Neckpinch — Curvature pinching

Entropy and round-point convergence for the σₖ-curvature flow

Mon, 02 Nov 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We prove that every smooth closed strictly convex hypersurface in ℝⁿ⁺¹ contracts to a round point under the σₖ-curvature flow for 2≤ k≤ n-1, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.

Rigidity of shrinking Ricci solitons with low-dimensional focal sets

Mon, 02 Nov 2026 00:00:00 GMT

Matheus Andrade Ribeiro de Moura Horácio — We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.

Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold

Thu, 29 Oct 2026 00:00:00 GMT

Francesco Ruatta — We derive an interior planarity estimate for space-time proper mean curvature flows of n-dimensional submanifolds of ℝ^(n+p), n≥5, that are uniformly quadratically pinched up to the Andrews-Baker threshold 4/3n. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for n≥5, and by known results for 2≤ n≤4, every space-time proper ancient solution that is uniformly 4/3n-pinched is either a static affine n-plane or lies in a fixed affine (n+1)-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.

Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

Mon, 19 Oct 2026 00:00:00 GMT

Tsz-Kiu Aaron Chow — We develop a pinching method for mean curvature flow in higher codimension. Motivated by pinching constructions for Ricci flow, the method uses a nonlinear deformation of the second fundamental form. We introduce two tensorial curvature conditions for closed immersions in Euclidean space which extend beyond the scope of classical scalar pinching results. In dimensions at least ten, we apply the method to prove convergence to a round point under the first condition and a canonical neighborhood theorem under the second.

Local Convexity and Planarity Estimates for Mean Curvature Flow

Thu, 15 Oct 2026 00:00:00 GMT

Liang 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.

Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

Thu, 15 Oct 2026 00:00:00 GMT

Lei Zhang, Zhenlei Zhang — In this paper, we study heat kernel estimates along the Kähler-Ricci flow with finite time noncollapsing singularities over a compact Kähler manifold. The starting point is the type-II scalar-curvature estimate, which gives the natural parabolic scale r≲ T-t near the singular time. By combining the distance distortion and space-time cut-off estimates of Bamler–Zhang with entropy, mean-value inequality, and conjugate heat-kernel estimates, we obtain two-sided Gaussian bounds and gradient bounds that are uniform on this scale. Under the global type-I scalar-curvature estimate, the same estimates hold on every scale. We then use these estimates and the monotonicity of the W₁-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances dₜ (induced by ω(t)) converge uniformly to this pseudo-distance. Its metric quotient is the unique Gromov–Hausdorff limit. Finally, we apply Bamler's theory and the recent work of Hallgren–Zhang to obtain the following. Every fixed-base-point type-I blow-up admits a subsequential metric-flow limit whose time slices are singular shrinking Kähler–Ricci soliton spaces with normal complex-analytic and log-terminal structure, unique tangent cones, and a polarized Fano fibration. For moving base points, the limiting metric flow, equipped with its limiting conjugate heat flow, is a metric soliton precisely under an entropy-pinching condition.

Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

Thu, 08 Oct 2026 00:00:00 GMT

Matheus Horácio, Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete nonsteady gradient Kähler-Ricci soliton satisfying Ric+∇² f=λg, λ≠ 0. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as N^(2k)×ℂ^(m-k), where N^(2k) is Kähler-Einstein with Ric_(g_N)=λg_N. In the shrinking case the quotient is trivial; in the normalization λ=1/2 one has R≡ k. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if 𝓛_∇fRic is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching 0≤Ric≤λg forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

Ricci flow on manifolds with positive isotropic curvature in all dimensions

Thu, 01 Oct 2026 00:00:00 GMT

Simon Brendle, Raphael Tsiamis — We extend the classification of manifolds with positive isotropic curvature to all dimensions n ≥ 5. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact κ-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

Classification of quadratically pinched translators in higher codimension

Thu, 01 Oct 2026 00:00:00 GMT

Debora Impera, Michele Rimoldi, Francesco Ruatta — We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete n-dimensional connected translator isometrically immersed in (ℝ^(n+p),⟨·,·⟩) satisfies |B|²≤ (4/3n-ε₀)|H|² for some ε₀>0, then either it is an affine n-plane or |H|>0 everywhere and the translator is contained in an (n+1)-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition |B|²≤ (cₙ−ε₀)|H|², cₙ=min\4/3n,1/(n-2)\ for n≥3, with cₙ=2/3 when n=2, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an (n+1)-dimensional affine subspace. No entropy assumption is required.

The planarity estimate in dimensions five and six

Wed, 30 Sep 2026 00:00:00 GMT

Stephen Lynch — Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least 7, while in dimensions 5 and 6 a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions 5 and 6 throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

The rigidity theorems for complete self-expander of mean curvature flow

Thu, 17 Sep 2026 00:00:00 GMT

Zhi Li, Guoxin Wei — Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander x:M→ ℝ^(n+p)(n≥3) that satisfies both (∫_M|A|ⁿdμ)^(1/n)

Alexandrov–Fenchel inequalities for convex domains in the sphere

Sat, 12 Sep 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Rong Zhou — We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

Sectional Curvature Pinching of Two-Step Nilmanifolds

Fri, 11 Sep 2026 00:00:00 GMT

Tomoya Tatsuno — We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval [−3, −3/2]. The upper bound −3/2 is achieved by the complex Heisenberg group Heis₃(ℂ) with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold N has the pinching constant −3/2, then N admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant −3. This is derived by showing that there is an open neighborhood U of Heis₃(ℝ)× ℝⁿ⁻³ in the space of n-dimensional 2-step nilmanifolds such that the pinching constant is −3 on U, and any 2-step nilpotent Lie group N has a metric g such that (N,g) lies in U. In fact, if N is not isomorphic to Heis₃(ℝ)× ℝⁿ⁻³, then there is a curve gₜ of metrics on N with (N,gₜ)∈ U, showing that there are uncountably many left-invariant metrics on N such that the pinching constant is −3. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant −3/2 is also given, and the pinching constants of various examples are computed.

Pinching cones for positive isotropic curvature in dimensions seven and eight

Thu, 27 Aug 2026 00:00:00 GMT

Jae Ho Cho — We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE d/dtR=Q(R) in dimensions n=7,8, thereby extending the pinching estimate established by Brendle for n≥ 12 and by Chen for 9≤ n≤ 11. In dimension n=8, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at n=8 by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension n=7, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension n=8. The n=8 pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible (n-1)-dimensional space forms, extending a theorem of Brendle from n≥ 12. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact κ-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from n=4 or n≥ 12.

Positive τ-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

Wed, 26 Aug 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Rong Zhou — Let n≥3 and 0≤τ≤2. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive τ-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint n=3, τ=2, positive τ-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of Sⁿ⁻¹×S¹. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.

A Lean Formalization of Hamilton's Three-Manifold Theorem

Fri, 21 Aug 2026 00:00:00 GMT

Bennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

Tue, 11 Aug 2026 00:00:00 GMT

Sara Albert-Niclòs, Esther Cabezas-Rivas, Shujing Pan — We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential C^∞ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching

Mon, 20 Jul 2026 00:00:00 GMT

Jian Ge — Let V be an n-dimensional Euclidean vector space, ,where n≥ 4, and ℓ = lfloorn/2rfloor. We prove the sharp pointwise estimate q₂(E) ≥ −(2(ℓ −1))/3ℓ Scal(E) Id_(Λ²V^*) for every algebraic curvature tensor E on V with nonnegative sectional curvature. Applying this estimate to the decomposition Rm_g=KₘᵢₙI+E, we obtain the vanishing of H²(M; ℝ) under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields b₂(M)=0 in odd dimensions and b₂(M)≤ 1 in even dimensions. At even-dimensional endpoint, b₂(M)>0 forces (M, g) to be isometric, up to scaling, to ℂP}^(ℓ) with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion Kₘᵢₙ ≥ (n(n-1))/(n²−n+12)S₀ Longrightarrow PIC2. The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.

The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

Sun, 05 Jul 2026 00:00:00 GMT

Fagui Li, Yuhang Zhao — In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work. If λ_ρ(Σ)≥λ>0 and S=|A|²<1+λ, then Σ is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives λ_ρ(Σ)≥1/2. Consequently, the pointwise upper pinching S<3/2 forces Σ to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in ℝ³, we also obtain the endpoint case S≤3/2. These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding–Xin , Cheng–Wei , and Lei–Xu–Xu .

Spectral Properties of the Chen-Nagano Gauge on Einstein Manifolds

Mon, 25 May 2026 00:00:00 GMT

Sergey Stepanov — The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.

A mean curvature flow approach to Hamilton's pinching theorem

Fri, 17 Apr 2026 00:00:00 GMT

Liang Cheng, Zhenyu Lu — In this paper, we provide a proof of Hamilton's extrinsic pinching theorem using the mean curvature flow approach.

PIC1 pinched manifolds are flat or compact

Mon, 23 Mar 2026 00:00:00 GMT

Alix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping — Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

Classification of quadratically pinched self-shrinkers in higher codimension

Mon, 23 Feb 2026 00:00:00 GMT

Debora Impera, Michele Rimoldi, Francesco Ruatta — We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant 4/3n and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions.

High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

Mon, 23 Feb 2026 00:00:00 GMT

Ben Andrews, Qiyu Zhou — In the pseudo-Euclidean space ℝ^(n+1,k), we consider the mean curvature flow of n-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension k. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace ℝ^(n+1,0)⊂ ℝ^(n+1,k).

Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

Mon, 22 Dec 2025 00:00:00 GMT

Kuicheng Ma — In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike k-convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities.

Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

Tue, 14 Oct 2025 00:00:00 GMT

Mat Langford, James McCoy — In a recent article, a localization of the Huisken–Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.

Curvature pinching of asymptotically conical gradient expanding Ricci solitons

Mon, 06 Oct 2025 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

Thu, 25 Sep 2025 00:00:00 GMT

Xiaodong Cao, Ernani Ribeiro, Hosea Wondo — We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.

On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Tue, 16 Sep 2025 00:00:00 GMT

Zhengnan Chen — For all dimensions n≥5, let (M,g,f) be a n-dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that ∇²f is 2-nonnegative and the curvature tensor is WPIC1 at some point x∈ M. Then (M,g) must be a quotient of either Sⁿ or Sⁿ⁻¹×ℝ. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions n≥9. Namely that a complete ancient solution to the Ricci flow of dimension n≥9 with uniformly PIC must be weakly PIC2.

A gap Theorem on closed self-shrinkers of mean curvature flow

Sat, 01 Mar 2025 00:00:00 GMT

Yuhang Zhao — In this paper, we prove a pinching theorem for n-dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: | uppercaseexpandafterromannumeral2 |² ≤ 1 +1/(10 π(n+2)), then it must be the standard sphere Sⁿ(√n). This result may provide some evidence for the open problem 13.76 in.