gpt 6 astra ultra attempt on Math-405

Long-Time Existence of the Calabi Flow on Compact Kaehler Manifolds

Definitions and mathematical statement

Let $(X,J)$ be compact Kähler and fix a Kähler class $[\omega]$ containing a constant-scalar-curvature Kähler metric. Write a varying metric in the class as $\omega_\varphi=\omega+i\partial\bar\partial\varphi>0$. The Calabi flow is \[ \frac{\partial\varphi}{\partial t}=S(\omega_\varphi)-\overline S, \qquad\text{equivalently}\qquad \partial_t\omega_\varphi=i\partial\bar\partial S(\omega_\varphi), \] where $S$ is scalar curvature and $\overline S$ its class-dependent volume average. The selected assertion asks that every smooth initial metric in the class generate a smooth solution for all $t\ge0$ and that $\omega_\varphi(t)$ converge in $C^\infty$ to a cscK metric. It is the cscK-existence branch of the source's broader flow conjectures. Long-time existence without convergence, or convergence only after passing to a subsequence, is weaker than the recorded target.

Formulation and concept references: [S1].

Short English statement

Whenever a Kähler class contains a constant-scalar-curvature metric, does Calabi flow from every smooth metric in that class exist forever and converge smoothly to one?

Research attempt

Scope, attribution, and source check.

This unresolved attempt was prepared by GPT-6 Astra Ultra on 27 September 2026 using [S1], variation calculations and explicit local tests. The class already contains a cscK metric. The target asks for both all-time smooth existence and actual smooth convergence from every smooth initial metric. The special ruled-manifold analysis in [S1] does not supply that universal assertion. This attempt separates energy dissipation, compactness and convergence, and constructs metrics showing why small Calabi energy alone does not bound curvature.

The conserved cohomological average.

Let the complex dimension be $m$. Along the fixed class, \[ \int_X\omega_\varphi^m=\int_X\omega^m,\qquad \int_XS_\varphi\omega_\varphi^m =m\int_X\operatorname{Ric}(\omega_\varphi) \wedge\omega_\varphi^{m-1}, \] in the complex scalar-curvature convention. The Ricci form represents a fixed class, so $\overline S$ is independent of time. A real scalar-curvature convention rescales the numerical factors consistently, not the parabolic sign. Potentials have an additive time-dependent gauge; when discussing their convergence, fix their average against one reference volume. This does not change the metric equation.

What energy dissipation provides.

The Mabuchi functional is characterized up to an additive constant by \[ d\mathcal M_\varphi(u) =-\int_Xu(S_\varphi-\overline S)\frac{\omega_\varphi^m}{m!}. \] Closedness of this variational one-form and existence of the functional are established inputs underlying the flow in [S1]. Along a smooth solution, substitution gives \begin{equation}\label{wsref-0-61-34-30-31-3a-64-69-73-73-69-70-61-74-69-6f-6e} \frac{d}{dt}\mathcal M(\varphi(t))=-\mathcal C(t),\qquad \mathcal C(t)=\int_X(S_\varphi-\overline S)^2 \frac{\omega_\varphi^m}{m!}. \end{equation} On an all-time trajectory where $\mathcal M$ is bounded below, $\int_0^\infty\mathcal C(t)\,dt<\infty$. There are therefore times $t_j\to\infty$ with $\mathcal C(t_j)\to0$. This says nothing by itself about smooth compactness of the metrics.

Using only integrability, even pointwise decay of $\mathcal C(t)$ requires additional temporal control: a smooth nonnegative integrable function may have arbitrarily narrow spikes of fixed height. Uniform continuity excludes this. A fixed positive value at infinitely many late times would then force disjoint intervals of uniform length on which the function is at least half as large, contrary to integrability. This elementary argument explains one use of uniform geometric estimates.

A conditional convergence package.

Assume that an all-time solution has normalized potentials precompact in $C^\infty$, with uniformly positive metrics, and that $\mathcal M$ is bounded below on it. At the times above, take a subsequence converging smoothly to $\varphi_\infty$. Continuity of scalar curvature and volume gives $S_{\varphi_\infty}=\overline S$. This proves conditional subsequential convergence, not the full target.

For a precise upgrade, put $E(t)=\mathcal M(\varphi(t))-\mathcal M(\varphi_\infty)$. Monotonicity and continuity along the subsequence give $E(t)\downarrow0$. Suppose for all sufficiently large $t$ \begin{equation}\label{wsref-0-61-34-30-31-3a-67-72-61-64-69-65-6e-74} \sqrt{\mathcal C(t)}\geq cE(t)^{1-\theta}, \qquad c>0,\quad0<\theta<1. \end{equation} Then, wherever $E>0$, \[ -\frac{d}{dt}E^\theta =\theta E^{\theta-1}\mathcal C \geq c\theta\sqrt{\mathcal C}. \] Thus $\int_T^\infty\sqrt{\mathcal C(t)}\,dt<\infty$. If $E$ reaches zero in finite time, the dissipation identity makes the later metric stationary.

Uniform equivalence of volume forms, together with subtraction of the fixed-volume mean, bounds the normalized potential speed by $\|\partial_t\varphi\|_{L^2(\omega)} \leq C\sqrt{\mathcal C(t)}$. The potentials are consequently Cauchy in $L^2$. Their assumed $C^\infty$ precompactness makes every smooth cluster point equal to that unique $L^2$ limit. Every sequence has such a subsequence, so the whole flow converges smoothly to the cscK metric.

Inequality (a401:gradient) is an additional gradient inequality, including its validity along the whole late-time trajectory. Neither a global inequality, entry into its neighborhood, nor control of holomorphic automorphism directions is assumed to have been proved. Convergence only after time-dependent automorphisms would also be weaker than the fixed-metric conclusion here. The value of this package is that every missing analytic hypothesis can be identified separately.

The linearized equation has the correct sign.

At a flat complex torus, scalar-curvature variation is $DS(u)=-\Delta^2u$, using the complex Laplacian. Hence the linearized flow is \[ \partial_tu=-\Delta^2u. \] If $-\Delta e_k=\lambda_ke_k$, Fourier modes evolve as $u_k(t)=e^{-\lambda_k^2t}u_k(0)$. Nonconstant modes decay and the constant mode is the potential gauge. This verifies linear stability and fourth-order parabolicity, but not nonlinear global existence.

Fourth order prevents a direct second-order maximum-principle argument. On a circle, consider the scalar biharmonic heat equation and $u_0(x)=(1-\cos x)^2\geq0$. At zero, $u_0(0)=0$ and $u_0^{(4)}(0)=6$, giving $\partial_tu(0,0)=-6$. Positivity is immediately lost. This is a test of the leading operator, not an identification of Kähler positivity with positivity of a potential. A metric positivity argument needs additional structure.

Small energy does not bound curvature.

Work in the flat class of a complex torus. In a coordinate ball choose a smooth real bump $\psi$ whose linearized scalar curvature is not identically zero. For small $\varepsilon$ set \[ \varphi_\varepsilon(x) =\varepsilon^{7/2}\psi(x/\varepsilon), \] extended by zero. Its Hessian is $O(\varepsilon^{3/2})$, so $\omega+i\partial\bar\partial\varphi_\varepsilon$ is positive and uniformly close to the flat metric. Its fourth derivatives have size $\varepsilon^{-1/2}$. With the indicated choice of bump, scalar curvature, and hence the curvature tensor, attain a size bounded below by a positive multiple of this quantity.

The nonlinear errors are lower order: products of third derivatives are $O(\varepsilon)$, as is a second derivative multiplying a fourth derivative. The volume of the support in real dimension $d=2m$ is $O(\varepsilon^d)$. The class average is zero, and the metric-volume correction tends uniformly to zero. Consequently \[ \mathcal C(\varphi_\varepsilon) =O(\varepsilon^{d-1})\longrightarrow0, \qquad \sup|\operatorname{Rm}|\longrightarrow\infty. \] These are smooth metrics in one fixed class containing a cscK metric. They are not claimed to lie on one Calabi trajectory or to cause a finite-time singularity. They refute the specific a priori implication from small Calabi energy alone to bounded curvature.

The example also explains why smoothness of each initial condition cannot be replaced by a uniform estimate over all initial conditions using just this energy. The small support allows high derivatives to escape an integral scalar-curvature bound. Higher regularity, nonconcentration or an independent compactness mechanism is needed.

Verification, first gap, and outcome.

The derivative orders, bump exponents and Fourier decay were checked independently. The convergence deduction uses exactly the displayed compactness and gradient hypotheses. Finite-time continuation would need uniform positivity and sufficient higher spatial bounds to invoke fourth-order parabolic regularity; the energy identity does not provide them.

The first general gap is to exclude metric degeneration or curvature concentration for arbitrary initial metrics in the class. Even after all-time existence, full convergence needs a mechanism excluding multiple cluster points and uncontrolled drift. The stated package identifies one such sufficient mechanism. UNRESOLVED. Dissipation consequences, conditional convergence and explicit failure tests are proved, but neither part of the unrestricted target is established or claimed as new.

Sources

• S1 Benjamin Sibley, Long-Time Existence for the Calabi Flow on Ruled Manifolds over Riemann Surfaces, Conjectures 1.1-1.2 (2022 preprint). https://geometry.ulb.ac.be/wp-content/uploads/2022/04/Calabi-Flow-Version-VII.pdf. Formulation source. Consulted 24 September 2026: Calabi-flow formulation and the constant-scalar-curvature convergence branch.