Neckpinch — Mean curvature flow

On knotted self expanders of the mean curvature flow in R³

Mon, 02 Nov 2026 00:00:00 GMT

Alexander Mramor, Shengwen Wang — In this article we produce many examples of knotted self expanders. For example, there is an expander isotopic to a trefoil knot with three half cylinders attached.

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.

Mean-convex horizontal mean curvature flow in the Heisenberg group

Wed, 28 Oct 2026 00:00:00 GMT

Gaia Bombardieri, Mattia Fogagnolo, Valentina Franceschi — We study the arrival time equation of the mean-convex horizontal mean curvature flow in the Heisenberg group. We prove existence of viscosity solutions when the initial set is horizontally mean-convex and either non-characteristic, or axisymmetric and with horizontal mean curvature extending continuously across the characteristic set. A key ingredient is an approximation scheme by sets with prescribed Riemannian mean curvature, which preserves the mean-curvature bounds up to an arbitrarily small error. We further show that, when the evolving level sets form a generalized C²-foliation, the horizontal mean curvature blows up around characteristic points. This provides an obstruction to self-similar evolution in the axisymmetric class considered above and, in particular, shows that the Pansu sphere cannot evolve self-similarly.

Genus Increase in Closed Mean Curvature Flows

Mon, 26 Oct 2026 00:00:00 GMT

Jingwen Chen — We construct closed mean curvature flows with arbitrarily large genus increase in every smooth Riemannian three-manifold, giving counterexamples to Ilmanen's genus-reduction conjecture. We prove that a compact hypersurface with a single isolated conical singularity and a power rate of convergence to its cone admits a mean curvature flow whose forward rescaled limit is a prescribed expander in the tangent space at the singular point, smoothly asymptotic to that cone. Combining this theorem with closed mean curvature flows with prescribed tangent flows (arXiv:2405.15577, arXiv:2607.23879) and the noncompact counterexample (arXiv:2605.09657) gives the desired examples.

Instability of mean curvature flow in expander coordinates

Fri, 23 Oct 2026 00:00:00 GMT

Ben S. Lambert, Oliver C. Schnürer — We study rotationally symmetric graphical solutions to mean curvature flow that are initially close to a homothetic reference expander, in particular near infinity. Based on a computer algebra aided proof, we show that under appropriate conditions, at large times, the solutions are closer to a different expander than to the reference expander.

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.

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.

Cohomogeneity one Hermann actions and backward mean curvature flows

Thu, 15 Oct 2026 00:00:00 GMT

Kurando Baba, Naoyuki Koike — Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.

Generic Mean Curvature Flow With Rough Weights

Thu, 15 Oct 2026 00:00:00 GMT

Fabius Krämer, Tim Laux, Keisuke Takasao — We prove the generic existence of mean curvature flows in spaces with rough weights. Our weak solutions are both unit-density Brakke flows and distributional solutions. Such a generic statement is optimal in the sense that we consider weights that are too rough to have well-defined traces on every hypersurface.

Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators

Wed, 14 Oct 2026 00:00:00 GMT

Jingwen Chen — In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions n≥2. We construct eternal Allen–Cahn flows on Sⁿ⁺¹ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.

Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow

Wed, 14 Oct 2026 00:00:00 GMT

Esther Cabezas-Rivas, Carlo Sinestrari — We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group

Wed, 14 Oct 2026 00:00:00 GMT

A. Cesaroni, M. Novaga, P. Pozzi — The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group ℍ, and prove the existence of a flat flow which is locally 1/2-Hölder continuous in time, with respect to the L¹-distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.

Singularities of Mean Curvature Flow with Bounded Total Curvature

Tue, 13 Oct 2026 00:00:00 GMT

Yongheng Han, Zhen Wang — We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies sup_(t∈[0,T))∫_(Mₜ)|A|ⁿ dμₜ<∞. This extends Ilmanen's smoothness result for tangent flows to higher dimensions.

Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness

Fri, 09 Oct 2026 00:00:00 GMT

Hilário Alencar, Márcio Batista, Gregório Silva Neto — For 1≤ r≤ n-1 we study shrinking self-similar ends of the r-mean curvature flow, Sᵣ=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x^(−(r+1)), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.

Compact Mean Curvature Flows from Non-Compact Initial Data

Thu, 08 Oct 2026 00:00:00 GMT

Theodora Bourni, Nathan Burns, Marcus Flook — We construct compact, embedded solutions to the mean curvature flow in ℝⁿ⁺¹ that arise from various non-compact initial data and exist for a finite time.

Generic mean curvature flow has finitely many singularities

Thu, 08 Oct 2026 00:00:00 GMT

J. M. Daniels-Holgate, Or Hershkovits, Gábor Székelyhidi — We show that for a generic compact initial surface in ℝ³, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

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.

On the Gage-Hamilton-Grayson Theorem for free boundary curves

Wed, 07 Oct 2026 00:00:00 GMT

Yueheng Bao, Lorenzo Sarnataro — We provide an alternative and independent proof of the analogue of the Gage-Hamilton-Grayson Theorem for embedded free boundary curve shortening flows in a convex planar domain established by Langford-Zhu. Our proof is based on blowup arguments introduced by B. White in his work on mean-convex mean curvature flow.

Increase in topological complexity along the mean curvature flow

Tue, 06 Oct 2026 00:00:00 GMT

Natasha Diederen — We construct an example of a smooth embedding of S^(p+q-1) in ℝ^(p+q) whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of S^(p-1)× S^q. In doing so, we prove some general results about SO(p)× SO(q)-invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.

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.

Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

Tue, 06 Oct 2026 00:00:00 GMT

Tianhong Pu, Xiaoqian Xin, Lixia Yuan, Xiaoyu Zhang — We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by V=H+b, where b>0 is constant and H denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope uₓ/u converges to one away from the symmetry axis. The normalized velocity uₜ/u converges to the forcing parameter b. The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.

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.

Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

Wed, 30 Sep 2026 00:00:00 GMT

Andrea Poiatti — The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.

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.

Lagrangian mean curvature flow in the Kummer K3 surface

Mon, 28 Sep 2026 00:00:00 GMT

Carlos Alberto Ochoa Flores — A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.

Counterexamples to preservation of flat normal bundles under mean curvature flow

Mon, 28 Sep 2026 00:00:00 GMT

Keita Kunikawa — We construct an embedded torus and an entire graph in ℝ⁴ whose normal bundles are initially flat but lose this property instantaneously under mean curvature flow. We also give an example showing that the parallel principal normal condition is not preserved under mean curvature flow, even though the normal bundle remains flat along the flow.

Optimal regularity for the arrival time equation

Fri, 25 Sep 2026 00:00:00 GMT

Yiqi Huang, Jingze Zhu — For a convex mean curvature flow in ℝⁿ⁺¹, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is C^(24,α) for any 0<α<1 in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for n≥ 2, this also gives the sharp C^(2,2/n) regularity for convex arrival time functions in higher dimensions.

General regularity obstructions for the arrival time equation

Fri, 25 Sep 2026 00:00:00 GMT

Yiqi Huang, Jingze Zhu — We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be C²⁵. Our companion paper proves that every such arrival time is C^(24,α) for any 0<α<1. Hence the obstruction at order 25 gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.