https://neckpinch.com/authors/zilu-ma/New arXiv papers by Zilu Ma, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2502.19804arxiv:2502.19804Thu, 27 Feb 2025 00:00:00 GMTRonan J. Conlon, Max Hallgren, Zilu Ma — We show that any non-collapsed finite time singularity of the Ricci flow on a compact Kähler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle 𝓞_(ℙ¹)(−1)→ℙ¹.Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2502.08500arxiv:2502.08500Wed, 12 Feb 2025 00:00:00 GMTJames Isenberg, Dan Knopf, Zilu Ma, Natasa Sesum — We demonstrate that any four-dimensional shrinking Ricci soliton (𝓑 × S², g), where 𝓑 is any two-dimensional complete noncompact surface and g is a warped product metric over the base 𝓑, has to be isometric to the generalized cylinder ℝ²×S² equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products – but not products – and provide rigorous examples of the formation of generalized cylinder singularity models ℝ^k×S^ℓ.Ricci flowRicci solitonsSingularities4-manifoldsSurfacesBoundaries & non-compactRotational symmetry & warped products 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/2311.09405arxiv:2311.09405Wed, 15 Nov 2023 00:00:00 GMTZilu Ma, Hamidreza Mahmoudian, Natasa Sesum — In this paper we study 4d gradient steady Ricci solitons, which are weak κ-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact κ-solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.Ricci flowRicci solitonsAncient solutions3-manifolds4-manifolds 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/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/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