Neckpinch — Ricci-flat & Calabi–Yau

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.

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.

Pseudolocality and strong uniqueness for complete Ricci flows

Tue, 13 Oct 2026 00:00:00 GMT

Shaochuang Huang, Zhuo Peng — In this paper, we prove a uniform pseudolocality theorem for smooth complete Ricci flows whose initial metrics have Ricci curvature bounded below and uniformly positive injectivity radius. No a priori curvature bound is assumed on the flow. As a consequence, we obtain strong uniqueness for smooth complete Ricci flows from a complete metric with the above geometric bounds. We also prove that every smooth complete Ricci flow from a complete Ricci-flat metric with uniformly positive injectivity radius is stationary. The same conclusion holds for any complete flat initial metric. In particular, we give an affirmative answer to Chen's question on the strong uniqueness of Ricci flow from Euclidean space in every dimension.

HKT geometry on locally conformally hyperkähler manifolds

Tue, 06 Oct 2026 00:00:00 GMT

Giovanni Gentili, Luigi Vezzoni — We study the HKT geometry of compact locally conformally hyperkähler manifolds. We prove that on these manifolds the quaternionic Monge-Ampère equation is always solvable, extending results of Alesker and of Dinew-Sroka, and we confirm the conjecture on the maximal existence time of the HKT-Ricci flow. On quaternionic Hopf surfaces we construct non-trivial HKT solitons and prove convergence of the normalized flow for symmetric initial data.

Generalized scalar curvature and Ricci flow on compact Lie groups

Thu, 01 Oct 2026 00:00:00 GMT

Zhiqi Chen, Jianhua Yu, Huibin Chen, Junna Ni — Let G be a compact connected semisimple Lie group, let Q be a bi-invariant metric and let ρ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with [H₀]=[ρ] exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form [H₀]=[ρ] and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

Thu, 01 Oct 2026 00:00:00 GMT

Jiyuan Han — We prove a uniform L^∞-estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

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.

Cohomogeneity one solutions of the IIB system

Wed, 23 Sep 2026 00:00:00 GMT

Lorenzo Foscolo, Mario Garcia-Fernandez — The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

Topology of low-dimensional generalized Ricci solitons and string backgrounds

Wed, 26 Aug 2026 00:00:00 GMT

Jeffrey Streets — Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to S³ / Γ or S³ × S¹ / Γ. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion G₂, strong torsion Spin(7)-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered 5-manifolds over complex orbifolds.

Homogeneous Generalized Ricci flows II

Wed, 26 Aug 2026 00:00:00 GMT

Elia Fusi, Ramiro A. Lafuente, James Stanfield — We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.

Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow

Fri, 14 Aug 2026 00:00:00 GMT

Dongwoo Gang — For a closed connected Riemannian manifold (M,g), the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding x↦ pₜ(x,·) dvol_g. The resulting family gₜ agrees with Ricci flow to first order in t, but in general not to second order. We prove that gₜ =g-2tRic_g +t²(−ΔRic_g +2Ric_g²−2/3𝓠_g) +O_(C⁰)(t³), where 𝓠_g is quadratic in the full curvature tensor. The term −ΔRic_g also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while gₜ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is O(t²), and round spheres show that this estimate is sharp.

Integrable Deformations and Stability of the Ricci Flow

Thu, 16 Apr 2026 00:00:00 GMT

Maxwell Stolarski, Alex Waldron — We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the L^p-stability theorems of Deruelle-Kroncke and Kroncke-Petersen.

Ancient Ricci flows with nonnegative Ricci curvature

Mon, 30 Mar 2026 00:00:00 GMT

Yuxing Deng, Ganqi Wang, Yongjia Zhang — In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity – a noncollapsed F-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed F-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature (Ric≥ εR g), either it is compact, or every tangent flow at infinity is a Ricci flat cone.

Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

Fri, 20 Mar 2026 00:00:00 GMT

Arunima Bhattacharya, Micah Warren, Daniel Weser — We express the mean curvature flow of Lagrangian submanifolds in pseudo-Riemannian manifolds endowed with the Kim-McCann-Warren metric within the framework of generalized mean curvature flow on Kim-McCann manifolds. While generalized mean curvature flow has been studied in Kähler geometry, our work shows that techniques from para-Kähler geometry arise naturally in the Kim-McCann setting. Using this perspective, we prove that the Lagrangian condition is preserved along the flow. By identifying generalized mean curvature flow with Lagrangian mean curvature flow, we show that the Ma-Trudinger-Wang regularity theory applies to this setting. In particular, the cross-curvature positivity condition of Kim-McCann yields smoothly converging flows of Lagrangian submanifolds. Under the cross-curvature condition, any Lagrangian submanifold avoiding the cut locus converges exponentially to a stationary submanifold, which locally arises as the graph of an optimal transport map. Our framework substantiates the analogy between special Lagrangian geometry in almost Calabi-Yau manifolds and optimal transport theory in the Kim-McCann setting. In particular, we show that Kim-McCann manifolds equipped with a para-holomorphic volume form serve as the natural counterpart to almost Calabi-Yau manifolds.

Generic neck pinch singularities along 2D Lagrangian mean curvature flow

Tue, 17 Feb 2026 00:00:00 GMT

Gábor Székelyhidi — We introduce a notion of nondegenerate neck pinch singularity along the Lagrangian mean curvature flow of surfaces in a Calabi-Yau surface. We show that such singularities can occur, are stable under small perturbations, and any neck pinch singularity can be perturbed to such a nondegenerate singularity near the singular time. Using this we answer some questions raised by Neves and Joyce. We also introduce nondegenerate teardrop singularities and show that these cannot occur for embedded flows.

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.

Dynamical functionals on ancient ARF Ricci flows

Thu, 13 Nov 2025 00:00:00 GMT

Isaac M. Lopez, Rio Schillmoeller — We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary λ-functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.

Ricci Flow on ALF manifolds

Fri, 24 Oct 2025 00:00:00 GMT

Dain Kim, Tristan Ozuch — We prove that on ALF n-manifolds with n≥ 4 the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's λ-functional, we define a renormalized functional λ_ALF whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF 4-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of λ_ALF to positive relative mass statements for ALF metrics.

Remarks on Singular Kähler-Einstein Metrics

Sat, 03 May 2025 00:00:00 GMT

Max Hallgren, Gábor Székelyhidi — We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular Kähler-Einstein spaces as well, assuming that they define RCD spaces.

Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

Wed, 23 Apr 2025 00:00:00 GMT

Pinsard Morel Lucas — Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

Chern-Ricci flow and t-Gauduchon Ricci-flat condition

Sat, 12 Apr 2025 00:00:00 GMT

Eder M. Correa, Giovane Galindo, Lino Grama — In this paper, we study the t-Gauduchon Ricci-flat condition under the Chern-Ricci flow. In this setting, we provide examples of Chern-Ricci flow on compact non-Kähler Calabi-Yau manifolds which do not preserve the t-Gauduchon Ricci-flat condition for t<1. The approach presented generalizes some previous constructions on Hopf manifolds. Also, we provide non-trivial new examples of balanced non-pluriclosed solution to the pluriclosed flow on non-Kähler manifolds. Further, we describe the limiting behavior, in the Gromov-Hausdorff sense, of geometric flows of Hermitian metrics (including the Chern-Ricci flow and the pluriclosed flow) on certain principal torus bundles over flag manifolds. In this last setting, we describe explicitly the Gromov-Hausdorff limit of the pluriclosed flow on principal T²-bundles over the Fano threefold ℙ(T_ℙ²).

Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

Sat, 15 Mar 2025 00:00:00 GMT

Sigurd B. Angenent, Dan Knopf — Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have uncountably many unstable perturbations.

Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons

Wed, 19 Feb 2025 00:00:00 GMT

Carlos Esparza — We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient Kähler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of the metric, nor fixing the soliton vector field. The method used to prove the uniqueness of the soliton vector field can be applied more widely, for example to show that conical Calabi-Yau metrics on a given complex manifold are unique up to biholomorphism. We also use it to prove that if two polarized Fano fibrations, as introduced by Sun-Zhang, are biholomorphic, then they are isomorphic as algebraic varieties.

Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons

Mon, 17 Feb 2025 00:00:00 GMT

M. M. Akbar, M. Self — We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in d ≥ 5 dimensions is invariant under the action of a one-parameter Lie group, and the group action on any metric can be expressed in a closed, universal form. This enables the generation of a one-parameter family of solutions from any given "seed" solution of the system without solving additional equations, as well as one-parameter families of local steady Ricci solitons. This extends the Lie point symmetry in four dimensions, found earlier for axisymmetric static vacuum systems, and provides the first example of solution generation in higher-dimensional vacuum Einstein equations that can be realized purely algebraically.

Some remarks on strong G₂-structures with torsion

Sun, 09 Feb 2025 00:00:00 GMT

Anna Fino, Udhav Fowdar — A G₂-structure on a 7-manifold M is called a G₂T-structure if M admits a G₂-connection ∇^T with totally skew-symmetric torsion T_φ. If furthermore, T_φ is closed then it is called a strong G₂T-structure. In this paper we investigate the geometry of (strong) G₂T-manifolds in relation to its curvature, S¹ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ∇^T and give an equivalent characterisation in terms of geometric properties of the G₂ Lee form. Analogous results are also obtained for almost Hermitian 6-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the S¹ reduction by the dual of the G₂ Lee form, we show that Ricci-flat strong G₂T-structures correspond to solutions of the SU(3) heterotic system on certain almost Hermitian half-flat 6-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong G₂T-structures with ∇^T not Ricci flat. Lastly, we classify G₂-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

Deriving Perelman's entropy from Colding's monotonic volume

Wed, 22 Jan 2025 00:00:00 GMT

Ignacio Bustamante, Martin Reiris — In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed N-space, which becomes Ricci-flat as N→ ∞, Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's N-space.

Infinitely many non-collapsed steady Ricci solitons on complex line bundles

Sun, 22 Dec 2024 00:00:00 GMT

Hanci Chi — We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles O(k) over ℂP}^(2m+1), where the base space is not necessarily Kähler–Einstein. Each O(k) with k∈ [3,2m+1] admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each O(k) with k≥ 3, the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

From Kähler Ricci solitons to Calabi-Yau Kähler cones

Tue, 03 Dec 2024 00:00:00 GMT

Vestislav Apostolov, Abdellah Lahdili, Eveline Legendre — We show that if X is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of X with a complex projective space of sufficiently large dimension is a Calabi–Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions v over the momentum polytope of a given smooth Fano manifold, for which a v-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a v-soliton.

Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds

Sun, 10 Nov 2024 00:00:00 GMT

Mathew George — A complex Monge-Ampère equation for differential (p,p)-forms is introduced on compact Kähler manifolds. For any 1 ≤ p < n, we show the existence of smooth solutions unique up to adding constants. For p=1, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for p=n-1, this gives the Monge-Ampère equation for (n-1) plurisubharmonic functions studied by Tosatti-Weinkove. For other p values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to (p,p)-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

Thu, 07 Nov 2024 00:00:00 GMT

Daheng Min — We first determine the asymptotic cone of the steady gradient Kähler-Ricci soliton of the Taub-NUT type constructed by Apostolov and Cifarell. Then we study a special case and prove that it is an ALF Calabi-Yau metric in a certain sense. Finally we construct new ALF Calabi-Yau metrics on crepant resolution of its quotients modeled on it using the method of Tian-Yau-Hein.