1 · The essentials
A geometric flow is a rule for changing the shape of a space over time. The manuscript studies one such rule, called Calabi flow, and asks whether it can always keep running smoothly. It presents a smooth starting shape for which a measure of curvature becomes unbounded in a finite amount of time. This contradicts the smooth long-time existence conjecture attributed to Chen in the source. The starting shape belongs to a collection that also contains a shape whose curvature measure is constant everywhere, so having such a shape available does not prevent this failure. The result is shown in the manuscript, but it is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine adjusting a landscape according to a fixed rule, only for its sharpness to grow without bound before a finite deadline. The presence of a uniformly curved landscape among the allowed alternatives does not guarantee that this particular adjustment process will reach it smoothly.
Why it matters
The construction shown in the manuscript gives a counterexample to Chen's smooth long-time existence conjecture for Calabi flow. Its significance is sharpened by where the starting metric lies: the same class already contains a constant-scalar-curvature metric. Thus, the existence of such a metric in the class is not enough to guarantee smooth long-time evolution from every smooth starting metric there.
2 · Going further
The question concerns smooth long-time existence for Calabi flow, an evolution of Kähler metrics. The main manuscript constructs a smooth Kähler metric on complex projective space whose scalar curvature becomes unbounded in finite time under this flow. The repository summary additionally specifies that the starting metric is -invariant.
The metric lies in the Fubini–Study class. As the source emphasizes, this class contains a constant-scalar-curvature metric, yet the constructed starting metric still develops a finite-time singularity. The manuscript therefore presents a counterexample to the smooth long-time existence conjecture attributed to Chen, even with this additional property of the class.
The argument described by the abstract is constructive: supply a smooth initial metric and show that its evolving scalar curvature becomes unbounded within finite time. The supplied text does not explain the construction or the estimates establishing this behavior, so a more detailed account of the mechanism is not available here. Only one manuscript is supplied; no relationship to companion manuscripts is described.
Useful notions
- Smooth metrics and curvature
- Kähler geometry
- Complex projective space
- Geometric evolution equations
- Constant scalar curvature
What was known before / what is new
Before
The source identifies Chen's smooth long-time existence conjecture as the assertion being challenged. It also states that the Fubini–Study class in question contains a constant-scalar-curvature metric. No further history or earlier partial results are supplied.
New
The manuscript presents a smooth metric in the Fubini–Study class of whose Calabi flow develops unbounded scalar curvature in finite time. The repository summary specifies symmetry for this metric. This construction is shown in the manuscript to contradict the stated smooth long-time existence conjecture.
Limits and assumptions
The supplied construction concerns a specific smooth, -invariant Kähler metric on in the Fubini–Study class; it does not assert that every metric in that class develops a singularity. The reported failure is unbounded scalar curvature in finite time. The abstract provides neither the construction's details nor a classification of which other initial metrics have smooth long-time flows, and it states no further open problems. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Calabi flow —
- The geometric evolution of Kähler metrics studied in the manuscript.
- Scalar curvature —
- A numerical measure of curvature at each point of a geometric space.
- Constant-scalar-curvature metric —
- A metric whose scalar curvature has the same value at every point.
- Finite-time singularity —
- A failure of smooth evolution within finite time, exhibited here by scalar curvature becoming unbounded.
Keywords : Calabi flow · Finite-time singularity · Kähler geometry
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth -invariant Kähler metric on whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A finite-time singularity of Calabi flow on projective space
24/09/2026We construct a smooth Kähler metric in the Fubini–Study class of whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.