https://neckpinch.com/authors/bennett-chow/New arXiv papers by Bennett Chow, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.13897arxiv:2610.13897Fri, 09 Oct 2026 00:00:00 GMTYuan Liao, Ziyang Qin, Jui-Hui Chung, Ayush Khaitan, Zhengyang Zhang, Hongzhou Lin, Zijian Zhang, Ligeng Zhu, Peng Lu, Bennett Chow — We formalize the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture. The public Lean declaration is reproduced in this paper. Its geometric interpretation follows, and verification is recorded in 4.Ricci flowGeometrization https://arxiv.org/abs/2609.33842arxiv:2609.33842Sun, 27 Sep 2026 00:00:00 GMTZiyang Qin, Yuan Liao, Ayush Khaitan, Bennett Chow — We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.Ricci flowSingularitiesSurgeryPoincaré conjecture3-manifolds https://arxiv.org/abs/2608.21502arxiv:2608.21502Fri, 21 Aug 2026 00:00:00 GMTBennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.Ricci flowSingularities(Non)collapsingCurvature pinching3-manifoldsLimit spaces & RCDHarmonic maps & DeTurck trick https://arxiv.org/abs/2608.14478arxiv:2608.14478Fri, 14 Aug 2026 00:00:00 GMTBennett Chow, Robert Koirala — We introduce and study a Fisher information metric g^F_τ associated to the conjugate heat kernel of a Ricci flow (Mⁿ,gₜ). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that g^F_τ is monotone in scale and satisfies g^F_τ≤ gₜ. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect gₜ−g^F_τ. This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities 0<g^F_τ<gₜ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for φ-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to 0 to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's ε-regularity theorem.Ricci flowPerelman entropyHeat kernelStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2505.03823arxiv:2505.03823Sat, 03 May 2025 00:00:00 GMTBennett Chow, Michael H. Freedman, Henry Shin, Yongjia Zhang — This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if M is a compact connected oriented 4-manifold with connected boundary ∂M, and if an unbounded number of disjoint copies of M embed topologically and locally flatly in the interior of a compact 4-manifold N, then TorH₁(∂M;ℤ) is a direct double, i.e., TorH₁(∂M;ℤ)≅ A ⊕ A, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed 3-manifold that embeds in S⁴ is hyperbolic.Ricci solitonsSingularities4-manifolds https://arxiv.org/abs/2208.13206arxiv:2208.13206Sun, 28 Aug 2022 00:00:00 GMTPak-Yeung Chan, Bennett Chow, Zilu Ma, Yongjia Zhang — In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger–Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of 4-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.Ricci flowRicci solitonsRicci-flat & Calabi–YauSingularitiesScalar curvatureLimit spaces & RCD