https://neckpinch.com/topics/ancient-solutions/New arXiv papers on ancient solutions, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2610.39877/arxiv:2610.39877Thu, 29 Oct 2026 00:00:00 GMTFrancesco 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.Mean curvature flowAncient solutionsCurvature pinching https://neckpinch.com/papers/2610.34274/arxiv:2610.34274Sun, 25 Oct 2026 00:00:00 GMTBeomseok Goh, Taehun Lee — We study ancient solutions to curve shortening flow on a class of warped product surfaces. For closed embedded ancient solutions whose curves are not null-homotopic, we prove that graphicality at every time is equivalent to a uniform length bound as t→-∞, and that these properties hold whenever the length is o(√(−t)). Under an additional positivity condition on the linearized height equation, every such solution with bounded length is a family of slices. We derive backward convergence to a limiting circle from the length bound and prove slice rigidity by an L² estimate for the height perturbation.Curve shorteningAncient solutionsStability & uniquenessRotational symmetry & warped products https://neckpinch.com/papers/2610.30939/arxiv:2610.30939Thu, 22 Oct 2026 00:00:00 GMTYu Li — We study ancient Ricci flows whose tangent flow at backward infinity is a smooth compact Ricci shrinker. After normalization, every nontrivial solution either converges exponentially or has polynomial decay governed by neutral directions. We recover the finite-dimensional parametrization of the exponentially convergent solutions and establish finite-order asymptotic expansions for all slowly convergent solutions. For principal slow solutions, we obtain a locally bijective finite-dimensional parametrization compatible with diffeomorphisms, parabolic rescaling, and time translation.Ricci flowRicci solitonsAncient solutionsSingularitiesLong-time existence & convergence https://neckpinch.com/papers/2610.21903/arxiv: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://neckpinch.com/papers/2610.02325/arxiv:2610.02325Thu, 01 Oct 2026 00:00:00 GMTSimon 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.Ricci flowRicci solitonsAncient solutions(Non)collapsingCurvature pinchingSphere theoremsBoundaries & non-compact https://neckpinch.com/papers/2610.01713/arxiv:2610.01713Thu, 01 Oct 2026 00:00:00 GMTDebora 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.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsCurvature pinchingStability & uniquenessMinimal & CMC surfaces https://neckpinch.com/papers/2609.18064/arxiv:2609.18064Wed, 16 Sep 2026 00:00:00 GMTLi Xiang, Sun Jun — A λ-translating soliton is a hypersurface in ℝⁿ⁺¹ satisfying H=⟨ T,ν⟩+λ; equivalently, it has constant weighted mean curvature with respect to the log-linear density e^(⟨ T,X⟩), and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed λ-translating soliton with λ>0 and inf_ΣH>λ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic λ-translating solitons (λgeqslant 0) with bounded gradient, and two rigidity theorems.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsStability & uniqueness https://neckpinch.com/papers/2608.30787/arxiv:2608.30787Mon, 31 Aug 2026 00:00:00 GMTKyeongsu Choi, Robert Haslhofer — In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).Mean curvature flowAncient solutionsSingularities https://neckpinch.com/papers/2608.26619/arxiv:2608.26619Thu, 27 Aug 2026 00:00:00 GMTKyeongho Bang, Eric Chen, Wenkui Du — We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow (M², ∂M², g(t)) on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.Ricci flowAncient solutionsSingularitiesSurfacesStability & uniquenessRotational symmetry & warped products https://neckpinch.com/papers/2608.26598/arxiv:2608.26598Thu, 27 Aug 2026 00:00:00 GMTJae 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.Ricci flowAncient solutions(Non)collapsingCurvature pinchingSphere theoremsBoundaries & non-compact https://neckpinch.com/papers/2608.19565/arxiv:2608.19565Thu, 20 Aug 2026 00:00:00 GMTGanqi Wang, Yongjia Zhang — In this paper, we prove a gap theorem for F-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an ε-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).Ricci flowAncient solutionsSingularitiesScalar curvature https://neckpinch.com/papers/2608.11011/arxiv:2608.11011Tue, 11 Aug 2026 00:00:00 GMTJunyoung Park — In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an O(n)× O(m) symmetric minimal quadratic cone for n +m ≥ 10, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a singular minimal cone as the asymptotic model.Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces https://neckpinch.com/papers/2608.08533/arxiv:2608.08533Sun, 09 Aug 2026 00:00:00 GMTHenry Shin — We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively C^(2,α)-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder h^(2,α) neighborhood, a fixed positive-time restart yields the marked first-profile coordinate A₁=λ_∞^(−γ₁)V_∞∈ E₁. This amplitude is a split C¹ submersion and locally the projection onto E₁. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.Ricci flowRicci solitonsAncient solutionsSingularities4-manifoldsKähler geometryBoundaries & non-compactEigenvalues & spectrum https://neckpinch.com/papers/2607.03383/arxiv:2607.03383Fri, 03 Jul 2026 00:00:00 GMTPanagiota Daskalopoulos, Wenkui Du, Natasa Sesum, Ziyi Zhao — We obtain the unique asymptotics of SO(k)× SO(n-k+1)-invariant, compact, simply-connected, factorwisely non-self-similar n-dimensional κ-solutions of the Ricci flow (Mⁿ, g(t)), where n≥ 4 and 2≤ k≤ n-2. More precisely, these κ-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere Sⁿ, having a positive curvature operator metric g(t) and a cylindrical tangent flow at -∞, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric g(t) of every SO(k)× SO(n-k+1)-invariant ancient oval is represented in the form g(t)=dz⊗ dz + F²(z,t) g_(S^(k-1)) + G²(z,t)g_(S^(n-k)) (up to flipping k-1 and n-k). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function G(z, t), and prove that the uniqueness of G(z, t) implies the uniqueness of F(z, t). In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional κ-solutions of the Ricci flow.Ricci flowAncient solutionsSingularitiesSphere theoremsStability & uniqueness https://neckpinch.com/papers/2606.22355/arxiv:2606.22355Sun, 21 Jun 2026 00:00:00 GMTHaixia Chen, Seunghyeok Kim, Monica Musso — For every n ≥ 3, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round n-sphere Ssⁿ. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of Ssⁿ. These examples show that the collection of ancient Yamabe flows on Ssⁿ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on Ss², all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on Rⁿ, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner–outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on Rⁿ. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted Hölder estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.Ricci flowYamabe flowAncient solutionsSingularitiesRotational symmetry & warped products https://neckpinch.com/papers/2606.17353/arxiv:2606.17353Mon, 15 Jun 2026 00:00:00 GMTYueheng Bao, Robert Haslhofer — We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types ℝ^k× S^(n-k), ℝ^k_+× S^(n-k) or ℝ^k× S^(n-k)_+. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.Mean curvature flowAncient solutionsSingularitiesWeak & singular flowsFree boundary & capillary https://neckpinch.com/papers/2606.12379/arxiv:2606.12379Wed, 10 Jun 2026 00:00:00 GMTReto Buzano, Gianmichele Di Matteo — We continue our local singularity analysis for Ricci flow initiated in ArXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.Ricci flowAncient solutionsSingularitiesScalar curvature https://neckpinch.com/papers/2605.14905/arxiv:2605.14905Thu, 14 May 2026 00:00:00 GMTAprameya Girish Hebbar — We show that κ-solutions to the Ricci flow in dimensions n≥ 4 whose asymptotic shrinking Ricci soliton is the round cylinder Sⁿ⁻¹×ℝ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.Ricci flowRicci solitonsAncient solutionsSphere theoremsBoundaries & non-compact https://neckpinch.com/papers/2605.13023/arxiv:2605.13023Wed, 13 May 2026 00:00:00 GMTIvan Krznarić, Rafael López — We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when the curvature function is given by separation of variables. Lastly, we prove some area estimates for closed ancient solutions of the curve shortening flow.Curve shorteningAncient solutions https://neckpinch.com/papers/2604.14139/arxiv:2604.14139Wed, 15 Apr 2026 00:00:00 GMTRaphael Tsiamis — For every closed set K ⊂ ℝⁿ and every m ≥ 2, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in ℝ^(m+n), with respect to a smooth Riemannian metric arbitrarily C^∞-close to the Euclidean metric, whose first-time singular set is exactly K × \0\.Mean curvature flowAncient solutions https://neckpinch.com/papers/2603.28014/arxiv:2603.28014Mon, 30 Mar 2026 00:00:00 GMTYuxing 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.Ricci flowAncient solutionsRicci-flat & Calabi–YauSingularities(Non)collapsing https://neckpinch.com/papers/2603.21397/arxiv:2603.21397Sun, 22 Mar 2026 00:00:00 GMTAprameya Girish Hebbar, Natasa Sesum — In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak κ-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional steady solitons constructed by Lai.Ricci flowRicci solitonsAncient solutions4-manifoldsScalar curvatureBoundaries & non-compact https://neckpinch.com/papers/2603.16541/arxiv:2603.16541Tue, 17 Mar 2026 00:00:00 GMTXiangzhi Cao — In this paper, we are devoted to obtain some results on p-biharmonic map from gradient Ricci soliton, especially on two dimensional cigar soliton.Ricci solitonsAncient solutionsSurfacesHarmonic maps & DeTurck trick https://neckpinch.com/papers/2603.09112/arxiv:2603.09112Tue, 10 Mar 2026 00:00:00 GMTKyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao — We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together m translating grim reaper curves. For each m, there exists a (2m-1)-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.Curve shorteningAncient solutionsSelf-shrinkers & translatorsBoundaries & non-compact https://neckpinch.com/papers/2602.19681/arxiv:2602.19681Mon, 23 Feb 2026 00:00:00 GMTDebora 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.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsCurvature pinching https://neckpinch.com/papers/2602.15661/arxiv:2602.15661Tue, 17 Feb 2026 00:00:00 GMTAnusha M. Krishnan, Francesco Pediconi — We prove a precompactness theorem for invariant metrics on compact homogeneous spaces without injectivity radius bounds, assuming uniform bounds on the diameter and on all derivatives of the curvature tensor. As a consequence, we prove that every ancient homogeneous Ricci flow on a compact manifold admits a blow-down sequence that converges to a gradient shrinking Ricci soliton.Ricci flowRicci solitonsAncient solutionsLimit spaces & RCDHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2602.08850/arxiv:2602.08850Mon, 09 Feb 2026 00:00:00 GMTTheodora Bourni, Giada Franz — We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.Mean curvature flowAncient solutionsStability & uniquenessFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces https://neckpinch.com/papers/2601.09441/arxiv:2601.09441Wed, 14 Jan 2026 00:00:00 GMTBeomjun 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.Mean curvature flowAncient solutions(Non)collapsingEigenvalues & spectrum https://neckpinch.com/papers/2512.25050/arxiv:2512.25050Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the PDE–ODI principle, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.Mean curvature flowRicci solitonsAncient solutionsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://neckpinch.com/papers/2512.24524/arxiv:2512.24524Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel leading mode condition, together with a new "induction over thresholds" argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingRotational symmetry & warped products