https://neckpinch.com/authors/pak-yeung-chan/New arXiv papers by Pak-Yeung Chan, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2512.05625arxiv:2512.05625Fri, 05 Dec 2025 00:00:00 GMTPak-Yeung Chan, Ming Hsiao — In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically, then the curvature operator decays at the rate BigO(1/r²) when n=4 and BigO((log r)/r²) when n≥5. This refines the curvature bounds in a previous result by Cao-Liu-Xie, and removes the nonnegative Ricci curvature assumption in the estimates by Cao-Liu and Cao-Liu-Xie. As a geometric application, we establish the existence and uniqueness of C^(1,α) conical structure at infinity of Ricci expander with finite Ricci curvature ratio. In the steady case, using an integral estimate of the curvature, we prove that the curvature operator has at most polynomial growth when the potential function is proper and the Ricci curvature has linear decay. Moreover, we also confirm that the curvature is bounded if we further assume the Ricci curvature has super-linear decay BigO(r^(−1-ε)). As an application, we prove the existence and uniqueness of cylindrical structure at infinity of steady soliton with super-linear Ricci curvature decay and proper potential function.Ricci solitonsStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2510.23005arxiv:2510.23005Mon, 27 Oct 2025 00:00:00 GMTPak-Yeung Chan, Yi Lai, Man-Chun Lee — For any n≥ 4, we construct an (n-2)-parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at a critical point of the soliton potential. Among them lies an (n-3)-parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author and produced new examples of steady gradient Ricci solitons with non-negative curvature operator for n≥ 4. Our approach is based on constructing continuous families of Ricci flows smoothing emanating from continuous families of spherical polyhedra which still preserves symmetry. This is built upon a new stability result of Ricci flows with scaling invariant estimates. As another application of the method, we prove the stability of asymptotically conical expanding solitons constructed by Deruelle under L^∞ perturbation of links. In particular, the C⁰-convergence of smooth links implies the smooth convergence of the expanding solitons.Ricci flowRicci solitons(Non)collapsingStability & uniquenessBoundaries & non-compactEigenvalues & spectrum https://arxiv.org/abs/2503.12416arxiv:2503.12416Sun, 16 Mar 2025 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee — In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay. This answers a question proposed by Chow-Lu-Ni and also a question by Cao-Liu, respectively.Ricci solitons https://arxiv.org/abs/2404.12755arxiv:2404.12755Fri, 19 Apr 2024 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee, Luke T. Peachey — Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.Ricci flowRicci solitonsSphere theoremsKähler geometryWeak & singular flows https://arxiv.org/abs/2403.04089arxiv:2403.04089Wed, 06 Mar 2024 00:00:00 GMTPak-Yeung Chan, Ronan J. Conlon, Yi Lai — In 1996, H.-D. Cao constructed a U(n)-invariant steady gradient Kähler-Ricci soliton on ℂⁿ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on ℂⁿ is necessarily U(n)-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for n=2. Here, we construct a family of U(1)× U(n-1)-invariant, but not U(n)-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real (1, 1)-forms (in particular, with strictly positive sectional curvature) on ℂⁿ for n≥3, thereby answering Cao's question in the negative for n≥3. This family of steady Ricci solitons interpolates between Cao's U(n)-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's U(n-1)-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of ℙⁿ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by 2 on real (1, 1)-forms.Ricci solitonsKähler–Ricci solitonsAncient solutionsSphere theoremsKähler geometryStability & uniqueness 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/2305.01396arxiv:2305.01396Tue, 02 May 2023 00:00:00 GMTPak-Yeung Chan, Man-Chun Lee — In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence theory without non-collapsing assumption. We use it to show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay are necessarily flat. The second main result concerns three-manifolds with non-negative Ricci curvature of quadratic decay. By combining our newly established curvature estimate and method in Kähler geometry, we show that if the curvature decays slightly faster even in average sense, the manifold must be flat. This strengthens a result of Reiris. In the compact case, we use the Ricci flow regularization to generalize the celebrated Gromov-Ruh Theorem in this direction.Ricci flow(Non)collapsing3-manifoldsKähler geometryBoundaries & non-compact 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/2207.07495arxiv:2207.07495Fri, 15 Jul 2022 00:00:00 GMTPak-Yeung Chan, Shaochuang Huang, Man-Chun Lee — In this work, we construct distance like functions with integral hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean.Ricci flowScalar curvature 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 https://arxiv.org/abs/2108.05477arxiv:2108.05477Thu, 12 Aug 2021 00:00:00 GMTPak-Yeung Chan, Bo Zhu — We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.Ricci solitonsRicci-flat & Calabi–Yau4-manifoldsBoundaries & non-compact