https://neckpinch.com/authors/yu-li/New arXiv papers by Yu Li, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.41345arxiv:2610.41345Fri, 30 Oct 2026 00:00:00 GMTYu Li, Longhuan Yan, Wenjia Zhang — We prove strong uniqueness of tangent flows at the first singular time of a closed Ricci flow when one tangent flow is generated by a complete smooth asymptotically conical Ricci shrinker. The modified flow converges to this shrinker in a fixed gauge, with quantitative estimates on expanding domains. The main ingredients are a Łojasiewicz-Simon inequality for Perelman's entropy, based on the global weighted divergence-free gauge constructed in the preceding paper, and estimates controlling the geometric radius under Ricci flow.Ricci flowRicci solitonsSingularitiesPerelman entropyStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2610.39814arxiv:2610.39814Thu, 29 Oct 2026 00:00:00 GMTYu Li, Longhuan Yan, Wenjia Zhang — We develop the weighted Sobolev and Hölder analysis needed to study perturbations of complete asymptotically conical (AC) Ricci shrinkers. Weighted elliptic estimates yield a global weighted divergence-free slice in spaces that retain the asymptotic data at infinity. As a geometric application, we prove pointed Gromov-Hausdorff rigidity of nontrivial AC shrinkers with vanishing infinitesimal solitonic deformation space.Ricci solitonsLimit spaces & RCDStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2610.30939arxiv:2610.30939Thu, 22 Oct 2026 00:00:00 GMTYu Li — We study ancient Ricci flows whose tangent flow at backward infinity is a smooth compact Ricci shrinker. After normalization, every nontrivial solution either converges exponentially or has polynomial decay governed by neutral directions. We recover the finite-dimensional parametrization of the exponentially convergent solutions and establish finite-order asymptotic expansions for all slowly convergent solutions. For principal slow solutions, we obtain a locally bijective finite-dimensional parametrization compatible with diffeomorphisms, parabolic rescaling, and time translation.Ricci flowRicci solitonsAncient solutionsSingularitiesLong-time existence & convergence https://arxiv.org/abs/2610.05643arxiv:2610.05643Mon, 05 Oct 2026 00:00:00 GMTYu Li, Junsheng Zhang — We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler–Ricci shrinkers. We also prove that every noncompact Kähler–Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration. For shrinkers with maximal volume growth, this cone is isometric to the metric completion of the regular part equipped with the intrinsic distance induced by the limiting Kähler metric.Ricci solitonsKähler geometryScalar curvatureBoundaries & non-compactAnalytic minimal model program https://arxiv.org/abs/2610.05204arxiv:2610.05204Sun, 04 Oct 2026 00:00:00 GMTYu Li, Bing Wang — We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.Ricci flowSingularities(Non)collapsing https://arxiv.org/abs/2607.25644arxiv:2607.25644Tue, 28 Jul 2026 00:00:00 GMTYu Li, Junsheng Zhang — We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an S¹-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsing4-manifoldsKähler geometryScalar curvatureLimit spaces & RCDStability & uniqueness https://arxiv.org/abs/2606.13301arxiv:2606.13301Thu, 11 Jun 2026 00:00:00 GMTYu Li — Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any t₀>0 and any connected component Z_(t₀)' of the time-slice Z_(t₀) of the completion, we prove that (𝓡ₜ',d_(gₜ)) xrightarrow[t↗ t₀]mathrmGromov-Hausdorff (Z_(t₀)',d_(t₀)^Z), where 𝓡ₜ' is the corresponding connected component of the regular part and 𝓡ₜ' denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic 1-rectifiable, and its time image has vanishing 1/2-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most 1. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.Ricci flow(Non)collapsingHeat kernel3-manifoldsWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/2605.17001arxiv:2605.17001Sat, 16 May 2026 00:00:00 GMTHanbing Fang, Yu Li — In this paper, we extend the results of to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed 𝓦-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder ℝ^k × N^(n-k), where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset 𝓢^k_qc(N)⊂ 𝓢^k, consisting of points at which some tangent flow is given by ℝ^k × N^(n-k) or its quotient, is horizontally parabolic k-rectifiable.Ricci flowEinstein metricsSingularitiesStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2510.26317arxiv:2510.26317Thu, 30 Oct 2025 00:00:00 GMTHanbing Fang, Yu Li — In this paper, we study the singular set 𝓢 of a noncollapsed Ricci flow limit space, arising as the pointed Gromov–Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set 𝓢 admits a natural stratification: \beginequation* \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^n-2=\mathcal S, \endequation* where a point z ∈ 𝓢^k if and only if no tangent flow at z is (k+1)-symmetric. In general, the Hausdorff dimension of 𝓢^k with respect to the spacetime distance is at most k. We show that the subset 𝓢^k_qc ⊂ 𝓢^k, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic k-rectifiable. In dimension four, we prove the stronger statement that each stratum 𝓢^k is parabolic k-rectifiable for k ∈ \0, 1, 2\. Furthermore, we establish a sharp uniform 𝓗²-volume bound for 𝓢 and show that, up to a set of 𝓗²-measure zero, the tangent flow at any point in 𝓢 is backward unique. In addition, we derive L¹-curvature bounds for four-dimensional closed Ricci flows. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows.Ricci flowSingularities(Non)collapsing3-manifolds4-manifoldsLimit spaces & RCD https://arxiv.org/abs/2510.20320arxiv:2510.20320Thu, 23 Oct 2025 00:00:00 GMTHanbing Fang, Yu Li — In this paper, we establish a Lojasiewicz inequality for the pointed 𝓦-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder ℝ^k × S^(n-k) or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.Ricci flowSingularitiesStability & uniqueness https://arxiv.org/abs/2510.12398arxiv:2510.12398Tue, 14 Oct 2025 00:00:00 GMTHanbing Fang, Yu Li — We establish a weak compactness theorem for the moduli space of closed Ricci flows, each equipped with a natural spacetime distance, under pointed Gromov–Hausdorff convergence. For the subspace of flows with uniformly bounded entropy, we further develop a structure theory for the corresponding noncollapsed Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.Ricci flow(Non)collapsingLimit spaces & RCD https://arxiv.org/abs/2504.09811arxiv:2504.09811Mon, 14 Apr 2025 00:00:00 GMTHanbing Fang, Yu Li — In this paper, we establish uniform and sharp volume estimates for the singular set and the quantitative singular strata of mean curvature flows starting from a smooth, closed, mean-convex hypersurface in ℝⁿ⁺¹.Mean curvature flow https://arxiv.org/abs/2412.13546arxiv:2412.13546Wed, 18 Dec 2024 00:00:00 GMTConghan Dong, Yu Li — We establish a uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and a uniform curvature bound. Additionally, we extend the non-collapsing result to a broader class of smooth metric measure spaces satisfying Bakry-Émery conditions.Ricci solitons(Non)collapsingLimit spaces & RCD https://arxiv.org/abs/2410.21950arxiv:2410.21950Tue, 29 Oct 2024 00:00:00 GMTYu Li, Wenjia Zhang — In this paper, we prove the uniqueness of Kähler Ricci shrinkers on toric orbifolds, extending the corresponding results previously established for toric manifolds.Ricci solitonsKähler geometryStability & uniqueness https://arxiv.org/abs/2305.06143arxiv:2305.06143Wed, 10 May 2023 00:00:00 GMTYu Li, Wenjia Zhang — In this paper, we establish the rigidity of the generalized cylinder Nⁿ × ℝ^(m-n), or a quotient thereof, in the space of Ricci shrinkers equipped with the pointed-Gromov-Hausdorff topology. Here, N is a stable Einstein manifold that has an obstruction of order 3. The proof is based on a quantitative characterization of the rigidity of compact Ricci shrinkers, a rigidity inequality of mixed orders on generalized cylinders, and the method of contraction and extension. As an application, we prove the uniqueness of the tangent flow for general compact Ricci flows under the assumption that one tangent flow is a generalized cylinder.Ricci flowRicci solitonsEinstein metricsSingularitiesLimit spaces & RCDStability & uniqueness https://arxiv.org/abs/2301.09784arxiv:2301.09784Tue, 24 Jan 2023 00:00:00 GMTYu Li, Bing Wang — In this paper, we prove that any Kähler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all Kähler Ricci shrinker surfaces.Ricci solitonsKähler geometry https://arxiv.org/abs/2301.08430arxiv:2301.08430Fri, 20 Jan 2023 00:00:00 GMTYu Li, Bing Wang — This paper is the sequel to our study of heat kernels on Ricci shrinkers in. In this paper, we improve many estimates in and extend the recent progress of Bamler. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of IF-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.Ricci flowRicci solitonsHeat kernelLimit spaces & RCD https://arxiv.org/abs/2203.07762arxiv:2203.07762Tue, 15 Mar 2022 00:00:00 GMTYu Li, Wenjia Zhang — In this paper, we prove that any Ricci shrinker that is sufficiently close to (ℂ P^N,g_FS) in the Gromov-Hausdorff sense must itself be isometric to (ℂ P^N,g_FS).Ricci solitonsLimit spaces & RCDStability & uniqueness https://arxiv.org/abs/2108.03622arxiv:2108.03622Sun, 08 Aug 2021 00:00:00 GMTYu Li, Bing Wang — In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to Sⁿ⁻¹× ℝ in the pointed-Gromov-Hausdorff topology must itself be isometric to Sⁿ⁻¹× ℝ.Ricci solitonsLimit spaces & RCDStability & uniqueness