Paper · Nov 9, 2026
A geometric Kähler–Ricci flow proof of the Calabi–Yau theorem
As on arXiv
Abstract
We give a geometric proof that every Kähler class on a compact Kähler manifold with vanishing real first Chern class contains a unique Ricci-flat metric, and that the Kähler–Ricci flow converges smoothly to this metric. The fixed Kähler class and the bounded logarithmic volume density control the mean pairwise distance. Chow's scalar-curvature decay makes available, at a fixed spatial scale and all sufficiently late times, the relative-volume comparison of Jian–Song, derived from Bamler's pointed Nash entropy estimates. This comparison upgrades the mean-distance bound to a diameter bound without an absolute lower bound for entropy. A restarted heat-gradient estimate then gives exponential decay of the Ricci potential. A parabolic Schwarz estimate and the interior estimates of Sherman–Weinkove yield smooth exponential convergence. The proof does not invoke the Calabi–Yau existence theorem or global estimates for a scalar complex Monge–Ampère equation.
- Submitted
- Nov 9, 2026
- Subjects
- math.DG
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.
-
Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “Yau theorem”, “complex Monge”, “Ampère equation”
-
The Fisher Metric of the Ricci Flow Heat Kernel
Topics Perelman entropy · Same author Bennett Chow · Shared terms “pointed Nash”, “Nash entropy”, “Bamler”
-
Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Ricci-flat & Calabi–Yau, Scalar curvature · Same author Bennett Chow · Shared terms “lower bound”, “Bamler”
-
A Lean Formalization of Hamilton's Three-Manifold Theorem
Topics Ricci flow · Same authors Bennett Chow, Yuan Liao · Shared terms “flow proof”, “existence theorem”
-
Diameter estimates in Kähler geometry
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “diameter bound”, “Ampère equation”, “Monge”
-
A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture
Topics Ricci flow · Same authors Bennett Chow, Yuan Liao · Shared terms “flow proof”
-
Flows of -Structures associated to Calabi-Yau Manifolds
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “complex Monge”, “Ampère equation”
-
A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture
Topics Ricci flow · Same authors Bennett Chow, Yuan Liao
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6B.16Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.