https://neckpinch.com/authors/hongbing-qiu/New arXiv papers by Hongbing Qiu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2509.01100arxiv:2509.01100Mon, 01 Sep 2025 00:00:00 GMTShu-Cheng Chang, Hongbing Qiu — We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded as the Sasaki analogue of the work of Munteanu and Sesum [15] on Ricci solitons.Ricci solitonsSasakian & contact geometry https://arxiv.org/abs/2508.07900arxiv:2508.07900Mon, 11 Aug 2025 00:00:00 GMTShu-Cheng Chang, Hongbing Qiu, Liuyang Zhang — We study the self-shrinkers of the Legendrian mean curvature flow in the contact Euclidean space ℝ²ⁿ⁺¹, that is, the Legendrian submanifolds which move under the flow by the contact dilations (x,y,z)↦(λx,λy,λ²z), and we explain why it is the generator of these dilations, and not the Euclidean position vector, which appears in the self-shrinker equation. Our main result is that a complete properly immersed Legendrian self-shrinker with bounded Legendrian angle is a Legendrian n-plane. On a Legendrian self-shrinker the Legendrian angle is a constant multiple of the coordinate z, so that this is a statement about the position of the submanifold, and it has two consequences. A complete properly immersed Legendrian self-shrinker which lies in a slab |z|≤ C is a Legendrian n-plane, and in particular there are no closed Legendrian self-shrinkers in ℝ²ⁿ⁺¹; this has no counterpart for Lagrangian self-shrinkers in ℂⁿ, among which there are closed ones. And the potential of an entire smooth Legendrian self-shrinking graph is a quadratic polynomial, without any further assumption. The proof uses a volume growth estimate for Legendrian self-shrinkers with respect to a Gaussian weight which is built from the horizontal part of the generator of the dilations.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2502.16148arxiv:2502.16148Sat, 22 Feb 2025 00:00:00 GMTShu-Cheng Chang, Fengjiang Li, Chien Lin, Hongbing Qiu — In this paper, we study several properties of Sasaki-Ricci solitons as singularity models of the Sasaki-Ricci flow. First, we establish several fundamental equations for Sasaki-Ricci solitons, which enable us to derive potential estimates and prove the positivity of the scalar curvature. Then we present two criteria for the transverse rigidity of Sasaki-Ricci solitons. As essential applications, we prove that any low-dimensional Sasaki-Ricci soliton with constant scalar curvature must be Sasaki-Einstein, and that any Sasaki-Ricci soliton with harmonic Weyl tensor is a finite quotient of the sphere.Ricci flowRicci solitonsSingularitiesScalar curvatureStability & uniquenessSasakian & contact geometry https://arxiv.org/abs/2412.08867arxiv:2412.08867Thu, 12 Dec 2024 00:00:00 GMTQun Chen, Hongbing Qiu — By carrying out a point-wise estimate for the second fundamental form, we prove a rigidity theorem of complete noncompact ancient solutions to the mean curvature flow in codimension one. Moreover, we derive an optimal growth condition.Mean curvature flowAncient solutionsStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2311.12400arxiv:2311.12400Tue, 21 Nov 2023 00:00:00 GMTHongbing Qiu, Y. L. Xin — By carrying out refined curvature estimates, we prove better rigidity theorems of complete noncompact ancient solutions to the mean curvature flow in higher codimension under various Gauss image restriction.Mean curvature flowAncient solutionsStability & uniquenessBoundaries & non-compact