A Calabi flow that becomes singular before reaching a canonical metric

← All digests

Family 352 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Suppose a space admits an especially well-balanced metric. Can a natural geometric evolution always reach that metric from arbitrary smooth initial data?

Calabi flow was introduced to seek canonical Kähler metrics by reducing variation in scalar curvature while keeping the complex structure and Kähler class fixed. A longstanding conjecture proposed that the smooth flow should exist for all positive time from every smooth Kähler metric on a compact complex manifold.

Family 352 claims a counterexample on complex projective ten-space. The starting metric is smooth, highly symmetric, and belongs to the same Kähler class as the standard Fubini–Study metric, which already has constant scalar curvature. Nevertheless, its Calabi flow develops unbounded scalar curvature in finite time.

The distinction is between the existence of a desirable endpoint somewhere in the class and the behavior of a particular evolution from every starting point. A class can contain an ideal metric while some smooth trajectories become singular before reaching it.

2. Background and essential definitions

Kähler metrics and their class

A Kähler manifold has a complex structure, a compatible Riemannian metric, and a closed two-form $\omega$ encoding the metric and complex structure together. Within a fixed Kähler class, nearby forms can be written using a real potential $\varphi$:

$$\omega_\varphi=\omega_0+i\partial\bar\partial\varphi>0.$$

The operators $\partial$ and $\bar\partial$ are the complex-coordinate parts of exterior differentiation. Positivity means that the form still defines a positive-definite metric. Adding this exact correction preserves integrals over closed surfaces, hence the cohomology class.

A constant-scalar-curvature Kähler metric has scalar curvature independent of position. It need not have constant sectional curvature. In this article, the scalar convention is the manuscript’s complex trace

$$S(\omega)=\operatorname{tr}_{\omega}\operatorname{Ric}(\omega).$$

Keeping this convention matters for the numerical constant appearing in the radial formula below.

The flow

Calabi flow evolves the potential by

$$\partial_t\varphi=S(\omega_\varphi)-\overline S, \qquad\varphi(0)=0.$$

The average scalar curvature $\overline S$ is fixed by the Kähler class. A constant-scalar-curvature metric is stationary for this normalization.

Scalar curvature involves second derivatives of the metric, and the metric already involves second derivatives of the potential. Thus Calabi flow is fourth order in $\varphi$. It has smoothing properties, but it lacks some elementary maximum-principle tools familiar from second-order heat equations.

The Fubini–Study form on $\mathbb{CP}^n$ is normalized here by the potential $\log(1+|Z|^2)$ on an affine coordinate chart. Its scalar curvature is $n(n+1)$, hence 110 for $n=10$.

Smooth and weak continuation

A smooth flow remains a smooth positive metric at every time. A weak flow may continue in a larger space of finite-energy potentials, where smoothness is not required. Convergence in a weak potential-space distance does not imply uniform control of curvature or smooth continuation through every intermediate time.

The manuscript challenges smooth global existence. It does not deny the existence of global weak evolutions.

3. Development history and previous partial results

Calabi introduced the flow as a route toward extremal and constant-scalar-curvature metrics. On compact Riemann surfaces, where the complex dimension is one, Chruściel proved global forward existence and convergence to constant curvature. Chen gave a different approach, and Struwe developed a unified treatment of Calabi and normalized Ricci flows on surfaces. Chen’s surface account.

Higher dimensions brought local existence, uniqueness, and stability near canonical metrics. Chen–He proved short-time theory and stability near constant-scalar-curvature metrics; He–Zeng extended short-time theory to rougher initial data. These results do not say that every smooth metric lies in the basin of smooth global convergence. Chen–He, He–Zeng.

Continuation criteria identify quantities that must become uncontrolled at a singularity. Chen–He obtained extension under a Ricci-curvature bound, and Chen–Cheng obtained extension under a scalar-curvature bound. Further results use spatial or spacetime integrability conditions on scalar curvature. The claimed example is consistent with those criteria precisely because its scalar curvature diverges. Chen–Cheng.

Symmetry has yielded global convergence in particular settings. Guan studied modified Calabi flow on almost homogeneous manifolds with two hypersurface ends. The action in the present example has a point orbit and a hyperplane orbit instead; the point has complex codimension ten, so the two-hypersurface hypothesis does not apply.

Meanwhile, Streets constructed global weak Calabi flows by minimizing movements. Berman–Darvas–Lu identified these with finite-energy trajectories and established convergence in the $d_1$ distance to a smooth constant-scalar-curvature potential when the class contains one. Here $d_1$ extends an $L^1$-type path-length distance on potentials. The classical and weak flows agree while the classical flow remains smooth, but the weak theory does not force smoothness at every finite time. Streets, Berman–Darvas–Lu.

4. What the new manuscript claims

There is a smooth $U(10)$-invariant Kähler form $\omega_0$ on $\mathbb{CP}^{10}$, in the Fubini–Study class, whose maximal smooth Calabi flow exists on

$$[0,T_*),\qquad0<T_*<\infty.$$

At a fixed point $p$, its scalar curvature satisfies

$$S(\omega(t))(p)\sim a(T_*-t)^{-1/2},\qquad a>0.$$

The symbol $\sim$ means that the ratio tends to one. This is an asymptotic at the specified point, not a complete global curvature profile.

The group $U(10)$ acts by unitary transformations on the affine coordinates. Its invariance reduces the construction to one spatial variable, but the resulting metric is smooth on the whole compact projective manifold. The example has real dimension twenty.

Because the same class contains Fubini–Study, the result disproves the proposed smooth global existence statement even in a class containing a constant-scalar-curvature metric. It also prevents a universal claim of smooth global existence followed by smooth convergence modulo holomorphic automorphisms from arbitrary initial data.

The paper extends the singularity by products. If $(Y,\eta)$ is a compact connected Kähler manifold with constant scalar curvature, then

$$\Omega(t)=\pi_X^*\omega(t)+\pi_Y^*\eta$$

is the corresponding product Calabi flow, with the same maximal smooth time and the same blowup rate at every $(p,y)$. Suitable stationary projective factors give examples in every complex dimension at least ten, in classes containing Kähler–Einstein metrics. The theorem does not claim such examples in every lower dimension.

5. Why the advance matters

Importance: 5/5. If correct, the result disproves a central smooth long-time existence conjecture for a major geometric flow. Its strength is that failure occurs in a familiar projective manifold and a class already known to contain a canonical metric.

The example would show that the existence theory of canonical metrics and the global smooth dynamics of Calabi flow cannot be identified. Weak convergence may remain available even when a particular classical trajectory encounters finite-time curvature blowup.

The explicit pointwise rate also gives more information than an abstract failure of continuation. It identifies the scalar quantity that diverges and the leading temporal scale of that divergence.

This rating measures the consequences of the stated construction, not verification of the profile matching or the nonlinear correction.

6. The main idea

Convert radial geometry into one fourth-order equation

For a radial Kähler potential $\Phi(\ell)$, with $\ell=\log|Z|^2$, introduce the momentum coordinate and profile

$$x=\Phi'(\ell),\qquad v=\Phi''(\ell).$$

In the fixed Fubini–Study class, $0<x<1$. The two endpoints correspond to the affine origin and the hyperplane at infinity. Write

$$v=x(1-x)+x^2(1-x)^2H.$$

The metric extends smoothly across both special orbits when $H$ is smooth on the closed interval and

$$1+x(1-x)H>0.$$

This representation packages the endpoint behavior into a smooth function and an explicit positivity condition. The manuscript derives a scalar fourth-order equation for $H$, including the affine normalization in the Legendre potential needed to recover the exact Calabi flow.

At the affine origin $p$, its scalar-curvature formula simplifies to

$$S(t,p)=110\bigl(1-H(t,0)\bigr).$$

Thus a sufficiently negative divergence of one profile value produces positive scalar-curvature blowup at a fixed geometric point.

Construct an exact shrinking profile in the flat limit

Shift the singular time to zero temporarily and take $t<0$. Set $\rho=\sqrt{-t}$. The approximate compact solution is built from

$$F(t,x)=\rho^{-1}h(x/\rho), \qquad h(y)=\frac{G(y)-1}{y}.$$

The function $G$ is positive, bounded above and away from zero, and satisfies $G(0)=1$ with $G'(0)<0$. The quotient defining $h$ extends smoothly at zero, giving

$$F(t,0)=\rho^{-1}G'(0).$$

The profile is chosen to solve the scale-invariant flat-limit equation exactly. It matches an analytic solution near the origin with a half-line solution tending to a positive constant at infinity. The manuscript supports this matching with finite exact-arithmetic inequalities, analytic enclosures, and nonlinear error estimates, providing an executable certificate in its appendices.

That certificate is part of the source’s claimed construction; it has not been run or verified for this digest.

Correct the compact error without assuming stability

The compact projective equation differs from the flat model. The paper seeks an exact solution

$$H=F+V$$

with the correction smaller at the origin:

$$V(t,0)=O(\rho^{-1+\delta}) \qquad\text{for some }\delta>0.$$

The leading negative term then survives, yielding the asserted scalar asymptotic.

To construct $V$, the paper develops a right inverse for the linearized equation on negative times. In similarity variables, its evolution is a compact perturbation of a damped constant-coefficient evolution. A spectral split determines finitely many components from future forcing.

This is permitted because the initial metric has not been prescribed in advance: the construction is free to choose a suitable interior slice afterward. It does not assume that the shrinking profile attracts a neighborhood of initial metrics or prove a general nonlinear stability theorem for it.

Join the shrinking core to the outer region

Near the core, the negative-time inverse controls the singular scale. Away from it, a short-time forward solution handles the compact outer region. The outer forcing cancels the error introduced by the inner cutoff, leaving perturbative operator errors.

Weighted parabolic estimates control spatial derivatives through order four and the needed temporal Hölder behavior. A fixed-point construction then supplies the exact nonlinear correction, its time derivative, and positivity of the Kähler form.

Finally, choose a smooth time slice before the singular time as the initial metric and shift time forward. The flow is smooth and positive at every earlier time, but the scalar asymptotic prevents smooth continuation at $T_*$. For products, scalar curvature adds and the stationary factor contributes only its constant value, so the same singularity persists.

7. Scope, limitations, and open questions

The theorem constructs one carefully chosen symmetric initial metric. It does not say that every metric in the Fubini–Study class becomes singular, that the behavior is generic, or that small perturbations of Fubini–Study produce the same singularity.

The construction begins in complex dimension ten and extends upward by products. It does not determine the smallest dimension admitting such a singularity. The established global theory on complex curves remains a distinct positive case.

The result concerns the classical smooth flow. It does not contradict global weak Calabi flow or convergence in a weak potential-space distance. A weak trajectory can pass through times at which no smooth metric continuation is available.

The pointwise scalar asymptotic is not a classification of every blowup model or of the limiting weak metric. Further questions include stability of the constructed singularity, its geometric limit, and which initial-data conditions guarantee smooth global evolution.

8. Sources and further reading

The primary sources linked above explain surface existence, higher-dimensional stability and extension criteria, and weak continuation. Optional background appears under Kähler manifold, Fubini–Study metric, and Calabi flow.

Source manuscripts

• M1. OpenAI, A finite-time singularity of Calabi flow on projective space (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.