https://neckpinch.com/authors/chih-wei-chen/New arXiv papers by Chih-Wei Chen, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1803.09591/arxiv:1803.09591Mon, 26 Mar 2018 00:00:00 GMTChih-Wei Chen, Zhenlei Zhang — Let g(t)\_(t∈ [0,T)) be the solution of the Ricci flow on a closed Riemannian manifold Mⁿ with n≥ 3. Without any assumption, we derive lower volume bounds of the form rm Vol_(g(t))≥ C (T-t)^(n/2), where C depends only on n, T and g(0). In particular, we show that rm Vol_(g(t)) ≥ e^(Tλ−n/2) (4/((A(λ−r)+4B)T))^(n/2)(T-t)^(n/2), where r:=inf_(\|φ\|₂²=1) ∫_M Rφ² drm vol_(g(0)), λ:=inf_(\|φ\|₂²=1) ∫_M 4|∇φ|²+Rφ² drm vol_(g(0)) and A,B are Sobolev constants of (M,g(0)). This estimate is sharp in the sense that it is achieved by the unit sphere with scalar curvature R_(g(0))=n(n-1) and A=4/(n(n-2))ωₙ^(−2/n), B=(n-1)/(n-2)ωₙ^(−2/n). On the other hand, if the diameter satisfies rm diam_(g(t))≤ c₁√(T-t) and there exist a point x₀∈ M such that R(x₀,t)≤ c₂(T-t)⁻¹, then we have rm Vol_(g(t))≤ C (T-t)^(n/2) for all t>T/2, where C depends only on c₁,c₂,n,T and g(0).Ricci flowScalar curvature https://neckpinch.com/papers/1709.00143/arxiv:1709.00143Fri, 01 Sep 2017 00:00:00 GMTChih-Wei Chen, Kuo-Wei Lee — Let (M³, g, f) be a nontrivial 3-dimensional steady gradient Ricci soliton. If the scalar curvature R satisfies c₁r^(−b)≤ R≤ c₂r^(−a) for some a∈(0,1], b≥ a, and c₁,c₂>0, then the umbilical ratio of the level sets of f satisfies (2|A|²−H²)/(H²)∈ O(r^(6a-(8a²)/b))∩ O(r^(2b-4a)).Ricci solitons3-manifoldsScalar curvatureWeak & singular flows https://neckpinch.com/papers/1602.01939/arxiv:1602.01939Fri, 05 Feb 2016 00:00:00 GMTChih-Wei Chen — We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with |Ric|≤ K must satisfy |Rm|≤ Ct⁻¹ for all t∈ (0,T], where C depends only on the dimension n and T depends on K and the injectivity radius inj_(g(t)). In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.Ricci flowRicci solitonsScalar curvatureLimit spaces & RCD