https://neckpinch.com/authors/liang-cheng/New arXiv papers by Liang Cheng, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.21903arxiv:2610.21903Thu, 15 Oct 2026 00:00:00 GMTLiang Cheng — We establish local convexity and cylindrical pinching estimates for hypersurface mean curvature flow under an intermediate convexity condition and integral bounds on the mean curvature. These estimates hold after a positive waiting time and do not require pointwise upper bounds for the mean curvature on the parabolic boundary. In arbitrary codimension, we also establish local planarity estimates for properly defined flows satisfying a uniform cₙ-pinching condition, without assuming bounded curvature. As applications, the hypersurface estimates yield convexity and cylindrical pinching for complete, spacetime-proper ancient hypersurface flows with uniform Euclidean area growth, without any bounded-curvature assumption. The planarity estimates imply that every complete, connected, spacetime-proper ancient flow satisfying the uniform cₙ-pinching condition is either a static affine n-plane or lies in a fixed affine (n+1)-plane for all time.Mean curvature flowAncient solutionsCurvature pinching https://arxiv.org/abs/2604.15792arxiv:2604.15792Fri, 17 Apr 2026 00:00:00 GMTLiang Cheng, Zhenyu Lu — In this paper, we provide a proof of Hamilton's extrinsic pinching theorem using the mean curvature flow approach.Mean curvature flowCurvature pinching https://arxiv.org/abs/2502.15219arxiv:2502.15219Fri, 21 Feb 2025 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this article, we prove an ε-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to 1, then the curvature radius at the base point cannot be too small.Ricci flowReduced volume https://arxiv.org/abs/2310.05011arxiv:2310.05011Sun, 08 Oct 2023 00:00:00 GMTLiang Cheng — By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of L² logarithmic Sobolev inequality. Precisely, we prove that if a metric g on an open set V in an n-dimensional Riemannian manifold satisfies ∫_V R(g) dvol_g ≥ 0 and I(V)≥ I(ℝⁿ), or ∫_V R(g) dvol_g ≥ 0 and S(V)≥ S(ℝⁿ), then g=g_(ℝⁿ) on V, where R(g) is the scalar curvature of g, ℝⁿ is Euclidean space, I(V) is the isoperimetric constant of V and S(V) is best constant of L² logarithmic Sobolev inequality of V. Moreover,we also obtain the local ℝⁿ-rigidity about local Perelman's ν-entropy, and local Sⁿ-rigidity (resp. ℍⁿ-rigidity) theorems regarding the cases concerning R(g)≥ n(n-1) (resp. R(g)≥ −n(n-1)), weighted isoperimetric constant and best constant of weighted L² logarithmic Sobolev inequality for the weighted metric (cos(d_g(p,x))/2)⁻⁴g (resp. (cosh(d_g(p,x))/2)⁻⁴g).Ricci flowScalar curvatureStability & uniquenessGeometric inequalities https://arxiv.org/abs/2303.06648arxiv:2303.06648Sun, 12 Mar 2023 00:00:00 GMTLiang Cheng — By using the Yamabe flow, we prove that if (Mⁿ,g), n≥3, is an n-dimensional locally conformally flat complete Riemannian manifold Rc≥ εRg>0, where ε>0 is a uniformly constant, then Mⁿ must be compact. Our result shows that Hamilton's pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.Yamabe flowCurvature pinching https://arxiv.org/abs/2210.15397arxiv:2210.15397Thu, 27 Oct 2022 00:00:00 GMTLiang Cheng — In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a ball with its closure contained in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean isoperimetric constant, or almost Euclidean local boldsymbolν constant, then we can construct a solution of Ricci flow in the ball which have the pseudolocality property. We also give two applications. First, we prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly and almost Euclidean isoperimetric inequality holds locally. Second, we show that any complete manifold with nonnegative scalar curvature and Euclidean isoperimetric inequality must be isometric to the Euclidean space.Ricci flowScalar curvatureBoundaries & non-compactHarmonic maps & DeTurck trickGeometric inequalities https://arxiv.org/abs/2112.07818arxiv:2112.07818Wed, 15 Dec 2021 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this paper, we prove a pseudolocality-type theorem for 𝓛-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the 𝓛-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen.Ricci flowStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2108.06052arxiv:2108.06052Fri, 13 Aug 2021 00:00:00 GMTLiang Cheng, Yongjia Zhang — In this article we study the breathers of the mean curvature flow in the Euclidean space. A breather is a solution to the mean curvature flow which repeats itself up to isometry and scaling once in a while. We prove several no breather theorems in the noncompact category, that is, under certain conditions, a breather of the mean curvature flow must be a solitonic solution (self-shrinker, self-expander, or translator).Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact