https://neckpinch.com/authors/weimin-sheng/New arXiv papers by Weimin Sheng, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.29852arxiv:2610.29852Wed, 21 Oct 2026 00:00:00 GMTWeimin Sheng, Ye Zhu — We give a finite negative-shift criterion for preserving a prescribed exterior noncollapsing bound under fully nonlinear curvature flows whose hypersurfaces need not be convex. The criterion uses the optimized two-point expression of Andrews and Langford. We verify it by a uniform global two-matrix estimate for the explicit speed F_ε(A)= tr A- εfracnormA^(∘)² tr A on the open cone Γ=A∈Sym(n): tr A>0, normA^(∘)< tr A\. Here A^∘ denotes the trace-free part of A. This cone contains the positive cone and also matrices with a negative eigenvalue, while the displayed speed is concave and non-convex. For an explicit range of parameters depending on the initial exterior bound, the cone and that bound are preserved throughout the smooth embedded existence interval. Together with the known interior estimate for concave speeds, this gives two-sided tangent-ball bounds for this nonlinear family, including closed nonconvex initial hypersurfaces.Other curvature flows(Non)collapsingEigenvalues & spectrum https://arxiv.org/abs/2609.31023arxiv:2609.31023Fri, 25 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.Other curvature flowsHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2609.29199arxiv:2609.29199Thu, 24 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let H^n+1 be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^betar; we treat both regimes: beta>1+kalpha and beta=1+kalpha. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_infty respectively in two different cases.Other curvature flowsHomogeneous spaces https://arxiv.org/abs/2509.21830arxiv:2509.21830Fri, 26 Sep 2025 00:00:00 GMTWeimin Sheng, Ye Zhu — We study closed, embedded hypersurfaces in Euclidean space evolving by fully nonlinear curvature flows, whose speed is given by a symmetric, monotone increasing, 1-homogeneous, positive underlying speed function F composed with a modulating function Ψ. Under the assumption that F is convex or inverse-concave and that Ψ satisfies the corresponding structural conditions, we establish exterior noncollapsing estimates for the flow. The main difficulty stems from the nonlinearity of the evolution equation satisfied in the viscosity sense by the exscribed curvature, whereas in previous works it is a solution to the linearized flow. Moreover, in the case where F is inverse-concave, we refine Andrews and Langford's argument for the interior case.Other curvature flows(Non)collapsingHomogeneous spaces https://arxiv.org/abs/2508.07361arxiv:2508.07361Sun, 10 Aug 2025 00:00:00 GMTWeimin Sheng, Jiazhuo Yang — In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in ℝⁿ⁺¹. More precisely, we consider a hypersurface M in ℝⁿ⁺¹ deformed by a flow along its unit normal with its speed f(r)σₖ^α where σₖ is the k-th elementary symmetric polynomial of M's principle curvatures, r is the distance of the point on M to the origin, f is a smooth nonnegative function on [0,∞) and α> 0. Under some suitable conditions on f, we prove that starting from a star-shaped and k-convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in.Other curvature flowsHomogeneous spacesLong-time existence & convergence https://arxiv.org/abs/2212.01822arxiv:2212.01822Sun, 04 Dec 2022 00:00:00 GMTWeimin Sheng, Ke Xue — In this paper, we study the Lₚ-Gaussian Minkowski problem, which arises in the Lₚ-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space ℝⁿ⁺¹, and prove its long-time existence and converges smoothly to a smooth solution of the normalized Lₚ Gaussian Minkowski problem in cases of p>0 and −n-1<p≤ 0 with even prescribed function respectively. We also provide a parabolic proof in the smooth category to the Lₚ-Gaussian Minkowski problem in cases of p≥ n+1 and 0<p<n+1 with even prescribed function, respectively.Other curvature flowsGauss curvature flowLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2110.12157arxiv:2110.12157Sat, 23 Oct 2021 00:00:00 GMTWenshuai Jiang, Weimin Sheng, Huaiyu Zhang — In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is W^(1,p) for some n<p≤ ∞. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in distributional sense, then the manifold is isometric to a Ricci flat manifold.Ricci flowRicci-flat & Calabi–YauScalar curvature https://arxiv.org/abs/2108.11657arxiv:2108.11657Thu, 26 Aug 2021 00:00:00 GMTQi-Rui Li, Weimin Sheng, Deping Ye, Caihong Yi — In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure C_(G,Ψ,λ)(Ω,·) of convex body Ω in ℝⁿ⁺¹ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.Other curvature flows