Neckpinch — Singularities

Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Sun, 01 Nov 2026 00:00:00 GMT

Haozhao Li, Bing Wang, Kai Zheng — We study global Calabi flows g(t) on compact Kähler surfaces with a uniform Sobolev constant. For any sequence tⱼ→∞, set gⱼ(t):=g(tⱼ+t). After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime hat C^∞-Cheeger–Gromov topology: it is uniform in the Gromov–Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.

Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow

Sun, 01 Nov 2026 00:00:00 GMT

Hongyu Chen, Li Ma, Li Sheng — On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern–Calabi flow with a finite-time Chern Type-I singularity at any prescribed point. The full spatial supremum has an exact asymptotic coefficient, and prescribed rescalings converge to a complete nonflat radial shrinking solution. The positive part of Chern scalar curvature can remain arbitrarily small throughout the flow, as can each prescribed subcritical spatial curvature norm. We determine the punctured terminal metric, its one-point completion and unique nonflat tangent cone, and the terminal energy asymptotics. We also exclude Chern Type-II under a logarithmically improved critical weak L³ bound on Chern scalar curvature, and prove a separate scale-invariant integral criterion involving first Chern–Ricci curvature. The constructed singular solutions satisfy both criteria.

On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

Fri, 30 Oct 2026 00:00:00 GMT

Yu Li, Longhuan Yan, Wenjia Zhang — We prove strong uniqueness of tangent flows at the first singular time of a closed Ricci flow when one tangent flow is generated by a complete smooth asymptotically conical Ricci shrinker. The modified flow converges to this shrinker in a fixed gauge, with quantitative estimates on expanding domains. The main ingredients are a Łojasiewicz-Simon inequality for Perelman's entropy, based on the global weighted divergence-free gauge constructed in the preceding paper, and estimates controlling the geometric radius under Ricci flow.

Blowdowns along Ricci flow in dimension four

Thu, 29 Oct 2026 00:00:00 GMT

Dain Kim, Tristan Ozuch — We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C¹⁴-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C¹⁴-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂP}²\#ℓℂP}², through finitely many blowdown singularities. (2) On the standard 2ℂP}²\#9ℂP}², there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.

The Yang-Mills flow in vector bundles over higher dimensional Riemannian manifolds

Wed, 28 Oct 2026 00:00:00 GMT

Jared Casey, Chak Hoi Chan, Min-Chun Hong — We develop a parabolic formulation of the Uhlenbeck gauge fixing theorem and use it to prove the maximal existence of the Yang-Mills flow in vector bundles over compact n-dimensional Riemannian manifolds for n ≥ 4. We apply Moser's iteration technique to give a new proof of the bound of the curvature. Using an idea of Hamilton on the Ricci flow, we also provide a new proof on uniform estimates of higher order derivatives. Furthermore, we investigate the blow-up of the Yang-Mills flow at the maximal existence time and improve an asymptotical result of Hong-Tian.

An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

Tue, 27 Oct 2026 00:00:00 GMT

Hongxin Guo, Guozhi Lin — For a rotationally symmetric Ricci flow g(t)=ds²+f(s,t)²g_(Sⁿ⁻¹), n≥3, we study the scale-invariant curvature anisotropy A=1-fₛ²+ffₛₛ introduced by Angenent–Knopf. At the critical power p_*=(n-1)/(n-2), with Φ=sgn(A)|A|^(p_*/2), we prove the distributional identity (∂ₜ−Δ)|A|^(p_*) =−4/n-1 |Φₛ+(n-1)(fₛ)/fΦ|². Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on Sⁿ: if the entropy associated with a singular-time kernel is at most 1 at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the 0/1 quantization of the singular-time anisotropic density.

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.

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.

Chern holonomy under the first and second Chern–Ricci flows

Mon, 26 Oct 2026 00:00:00 GMT

Yuan Liao — We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)²; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.

Continuation and convergence for the higher dimensional conformal Chern–Calabi flow

Thu, 22 Oct 2026 00:00:00 GMT

Hongyu Chen, Li Ma, Li Sheng — We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension n≥3. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to L² in time. A uniform curvature bound in the evolving L^p norm, for one fixed p>n, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial L^p curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern–Ricci form gives exponential convergence on Chern–Ricci-flat Hermitian and positive Kähler–Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.

Parametrization of ancient Ricci flows with smooth compact tangent shrinkers

Thu, 22 Oct 2026 00:00:00 GMT

Yu 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.

Type I estimates for the Kähler-Ricci flow II

Fri, 16 Oct 2026 00:00:00 GMT

Wangjian Jian, Jian Song — In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial C⁰ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

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.

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.

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.

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.

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.

Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Sun, 04 Oct 2026 00:00:00 GMT

Haozhao Li, Bing Wang, Kai Zheng — We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.

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.

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.

Type I estimates for the Kahler-Ricci flow I

Wed, 30 Sep 2026 00:00:00 GMT

Wangjian Jian, Jian Song — We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

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.

A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

Sun, 27 Sep 2026 00:00:00 GMT

Ziyang Qin, Yuan Liao, Ayush Khaitan, Bennett Chow — We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.

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.

The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

Fri, 25 Sep 2026 00:00:00 GMT

Jason D. Lotay, Goncalo Oliveira — We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.

On the singularity formation of gauge fields coupled to Ricci flow

Wed, 23 Sep 2026 00:00:00 GMT

Andoni Royo Abrego — We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a SU(2) bundle over S⁴.

A Positive Mass Type Theorem for a Singular Toroidal Static Model

Wed, 23 Sep 2026 00:00:00 GMT

Zhixin Wang — We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.