Neckpinch — Ricci solitons

Conjugate radius and parabolicity along the Ricci flow

Mon, 02 Nov 2026 00:00:00 GMT

Chandan Kumar Mondal — We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if (Mⁿ,g(t))_(t∈[0,T]) is a complete Ricci flow with nonnegative scalar curvature at t=0, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor Ric-1/2 Rg is bounded above, then parabolicity is preserved along the flow and the conjugate radius of g(t) is finite for every t∈(0,T]. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that liminf_(r→∞)Vol(B(p,r))/r²=0, has finite conjugate radius.

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.

Rigidity of shrinking Ricci solitons with low-dimensional focal sets

Mon, 02 Nov 2026 00:00:00 GMT

Matheus Andrade Ribeiro de Moura Horácio — We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.

Ricci flows and bracket flows on twisted principal bundles

Mon, 02 Nov 2026 00:00:00 GMT

Henry Wood — We show that Ricci flows of invariant metrics on twisted principal bundles with nilpotent fibres over S¹ are immortal and type III. After normalisation and lift to the universal cover, a sequence of metrics converges in the Cheeger–Gromov topology to a homogeneous expanding Ricci soliton. The proof extends the bracket flow technique from homogeneous Ricci flows to invariant Ricci flows on twisted principal bundles.

Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

Sun, 01 Nov 2026 00:00:00 GMT

Vicente Cortés, Shubham Dwivedi, Lorenz Jetter — We prove that on a connected, simply-connected nilpotent Lie group, the notions of left-invariant generalized Ricci soliton, semi-algebraic and algebraic generalized Ricci soliton coincide. Moreover, every such soliton has harmonic torsion. We classify all algebraic solitons with positive definite metric on the five-dimensional Heisenberg algebra, the four-dimensional oscillator algebra and the four-dimensional split-oscillator algebra up to automorphisms and rescalings. We find that there is only a finite number of algebraic solitons, all of which have harmonic torsion.

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.

Steady conformally Kähler Ricci solitons with isoparametric potential

Thu, 29 Oct 2026 00:00:00 GMT

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

On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis

Thu, 29 Oct 2026 00:00:00 GMT

Yu Li, Longhuan Yan, Wenjia Zhang — We develop the weighted Sobolev and Hölder analysis needed to study perturbations of complete asymptotically conical (AC) Ricci shrinkers. Weighted elliptic estimates yield a global weighted divergence-free slice in spaces that retain the asymptotic data at infinity. As a geometric application, we prove pointed Gromov-Hausdorff rigidity of nontrivial AC shrinkers with vanishing infinitesimal solitonic deformation space.

Solitons on Statistical Manifolds and Their Submanifolds

Wed, 28 Oct 2026 00:00:00 GMT

Aliya Naaz Siddiqui, Bang-Yen Chen — Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections (∇,∇^(*)) together with a Riemannian metric. In this chapter we survey the theory of statistical solitons, that is, vector fields and constants that solve the Ricci-soliton equation simultaneously for ∇ and ∇^(*), together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products R×_f N₂ and N₁×_fR for (pseudo) Riemannian manifolds N₁ and N₂, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field ξ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, η-Einstein or quasi-Einstein character of the submanifold. We further review the results of Blaga and Chen on gradient solitons and statistical structures, in particular the connections ∇^θ defined by a 1-form, volume formulas and bounds for the Ricci tensor. We also describe Chen-type inequalities for statistical warped product submanifolds, which give the extrinsic context in which such solitons live. Key steps of the proofs, remarks on the sharpness of hypotheses, worked examples, and open problems are included.

The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-

Wed, 28 Oct 2026 00:00:00 GMT

Tomoyuki Hisamoto, Masataka Iwai — Inspired by recent work of Inoue and Hallam-Lahdili, we prove an equivariant Miyaoka-Yau inequality with a correction term for smooth Fano manifolds admitting Kähler-Ricci solitons. It recovers the usual Miyaoka-Yau inequality in the Kähler-Einstein case. Our proof uses weighted equivariant Bogomolov-Gieseker and Langer inequalities for torsion-free sheaves, together with a weighted slope estimate for the canonical extension sheaf. We also construct toric examples showing that the existence of a soliton implies neither weighted semistability of the canonical extension sheaf nor nonnegativity of the equivariant Miyaoka-Yau number.

Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

Tue, 27 Oct 2026 00:00:00 GMT

Fengjiang Li, Wenqi Li, Jianyu Ou, Yuanyuan Qu, Guoqiang Wu, Detang Zhou — Let (Mⁿ,g,f) be a complete connected shrinking gradient Ricci soliton with constant scalar curvature, satisfying Ric+∇²f=1/2g. We prove that if the focal set of f is totally geodesic, then M is rigid. Equivalently, its universal cover is a product of a compact Einstein manifold and a Euclidean space.

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

Tue, 27 Oct 2026 00:00:00 GMT

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

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.

A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups

Mon, 26 Oct 2026 00:00:00 GMT

Fanqi Lin — We classify all left-invariant affine Ricci solitons associated with the Bott connections of two fixed rank-two distributions on the seven standard families of three-dimensional Lorentzian Lie groups. The distributions are defined by a fixed orthogonal change of the spacelike frame. A universal adapted-frame calculation provides explicit connection matrices, curvature operators and the symmetrized Ricci tensor. The equation in the orthogonal-complement direction forces every left-invariant solution to be steady. The remaining equations yield elementary parameter families and finitely many real algebraic branches: four normalized branches for G₂, six branches for each sign parameter in G₄, and two quartic branches in G₇ in addition to its elementary families. Resultants, subresultants, exact polynomial reduction and Sturm root isolation establish necessity, sufficiency and exhaustiveness. Degenerate cases are treated separately, including all families with nonunique soliton vector fields. An invertible frame transformation relates the two distribution problems. The classification is in the stated structure parameters; no quotient by isometries, automorphisms or scaling is taken.

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.

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.

Fixed geodesics of involutions on 3-spheres

Wed, 21 Oct 2026 00:00:00 GMT

Bin Shen, Hui Zhang — Let an isometric involution act on a smooth Riemannian 3-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most 2π, and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop–Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis–Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.

Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature

Tue, 20 Oct 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — We prove that every complete Kähler–Ricci shrinker with nonnegative Ricci curvature is holomorphically isometric to ℂ^k× N^(n-k) for some 0≤ k≤ n, where ℂ^k is the Gaussian shrinker and N is a compact Kähler–Ricci shrinker with nonnegative Ricci curvature. As a corollary, every complete Kähler–Ricci shrinker with quasi-positive Ricci curvature is compact.

Two-step nilpotent Lie algebras obtained by quivers and geometric structures

Mon, 19 Oct 2026 00:00:00 GMT

Fumika Mizoguchi — In the study of geometric structures on nilmanifolds, two-step nilpotent Lie algebras obtained by graphs play an important role. Recently, additional examples of nilpotent Lie algebras have been constructed from finite quivers without cycles. These latter examples can have arbitrarily high degrees of nilpotency, and admit Riemannian Ricci soliton metrics. In this paper, we study nilpotent Lie algebras obtained by finite quivers without cycles that are two-step nilpotent, and we prove that they can also be obtained by graphs. Using this relationship, we demonstrate that every two-step nilpotent Lie algebra obtained by a finite quiver without cycles admits a pseudo-Riemannian Ricci-flat metric. Additionally, we also classify these nilpotent Lie algebras that admit symplectic structures.

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.

Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups

Wed, 14 Oct 2026 00:00:00 GMT

Yuxin Ding — We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

Long-Time Behavior of the α-Ricci Flow on the 2-Torus

Tue, 13 Oct 2026 00:00:00 GMT

Yifan Guo, Xiaokang Wang — We study the α-Ricci flow (g(t),α(t)), which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus M. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the α-Ricci flow depends on the cohomology class [α] ∈ H¹(M,ℝ), which is invariant under the flow. If [α] is rational, then as t→∞, (M,t⁻¹g(t)) Gromov–Hausdorff converges to S¹ with length ℓ([α])/sqrt 2 where ℓ([α]) is the primitive length of [α]. On the universal cover, the rescaled flow (s⁻¹tilde g(st),tildeα(st)) smoothly converges to an expander (g^*(t),α^*(t)) on ℝ² as s→ ∞ where α^*(t) is nonzero and constant in spacetime and g^*(t) is flat and expanding in the α^* direction. If [α] is irrational, then there is a sequence sᵢ→∞ such that (M,sᵢ⁻¹g(sᵢt)) Gromov–Hausdorff converges to a point. Moreover, on the universal cover, (sᵢ⁻¹tilde g(sᵢt),tildeα(sᵢt)) smoothly converges to (g^*(t),α^*(t)).

Constant Rank Theorems under Intermediate Ricci Curvature

Tue, 13 Oct 2026 00:00:00 GMT

Mijia Lai, Guoqiang Wu — We prove two constant rank theorems for the Ricci tensor in two geometric settings: manifolds with harmonic curvature and gradient Ricci solitons. In both cases, if the k-intermediate Ricci curvature satisfies Ricₖ≥0 for some 1≤ k≤ n-2, then the Ricci tensor has constant rank and its nullity distribution is parallel. In the complete case, the universal cover splits isometrically along the Ricci-null directions. We also show that the range 1≤ k≤ n-2 is sharp: in every dimension n≥4, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

Direct proofs for spaces with cyclic parallel Ricci tensor

Fri, 09 Oct 2026 00:00:00 GMT

Fengyuan Zhang — We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial Q-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao–Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.

Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

Thu, 08 Oct 2026 00:00:00 GMT

Matheus Horácio, Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete nonsteady gradient Kähler-Ricci soliton satisfying Ric+∇² f=λg, λ≠ 0. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as N^(2k)×ℂ^(m-k), where N^(2k) is Kähler-Einstein with Ric_(g_N)=λg_N. In the shrinking case the quotient is trivial; in the normalization λ=1/2 one has R≡ k. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if 𝓛_∇fRic is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching 0≤Ric≤λg forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

η- Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip

Thu, 08 Oct 2026 00:00:00 GMT

Amadou Sy, Ameth Ndiaye, Ghodratallah Fasihi-Ramandi — We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip Sₐ=\(x,y)∈ℝ²:0

On Ricci solitons whose level hypersurfaces have parallel second fundamental form

Thu, 08 Oct 2026 00:00:00 GMT

Matheus Andrade Ribeiro de Moura Horácio — We study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.

Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

Mon, 05 Oct 2026 00:00:00 GMT

Huibin Chen, Zhiqi Chen, Fuhai Zhu — We prove harmonicity for nonshrinking algebraic generalized Ricci solitons and for all algebraic generalized Ricci solitons on solvable Lie algebras. In the solvable setting, we characterize expanding solitons by normal extensions of generalized nilsolitons, and derive a sharp rank bound.