Neckpinch — 4-manifolds

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.

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.

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.

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.

Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

Thu, 22 Oct 2026 00:00:00 GMT

G. Bonelli, S. Lacroix, T. Pedroni, A. Tanzini — We construct a family of integrable 2d sigma models through 4d Chern-Simons theory and show that their classical point-particle reduction corresponds to a Hitchin system. The integrable Hitchin quantisation then provides a natural candidate Hamiltonian for the one-loop reduced dynamics of the sigma models, which coincides with the Perelman operator in all cases examined. This allows us to study the corresponding spectral problems using 4d 𝓝=2 gauge theory and 2d conformal field theory methods. For the sausage model, we derive exact quantisation conditions and compute exponentially small corrections essential for the agreement with numerical finite-volume results.

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

L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

Mon, 05 Oct 2026 00:00:00 GMT

Xianbin Huang, Wancheng Zhang — In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform L² Riemannian curvature bounds and L⁴ Ricci curvature bounds up to dimension 7. These estimates improves L^(2-ε) curvature bounds established by Bamler and L² curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform L² curvature bounds on n-manifolds satisfying an integral L^p Ricci curvature bound for p> n/2, a uniform volume lower bound, and a strong ε-regularity assumption. This result extends the celebrated L² curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an L^p adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori L^(4-ε) curvature radius bound of Ricci flow established by Bamler, we obtain uniform L² curvature bounds on Ricci flows with bounded scalar curvature of dimension n ≤ 7.

Solvable Algebraic Generalized Ricci Solitons in low dimensions

Thu, 01 Oct 2026 00:00:00 GMT

Lorenzo Mucciante — We prove new structural results for algebraic generalized Ricci solitons on solvable Lie algebras and we discuss the existence of examples in low dimensions. We establish that all non-trivial solvable algebraic generalized Ricci solitons must be expanding and give necessary conditions for the existence of such solitons on one-dimensional extensions of abelian Lie algebras. Finally, we classify algebraic generalized Ricci solitons on three-dimensional Lie algebras, and algebraic generalized Ricci solitons on four-dimensional solvable unimodular Lie algebras, up to isometry and scaling.

Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity

Sun, 20 Sep 2026 00:00:00 GMT

Mohammadjavad Habibivostakolaei, Abbas M. Sherif, Yen-Kheng Lim — We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

Tue, 01 Sep 2026 00:00:00 GMT

Junming Xie — In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

Wed, 26 Aug 2026 00:00:00 GMT

JeongHyeong Park, Wooseok Shin — We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation |R|²∇f=0 for the soliton potential f. Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.

Ricci Soliton Classification on ℍ²×ℝ

Thu, 20 Aug 2026 00:00:00 GMT

Anton Khaliapin — We study Ricci solitons on the Riemannian manifold ℍ²×ℝ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra isom(ℍ²×ℝ). All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.

Hermitian curvature flow and HKT geometry

Fri, 14 Aug 2026 00:00:00 GMT

Beatrice Brienza, Jeffrey Streets — We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

Open full-metric formation and local marked first-order asymptotic moduli of FIK blowdown singularities

Sun, 09 Aug 2026 00:00:00 GMT

Henry Shin — We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively C^(2,α)-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder h^(2,α) neighborhood, a fixed positive-time restart yields the marked first-profile coordinate A₁=λ_∞^(−γ₁)V_∞∈ E₁. This amplitude is a split C¹ submersion and locally the projection onto E₁. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

Sat, 08 Aug 2026 00:00:00 GMT

Youssef Ayad — In this paper, we compute the canonical and Kobayashi-Nomizu connections, together with their curvature, on Lorentzian four-dimensional nilpotent Lie groups endowed with a product structure. We also classify the algebraic Ricci solitons associated with these connections.

Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

Tue, 28 Jul 2026 00:00:00 GMT

Yu Li, Junsheng Zhang — We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an S¹-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

On an invariant curvature cone along 4-dimensional Ricci flow

Mon, 11 May 2026 00:00:00 GMT

Hongting Ding, Shaochuang Huang, Zhuo Peng — In this paper, we study 4-dimensional complete noncompact manifolds (M,g) satisfying Rm(g) ∈C_(η,μ) via Ricci flow. Under the additional assumption of maximal volume growth, we prove topological and geometric gap theorems. We also study 4-dimensional complete manifolds satisfying a lower bound with respect to C_(η,μ) and obtain regularity results for Gromov-Hausdorff limits of complete volume non-collapsed manifolds satisfying such curvature lower bounds.

A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Tue, 28 Apr 2026 00:00:00 GMT

Chen Wang, Guoqiang Wu — Let (M⁴, g, f) be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f= 1/2g. If its scalar curvature is 1, Cheng-Zhou proved that it is a finite quotient of ℝ²× S². In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.

A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Mon, 27 Apr 2026 00:00:00 GMT

Chen Wang, Guoqiang Wu — Let (Mⁿ, g, f) be an n-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f= 1/2g. 1. If its scalar curvature is k/2, Ricci curvature is nonnegative and sectional curvature has upper bound 1/(2(k-1)), we prove that the Ricci shrinker is isometric to a finite quotient of ℝ^(n-k)× S^k. 2. If M has constant scalar curvature R=(n-2)/2, and each level set of f has vanishing Weyl curvature, we prove that it is a finite quotient of ℝ²× Sⁿ⁻². This can be seen a generalization of Cheng-Zhou's four dimensional result to high dimension, since the level set of the potential function f has vanishing Weyl curvature automatically when n=4.

Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

Wed, 22 Apr 2026 00:00:00 GMT

Samuel Belo — We study compact m-quasi-Einstein manifolds and derive geometric estimates relating the oscillation of the potential function to the diameter of the manifold. We obtain lower bounds for the diameter in terms of the oscillation of the potential function. As an application in dimension four, we derive diameter conditions ensuring that compact m-quasi-Einstein manifolds satisfy the Hitchin–Thorpe inequality. Our results extend diameter estimates in smooth metric measure spaces and are consistent with known bounds in the limiting case corresponding to Ricci solitons. Finally, we provide a volume estimate involving the oscillation.

A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

Sun, 29 Mar 2026 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient steady (respectively, expanding) Ricci solitons with half-harmonic Weyl curvature. In addition, we obtain a partial classification of four-dimensional gradient expanding Ricci solitons with half-nonnegative isotropic curvature.

Asymptotic Geometry of Four-Dimensional Steady Solitons

Sun, 22 Mar 2026 00:00:00 GMT

Aprameya Girish Hebbar, Natasa Sesum — In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak κ-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional steady solitons constructed by Lai.

Prescribed T-curvature flow on the four-dimensional unit ball

Fri, 30 Jan 2026 00:00:00 GMT

Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun, Heming Wang — In this paper, we study the prescribed T-curvature problem on the unit ball B⁴ of ℝ ⁴ via the T-curvature flow approach. By combining Ache-Chang's inequality with the Morse-theoretic approach of Malchiodi-Struwe, we establish existence results under strong Morse-type inequalities at infinity. As a byproduct of our argument, we also prove the exponential convergence of the T-curvature flow on B⁴, starting from a Q-flat and minimal metric conformal to the standard Euclidean metric, to an extremal metric of Ache-Chang's inequality whose explicit expression was derived by Ndiaye-Sun.

Comment on "Multidimensional arrow of time" (arXiv:2601.14134)

Tue, 27 Jan 2026 00:00:00 GMT

Andrei Galiautdinov — In a recent preprint [arXiv:2601.14134v1], Rubin argues that the arrow of time originates from the monotonic growth of the volume of extra dimensions. While the identification of a geometric origin for time's arrow is compelling in the case of brane-world models, we point out a possible tension between the proposed volume growth and the observational stability of the effective four-dimensional Newton's gravitational constant, G, that may arise in Kaluza-Klein (KK) theory. In standard KK approaches, such volume growth induces a time-variation of G that exceeds Big Bang Nucleosynthesis (BBN) and Lunar Laser Ranging (LLR) bounds by many orders of magnitude. To resolve this tension while preserving the author's key insight in the Kaluza-Klein case, we propose an extension: the "shape-dynamic arrow of time". By utilizing the scale-invariant monotonicity of Perelman's nu-entropy under normalized Ricci flow, we demonstrate how an arrow of time can emerge from the geometric smoothing of extra dimensions at fixed volume, thereby satisfying observational constraints on fundamental constants.

Four-dimensional Lorentzian algebraic Ricci solitons

Thu, 22 Jan 2026 00:00:00 GMT

Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey, Ramon Vazquez-Lorenzo — We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.

Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

Wed, 21 Jan 2026 00:00:00 GMT

Youssef Ayad — We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie algebra, we obtain a complete description of when such pseudo-Riemannian algebraic Ricci solitons arise in dimension four. We present two applications of our formalism on a chosen four-dimensional Lie algebra by exhibiting a pseudo-Riemannian algebraic Ricci soliton and a flat pseudo-Riemannian inner product, which is a trivial algebraic Ricci soliton.

Stability of Hyperkähler Flow

Tue, 06 Jan 2026 00:00:00 GMT

Kuan-Hui Lee — In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperkähler 4-manifold. In, Wang and Tsai proved a uniqueness theorem and C¹ dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperkähler flow.

Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

Mon, 10 Nov 2025 00:00:00 GMT

Giovanni Calvaruso, Lorenzo Pellegrino, Amirhesam Zaeim — In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

Rigidity of Gradient Shrinking Ricci Solitons with a Vanishing Bach-like Tensor and Related Variational Formulas

Sat, 08 Nov 2025 00:00:00 GMT

James Siene — The classical Bach tensor in four dimensions can be expressed as a linear combination of two independent, symmetric, divergence-free, quadratic-in-curvature tensors U and V. Several classification results for gradient-shrinking Ricci solitons have been obtained under the assumption that the Bach tensor vanishes. We define a Bach-like tensor to be any other linear combination of U and V. We prove that within a certain cone of parameters, the vanishing of a Bach-like tensor forces a four-dimensional complete gradient-shrinking Ricci soliton to be either Einstein or isometric to the Gaussian soliton, extending the results of Cao–Chen (2013). The special case where U=0 forces fₘᵢₙ∈\0,1,2\, with rigidity holding when fₘᵢₙ=0,2. The remaining case fₘᵢₙ=1 is the central open problem, with a cylinder as the conjectured exceptional geometry. Finally, we show that Bach-like tensors arise as Euler–Lagrange equations of a two-parameter family of quadratic curvature functionals and compute the corresponding first and second variation formulas.