https://neckpinch.com/authors/hanbing-fang/New arXiv papers by Hanbing Fang, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.36563arxiv:2610.36563Tue, 27 Oct 2026 00:00:00 GMTHanbing Fang — We prove that a complete connected four-dimensional gradient steady Ricci soliton with positive curvature operator that is κ-noncollapsed at all scales admits an effective isometric O(3) action preserving the potential function and having principal orbits diffeomorphic to S².Ricci solitons(Non)collapsing4-manifoldsSphere theorems 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