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Bennett Chow

Publications and source records attributed to Bennett Chow.

At least 19 recordsLinked to original sources
arXiv2026

A Lean Formalization of Hamilton's Three-Manifold Theorem

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.

math.DG↗
arXiv2026

The Fisher Metric of the Ricci Flow Heat Kernel

We introduce and study a Fisher information metric \(g^F_\tau\) associated to the conjugate heat kernel of a Ricci flow \((M^n,g_t)\). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that \(g^F_\tau\) is monotone in scale and satisfies \(g^F_\tau\le g_t\). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect \(g_t-g^F_\tau.\) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities \(0<g^F_\tau<g_t\) 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 Poincar\'e inequality for the heat semigroup, a contraction formula for \(\varphi\)-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 \(\varepsilon\)-regularity theorem.

math.DG↗
arXiv2026

Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold

Let \(M\) be a closed connected oriented topological \(4\)-manifold. We prove that if \(F_1,\dots,F_r\subset M\) are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera \(g_i\), same-sign twisted normal Euler numbers \(e_i\), and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on \(M\). Thus same-sign mod-\(2\)-null families of disjoint nonorientable surfaces in a fixed ambient \(4\)-manifold have uniformly bounded total excess over Massey's \(S^4\) bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for \(2\)-fold branched covers. As corollaries, every closed oriented topological \(4\)-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of \(\RP^2\) with \(\abs{e}>2\), and only finitely many pairwise disjoint tubular neighborhoods modeled on real \(2\)-plane bundles over \(\RP^2\) whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than \(2\). When \(M\) is a homology \(4\)-sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in \(S^4\).

math.GT↗
arXiv2025

Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

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 $\partial M$, 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 $\operatorname{Tor}H_1(\partial M;\mathbb{Z})$ is a direct double, i.e., $\operatorname{Tor}H_1(\partial M;\mathbb{Z})\cong A \oplus A$, 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 $S^4$ is hyperbolic.

math.DG↗
arXiv2022

Lower bounds for the scalar curvatures of Ricci flow singularity models

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.

math.DG↗
arXiv2021

Four-Dimensional Steady Gradient Ricci Solitons with $3$-Cylindrical Tangent Flows at Infinity

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $\mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general.

math.DG↗
arXiv2020

On Four-dimensional Steady gradient Ricci solitons that dimension reduce

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form $\mathbb{S}^3/\Gamma$ or weakly dimension reduce to the $3$-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensional manifolds. As an application, we prove that any steady gradient K\"{a}hler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperk\"{a}hler ALE Ricc-flat $4$-manifolds.

math.DG↗
arXiv2016

Three-dimensional noncompact $\kappa$-solutions that are Type I forward and backward

As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [9] and is covered by [19]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional $\kappa$-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional $\kappa$-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [2].

math.DG↗
arXiv2012

On a formula of Daskalopoulos, Hamilton and Sesum

We give an exposition of a formula of Daskalopoulos, Hamilton and Sesum for solutions to the Ricci flow on the 2-sphere. This is one of several estimates used by them to classify ancient solutions on the 2-sphere.

math.DG↗
arXiv2012

On a paper of Daskalopoulos and Sesum

This is an exposition of aspects of the result of Daskalopoulos and Sesum that any 2-dimensional complete noncompact ancient solution to Ricci flow with bounded positive scalar curvature and finite width must be the cigar soliton.

math.DG↗