate solutions, leveraging linearized operator invertibility and sufficient a priori estimates in Hölder norms (He et al., 4 Dec 2025).
• Concentration-Compactness: For extrinsic Chen flow, a combination of Michael–Simon Sobolev, Gagliardo–Nirenberg inequalities, and continuity methods captures the precise mechanism for singularity formation (Bernard et al., 2017).

5. Implications, Current Status, and Open Problems

The results in (Feng et al., 2012, He et al., 4 Dec 2025, Bernard et al., 2017) delineate the state of the conjecture and related flows:

• For torus-invariant Calabi flow, global existence and regularity are confirmed for complex dimension $n=2$, with the conjecture reduced in higher dimensions to curvature estimate questions (Feng et al., 2012).
• In the general cscK context, long-time existence and convergence for the full Calabi flow remain open; juxtaposed against the J-flow case, the set of parameter values for which smooth evolution exists is open but not yet closed (He et al., 4 Dec 2025).
• For extrinsic Chen flow, the expectation and proof is that finite-time singularity is unavoidable except at the minimal stationary point. The singularity structure is quantified and, in small energy/codimension-one scenarios, leads to unique round sphere extinction (Bernard et al., 2017).
• The closure of continuation sets for twisted Calabi flows, robust maximum principles for curvature, and the extension of uniform estimates beyond torus-invariant scenarios are highlighted as significant open technical challenges.

6. Table: Summary of Outcomes in Key Settings

Geometric Flow	Long-Time Existence Conjecture	Proven Results
Calabi flow on Kähler	All time existence for any initial data	True for $n=2$ torus-invariant (Feng et al., 2012); open in higher $n$, general cscK (He et al., 4 Dec 2025)
Twisted Calabi flow	Openness of existence in parameter $s$	Openness shown near $s=0$ (He et al., 4 Dec 2025)
Extrinsic Chen flow	No nonminimal stationary solution	Extinction in finite time for all closed data (Bernard et al., 2017)

These results collectively provide a framework for future progress, including the extension of curvature estimates, closure properties in the continuity method, and advances in the analytic theory for higher-order geometric flows.