Neckpinch — Self-shrinkers & translators

Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

Mon, 26 Oct 2026 00:00:00 GMT

Ze-Yu Ye, Xiao-Liu Wang — In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.

The global structure of bi-rotational self-expanders

Mon, 19 Oct 2026 00:00:00 GMT

Gregŕio Silva Neto — We describe the self-expanders of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1). Apart from the Clifford minimal cone, every such hypersurface is complete, proper and noncompact, and belongs to one of three families: two one-parameter families of embedded hypersurfaces diffeomorphic to ℝ^(p+1)×S^(q) or to ℝ^(q+1)×S^(p), whose profile curves leave an axis, and a two-parameter family diffeomorphic to ℝ×S^(p)×S^(q), parametrized by the point of the profile curve closest to the origin, which is the generic family: its set of initial conditions is open and of full measure in the phase space. Every end is asymptotically conical, with explicit decay rates, and every such hypersurface has Euclidean volume growth. A profile curve of the third family has at most one loop, and the point of the curve closest to the origin lies on that loop.

The global structure of bi-rotational self-shrinkers

Mon, 19 Oct 2026 00:00:00 GMT

Gregório Silva Neto — We describe the global structure of the self-shrinkers of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1), without assuming properness or embeddedness. The main tool is a pair of piecewise monotone quantities along the profile curves, in the spirit of Hsiang's analysis of constant mean curvature hypersurfaces. We prove that every such self-shrinker other than the two invariant cylinders intersects both of them transversely, and all of them except the Clifford minimal cone are complete. Apart from those of the cylinders, each end is either asymptotic to a cone or meets both cylinders infinitely many times. Consequently, a complete noncompact example is proper if and only if all its ends are asymptotically conical, and this is equivalent to polynomial volume growth, to finite entropy, and to finitely many intersections with the cylinders. The reduced flow preserves a finite measure, and for almost every initial condition the profile curve is either closed or generates a complete nonproper self-shrinker of infinite entropy. We also bound the winding of the profile curve by its number of cylinder crossings and localize its self-intersections.

Sharp dynamical isoperimetric principle for mean curvature flow

Wed, 07 Oct 2026 00:00:00 GMT

Beomjun Choi, Wenkui Du, Seung Chul Park, Junseo Youn — In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow Kₜ starting from the boundary of a smooth bounded domain Ω⊂ℝⁿ⁺¹, as well as for the integral Brakke flow \μₜ\ with initial Radon measure μ₀=𝓗ⁿllcorner∂Ω, we establish the corresponding optimal extinction time estimates \beginequation* T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac12n \left(\fracP( Ω){|{\mathbbS^n}|}\right)^{\frac2n}, \endequation* where P(Ω) is the perimeter of Ω representing the area of ∂Ω, and the equality holds if and only if Ω is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp L^p-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling X of outward minimizing level set flow starting from the boundary of a smooth bounded domain Ω⊂ ℝⁿ⁺¹: \beginequation* \mathcal H_par^n+2(X) \leq \fracπ {2n(n+2)^2|{\mathbbS^n}|^{{\frac2n}}} P(Ω)^{\fracn+2n}, \endequation* where the equality holds if and only if Ω is a round ball and the flow is standard smoothly shrinking round sphere.

Radial Powered Mean Curvature Flow with Robin Boundary Conditions

Tue, 06 Oct 2026 00:00:00 GMT

Tianhong Pu, Xiaoqian Xin, Lixia Yuan — We study a positive radial graph evolving by V=H^α+b, with α>0 and b<0, in the unit cylinder and with uᵣ(1,t)=u(1,t). We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for H, uᵣ/r, and uᵣᵣ. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

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 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)

A stable self-shrinking Möbius bundle in ℝ⁴

Thu, 17 Sep 2026 00:00:00 GMT

Kahnrad Braxton, Tang-Kai Lee, Jonathan J. Zhu — We prove the linear stability of an embedded mean curvature flow shrinker in ℝ⁴ with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.

Rigidity and Non-existence Results of λ-Translating Solitons

Wed, 16 Sep 2026 00:00:00 GMT

Li 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.

Formation and structural stability of nondegenerate neckpinches

Tue, 08 Sep 2026 00:00:00 GMT

Kyeongsu Choi, Panagiota Daskalopoulos, Natasa Sesum — In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.

A Bernstein problem for translating solutions to the mean curvature flow

Fri, 04 Sep 2026 00:00:00 GMT

Juan Davila, Manuel del Pino, Juncheng Wei — We study entire graphical translating solutions of the mean curvature flow, div (frac∇G√(1+|∇G|²)) =frac1√(1+|∇G|²) in ℝ^N. Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every N≥ 8 there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri–De Giorgi–Giusti (BDG) entire minimal graph in ℝ⁸ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.

Uniqueness of tangent flows in free boundary flow

Wed, 02 Sep 2026 00:00:00 GMT

Sourav Ghosh — We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form S^(n-k) × ℝ^k_+ or S^(n-k)_+ × ℝ^k, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

Sun, 23 Aug 2026 00:00:00 GMT

Hunter Liu, Sebastian Munoz, Zihui Zhao — We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees 0,1,2,3. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions 2≤ n ≤ 4000 is verified by a rigorous computer-assisted argument using interval arithmetic.

Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

Thu, 20 Aug 2026 00:00:00 GMT

Dake Li, Zhizhang Wang, Shiqi Yin — This paper studies self-shrinkers and the long-time behavior of the inverse σₖ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse σₖ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its σₖ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.

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 .

Invariant λ-translators in the Heisenberg group

Tue, 30 Jun 2026 00:00:00 GMT

Antonio Bueno — We study oriented surfaces in the Heisenberg space Nil₃ whose mean curvature H at each point is H=⟨ N,∂_z⟩+λ, where N is the unit normal, ∂_z is the vertical Killing vector field and λ∈ℝ. These surfaces are known as λ-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify λ-translators invariant by the following one-parameter groups of isometries of Nil₃: left-translations, rotations and helicoidal motions.

Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

Fri, 19 Jun 2026 00:00:00 GMT

Maxwell Stolarski, Wei-Bo Su — Let 𝓒 be a G-invariant special Lagrangian cone admitting a scaled family of G-invariant special Lagrangian desingularizations a overline L which converge to 𝓒 as a↘ 0. We study the linearized self-shrinker operator on aoverline L in a Gaussian weighted L² space of G-equivariant functions. For 0

Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

Thu, 18 Jun 2026 00:00:00 GMT

Peng Peng — Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type S¹× Sⁿ⁻¹ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker x²+r²=2n, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.

Topology of gradient Ricci shrinkers via weighted L² cohomology

Wed, 06 May 2026 00:00:00 GMT

Fei He — This paper proves several topological results for smooth gradient Ricci shrinkers. We establish upper bounds for the Betti numbers, a vanishing theorem for cohomology, and a dichotomy for the number of ends. We also prove a full Hodge theorem for a large class of shrinkers. The methods are based on weighted L² cohomology and extend to self-shrinkers of the mean curvature flow.

Translating solitons to higher order mean curvature flows in Riemannian products

Tue, 21 Apr 2026 00:00:00 GMT

Jorge Herbert Soares de Lira, Rafael Rocha de Farias — In this paper we prove existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows that are invariant by translations in warped product manifolds ℙ×_χℝ. Here, ℙ is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and χ is a radial function defined in ℙ. In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor ℝ in the warped product. This setting encompasses the cases of translating solitons in ℝⁿ⁺¹, ℍⁿ × ℝ and ℍⁿ⁺¹ studied in recent papers. In particular we prove the existence of families of bowl-type and catenoid-type translating solitons under mild assumptions about the curvature of the warped product. We also describe the asymptotic behavior for those solitons in terms of the geometry at infinity of ℙ. Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.

Classification of ancient finite-entropy curve shortening flows

Tue, 10 Mar 2026 00:00:00 GMT

Kyeongsu 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.

High Codimension Curve Shortening Flow with Free Boundary

Tue, 24 Feb 2026 00:00:00 GMT

Huy The Nguyen, Artemis A. Vogiatzi — We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound <2. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.

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.

Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

Mon, 02 Feb 2026 00:00:00 GMT

Ronaldo F. de Lima, Álvaro K. Ramos — We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension n+1≥ 3. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.

Dynamical stability of various convex graphical translators

Thu, 29 Jan 2026 00:00:00 GMT

Junyoung Park — In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

Half-space theorems for translating solitons of the r-mean curvature flow

Tue, 13 Jan 2026 00:00:00 GMT

Hilário Alencar, G. Pacelli Bessa, Gregório Silva Neto — In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the (r-1)-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the (r-1)-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.

The PDE-ODI principle and cylindrical mean curvature flows

Wed, 31 Dec 2025 00:00:00 GMT

Richard 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.

Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

Wed, 31 Dec 2025 00:00:00 GMT

Richard 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.

Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

Mon, 29 Dec 2025 00:00:00 GMT

José Torres Santaella — We develop a rotational theory for translating solitons of fully nonlinear extrinsic curvature flows in Euclidean space. Furthermore, we obtain fine asymptotic expansions for bowl-type translators in nondegenerate and degenerate regimes. On the other hand, we also introduce a signed-neck framework which yields the construction and classification of catenoidal-type translators, distinguishing complete embedded families from maximal admissible pieces according to the selected signed branch. As applications, we prove uniqueness results for strictly convex entire graphical translators with prescribed bowl-type asymptotics and obtain catenoidal-barrier nonexistence results for bounded graphical translators.

Mean curvature flow near a peanut solution

Thu, 04 Dec 2025 00:00:00 GMT

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum — It was shown by Angenent, Altschuler and Giga, and by Angenent and Velazquez that there exist closed mean curvature flow solutions that extinct to a point in finite time, without ever becoming convex prior to their extinction. These solutions develop a degenerate neckpinch singularity, meaning that the tangent flow at a singularity is a round cylinder, but at the same time for each of these solutions there exists a sequence of points in space and time, so that the pointed blow up limit around this sequence is the Bowl soliton. These solutions are called peanut solutions and they were first conjectured to exist by Richard Hamilton, while the existence of those solutions was shown by Angenent, Altschuler and Giga. In this paper we show that this type of solutions are highly unstable, in the sense that in every small neighborhood of any such peanut solution we can find a perturbation so that the mean curvature flow starting at that perturbation develops spherical singularity, and at the same time we can find a perturbation so that the mean curvature flow starting at that perturbation develops a nondegenerate neckpinch singularity. We also show that appropriately rescaled subsequence of any sequence of solutions whose initial data converge to the peanut solution, and all of which develop spherical singularities, converges to the Ancient oval solution.