https://neckpinch.com/authors/yongjia-zhang/New arXiv papers by Yongjia Zhang, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2608.19565arxiv:2608.19565Thu, 20 Aug 2026 00:00:00 GMTGanqi Wang, Yongjia Zhang — In this paper, we prove a gap theorem for F-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an ε-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).Ricci flowAncient solutionsSingularitiesScalar curvature https://arxiv.org/abs/2605.08884arxiv:2605.08884Sat, 09 May 2026 00:00:00 GMTPak-Yeung Chan, Yongjia Zhang — We prove local versions of the Ricci curvature and ν-entropy gap theorems for Ricci shrinkers, which respectively generalize a previous result of Munteanu-Wang and a prior result of the authors with Ma. The key point is that these local gaps depend only on the dimension and not on the global entropy or any other geometric information of the Ricci shrinker. As an application, we provide a local criterion for removable Type I singularities of the Ricci flow.Ricci flowRicci solitonsSingularitiesStability & uniqueness https://arxiv.org/abs/2603.28014arxiv:2603.28014Mon, 30 Mar 2026 00:00:00 GMTYuxing Deng, Ganqi Wang, Yongjia Zhang — In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity – a noncollapsed F-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed F-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature (Ric≥ εR g), either it is compact, or every tangent flow at infinity is a Ricci flat cone.Ricci flowAncient solutionsRicci-flat & Calabi–YauSingularities(Non)collapsing 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/2502.15219arxiv:2502.15219Fri, 21 Feb 2025 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this article, we prove an ε-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to 1, then the curvature radius at the base point cannot be too small.Ricci flowReduced volume https://arxiv.org/abs/2401.03387arxiv:2401.03387Sun, 07 Jan 2024 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — A noncollapsed F-limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of F-convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed F-limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.Ricci flowRicci solitonsSingularities(Non)collapsingScalar curvature https://arxiv.org/abs/2310.14020arxiv:2310.14020Sat, 21 Oct 2023 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions.Ricci solitonsSingularities(Non)collapsing4-manifolds https://arxiv.org/abs/2310.14007arxiv:2310.14007Sat, 21 Oct 2023 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — The metric flow is introduced and extensively studied by Bamler [Bam20b, Bam20c], especially as an F-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the F-convergence of a sequence of F-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in a forthcoming work of the authors [CMZ23].Ricci flowPerelman entropy https://arxiv.org/abs/2212.09203arxiv:2212.09203Mon, 19 Dec 2022 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — We prove a local gap theorem for Ricci shrinkers, which states that if the local μ-functional at scale 1 on a large ball centered at the minimum point of the potential function is close enough to 0, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global μ-functional to be close enough to 0. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].Ricci solitons 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 https://arxiv.org/abs/2202.13302arxiv:2202.13302Sun, 27 Feb 2022 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — In this paper, we survey the volume growth estimates for shrinking, steady, and expanding gradient Ricci solitons. Together with the known results, we also prove some new volume growth estimates for expanding gradient Ricci solitons.Ricci solitons https://arxiv.org/abs/2202.02421arxiv:2202.02421Fri, 04 Feb 2022 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of t^(−β) for some β>0. In this article, we calculate the corresponding F-convergence rate: after being scaled by a factor λ>0, a Ricci flow with closed and smooth tangent flow is |log λ|^(−θ) close to its tangent flow in the F-sense, where θ is a positive number, λ≫ 1 in the blow-up case, and λ≪ 1 in the blow-down case.Ricci flowSingularities https://arxiv.org/abs/2201.09387arxiv:2201.09387Sun, 23 Jan 2022 00:00:00 GMTTimothy Buttsworth, Maximilien Hallgren, Yongjia Zhang — We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.Ricci flowSingularitiesSurgeryStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2112.11025arxiv:2112.11025Tue, 21 Dec 2021 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy |Rm|≤ cR for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any 3-dimensional gradient Ricci expander C² asymptotic to (C(S²), dt²+αt² g_(S²)) is rotationally symmetric, where α∈ (0,1] is a constant and g_(S²) is the standard metric on S² with constant curvature 1.Ricci solitonsRicci-flat & Calabi–YauSingularitiesCurvature pinching3-manifoldsRotational symmetry & warped products https://arxiv.org/abs/2112.07818arxiv:2112.07818Wed, 15 Dec 2021 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this paper, we prove a pseudolocality-type theorem for 𝓛-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the 𝓛-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen.Ricci flowStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2111.05517arxiv:2111.05517Wed, 10 Nov 2021 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local ν-functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.Ricci flowPerelman entropy https://arxiv.org/abs/2110.04661arxiv:2110.04661Sat, 09 Oct 2021 00:00:00 GMTRichard H. Bamler, Pak-Yeung Chan, Zilu Ma, Yongjia Zhang — In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than r^((n+1)/2). This result not only improves the estimate in [CMZ21b, Theorem 1.3], but also is optimal since the Bryant soliton and Appleton's solitons [Ap17] have exactly this growth rate.Ricci solitonsAncient solutionsPerelman entropy(Non)collapsing https://arxiv.org/abs/2109.14763arxiv:2109.14763Wed, 29 Sep 2021 00:00:00 GMTPak-Yeung Chan, Zilu Ma, Yongjia Zhang — In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in [CCGGIIKLLN10], (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.Ricci flowAncient solutionsSingularities https://arxiv.org/abs/2108.06052arxiv:2108.06052Fri, 13 Aug 2021 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this article we study the breathers of the mean curvature flow in the Euclidean space. A breather is a solution to the mean curvature flow which repeats itself up to isometry and scaling once in a while. We prove several no breather theorems in the noncompact category, that is, under certain conditions, a breather of the mean curvature flow must be a solitonic solution (self-shrinker, self-expander, or translator).Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact