Neckpinch — (Non)collapsing

Steady conformally Kähler Ricci solitons with isoparametric potential

Thu, 29 Oct 2026 00:00:00 GMT

Ernani Ribeiro, Hung Tran — We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are U(2)-invariant nuts on ℝ⁴, including the Bryant and Cao solitons, and bolts on S² × ℝ² and on 𝓞(−m)→ℂP}¹. For each m≥3 there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.

O(3)-symmetry of four-dimensional steady Ricci solitons

Tue, 27 Oct 2026 00:00:00 GMT

Hanbing Fang — We prove that a complete connected four-dimensional gradient steady Ricci soliton with positive curvature operator that is κ-noncollapsed at all scales admits an effective isometric O(3) action preserving the potential function and having principal orbits diffeomorphic to S².

Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces

Wed, 21 Oct 2026 00:00:00 GMT

Weimin 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.

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.

The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers

Mon, 12 Oct 2026 00:00:00 GMT

Shaosai Huang — Let compact RCD(κ,N) spaces converge in the measured Gromov–Hausdorff sense, and suppose that they have (δ,ρ)-Reifenberg local covers for one fixed ρ>0, with |κ|ρ²≤1 and δ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of O(n) whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with O(n), with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition Aₛ it is a closed aspherical manifold. Three appendices treat smooth manifolds with Ric≥−(n-1) and small local rewinding Reifenberg error: a uniform short-time Ricci flow with |Rm|≤α/t, also for complete non-compact data, where it is unique among complete flows with curvature at most A/t; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov–Hausdorff approximations with small error.

Scale-Invariant Non-Collapse from Raychaudhuri Comparison

Tue, 06 Oct 2026 00:00:00 GMT

Rohit Dhormare — We establish a scale-invariant spatial non-collapse estimate for hypersurface-orthogonal timelike geodesic congruences. The result extends a previous Raychaudhuri-based construction in which a comparison estimate for the expansion was used to control the volume of domains transported along the congruence. The additional ingredient here is a quantitative estimate of the spatial distance distortion induced by the expansion and shear. Assuming that the initial spatial geometry is non-collapsed up to a scale r₀, and that the expansion, shear, and timelike Ricci curvature satisfy suitable uniform bounds, we obtain, before the first blow-up time of the Raychaudhuri comparison solution, \beginequation* \operatornameVol_h_τ\bigl(B_h_τ(p_τ,r)\bigr) \geq κ_L(τ)\, r^3, \endequation* where \beginequation* κ_L(τ) = κ_0 e^-3Kτ \exp\!\left(\int_0^τΘ(s)\,ds\right), \qquad K = \fracΘ_03 + Σ_0 . \endequation* Here Θ denotes the solution of the scalar Raychaudhuri comparison equation and K controls the spatial metric distortion. For every finite time strictly before the comparison barrier, κ_L(τ) is positive. When the comparison solution has a finite blow-up time, the explicit comparison coefficient approaches zero as that time is approached from below. The comparison barrier is a boundary of the estimate and is not, by itself, a statement that the physical congruence develops a focal point or spacetime singularity. The result is structurally analogous to scale-invariant non-collapse estimates in geometric flow, but is mathematically distinct from Perelman's κ-noncollapsing theorem for Ricci flow.

On the tangent flows of noncollapsed Ricci flow limit spaces

Sun, 04 Oct 2026 00:00:00 GMT

Yu Li, Bing Wang — We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.

On the uniqueness of Ricci flows from certain Alexandrov spaces

Fri, 02 Oct 2026 00:00:00 GMT

Laura Flower — We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon–Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.

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.

Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

Thu, 01 Oct 2026 00:00:00 GMT

Yeyun Xu, Linfeng Zhou — We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

Thu, 17 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.

Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

Wed, 16 Sep 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove that any finite-time collapsing Kähler-Ricci flow on compact Kähler surfaces develops a Type I singularity. Together with the previous results, this implies that any finite time singularity of the Kähler-Ricci flow on compact Kähler surfaces is of Type I.

Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

Tue, 15 Sep 2026 00:00:00 GMT

Charles Cifarelli, Ronan Conlon, Max Hallgren, Junsheng Zhang — For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.

Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

Sat, 12 Sep 2026 00:00:00 GMT

Shaochuang Huang, Zhuo Peng — In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

Wed, 02 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle Xⁿ→ Y^m, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as (C^m,g_(rm E),J₀)×(Z',d',J'). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to √(T-t). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder C^m×PP¹.

Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

Tue, 01 Sep 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker ℙ¹× ℂ. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

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.

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.

Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

Wed, 19 Aug 2026 00:00:00 GMT

Longteng Chen, Max Hallgren, Lucas Lavoyer — We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

Tue, 28 Jul 2026 00:00:00 GMT

Yu Li, Junsheng Zhang — We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an S¹-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

Entropy and Non-Collapse in Lorentzian Geometry

Fri, 10 Jul 2026 00:00:00 GMT

Rohit Dhormare — In this paper, we establish a geometric correspondence between the Lorentzian Raychaudhuri equation and Perelman's non-collapsing theorem for the Ricci flow. By interpreting the Raychaudhuri equation as a Lorentzian analogue of Ricci flow, we connect geodesic focusing in general relativity to the monotonicity and entropy functionals of geometric analysis. Using this correspondence, we derive a Lorentzian non-collapsing theorem and introduce a covariant entropy functional governing causal volume evolution. Finally, we propose the concept of geodesic entropy capacity, a curvature-bounded limit on the information that can be stored in spacetime regions, providing a unified geometric framework linking gravitation, thermodynamics, and information.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

Thu, 11 Jun 2026 00:00:00 GMT

Yu Li — Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any t₀>0 and any connected component Z_(t₀)' of the time-slice Z_(t₀) of the completion, we prove that (𝓡ₜ',d_(gₜ)) xrightarrow[t↗ t₀]mathrmGromov-Hausdorff (Z_(t₀)',d_(t₀)^Z), where 𝓡ₜ' is the corresponding connected component of the regular part and 𝓡ₜ' denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic 1-rectifiable, and its time image has vanishing 1/2-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most 1. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.

Singular Kähler–Ricci shrinkers and polarized Fano fibrations

Sun, 24 May 2026 00:00:00 GMT

Max Hallgren, Junsheng Zhang — We prove that any singular Kähler–Ricci shrinker X arising as a noncollapsed limit of Kähler–Ricci flows admits a natural structure of a polarized Fano fibration. We also show that it is simply connected, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As an application, we prove a new long-time pseudolocality theorem for almost-selfsimilar Kähler–Ricci flows.

Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

Thu, 21 May 2026 00:00:00 GMT

Haoyuan Sun — We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.

On an invariant curvature cone along 4-dimensional Ricci flow

Mon, 11 May 2026 00:00:00 GMT

Hongting Ding, Shaochuang Huang, Zhuo Peng — In this paper, we study 4-dimensional complete noncompact manifolds (M,g) satisfying Rm(g) ∈C_(η,μ) via Ricci flow. Under the additional assumption of maximal volume growth, we prove topological and geometric gap theorems. We also study 4-dimensional complete manifolds satisfying a lower bound with respect to C_(η,μ) and obtain regularity results for Gromov-Hausdorff limits of complete volume non-collapsed manifolds satisfying such curvature lower bounds.

Steady soliton with 𝓛¹ decay curvature

Wed, 22 Apr 2026 00:00:00 GMT

Ming Hsiao — In this paper, we establish a compactness theorem for gradient Ricci solitons with scalar curvature bounds and uniform lower bounds of harmonic coordinates. Our approach is to bootstrap regularity in harmonic coordinates by exploiting the soliton equation. As an application, we show that the regular part of any noncollapsed limit of gradient Ricci solitons with bounded Ricci curvature is smooth. Further, we show that a steady gradient Ricci soliton is asymptotically cylindrical under an 𝓛¹-decay assumption on its Ricci curvature.

Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

Mon, 06 Apr 2026 00:00:00 GMT

Haoyuan Sun — We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is Kähler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is Kähler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-Kähler torsion in the Green's formula by exploiting our local Kähler assumption, successfully adapting recent estimates of Kähler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.

Ancient Ricci flows with nonnegative Ricci curvature

Mon, 30 Mar 2026 00:00:00 GMT

Yuxing Deng, Ganqi Wang, Yongjia Zhang — In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity – a noncollapsed F-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed F-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature (Ric≥ εR g), either it is compact, or every tangent flow at infinity is a Ricci flat cone.

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).

Classification of ancient ovals in higher dimensional mean curvature flow

Wed, 14 Jan 2026 00:00:00 GMT

Beomjun Choi, Wenkui Du, Ziyi Zhao — We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in ℝⁿ⁺¹, called ancient ovals. Our main result is the classification of k-ovals: any k-oval (characterized by having cylindrical blow down ℝ^k× S^(n-k) and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open (k-1)-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of k-ovals in arbitrary dimensions based on a different spectral parametrization.