https://neckpinch.com/topics/homogeneous/New arXiv papers on homogeneous spaces, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2611.02674/arxiv:2611.02674Mon, 02 Nov 2026 00:00:00 GMTYa Gao — We study curvature flows for the dual Orlicz–Minkowski problem for strictly convex capillary hypersurfaces in the Euclidean half-space. The prescribed function has the form φ(h)G(X), where X=∇h+hν and G may be nonradial and nonhomogeneous. For acute contact angles, the evenness assumption together with either of the two integral conditions yields a smooth weighted-volume-preserving flow for all t>0 and a smooth solution to the dual Orlicz–Minkowski equation. Without the evenness assumption, two strict spherical-cap barriers yield the same conclusions for an unnormalized flow, whose limiting hypersurfaces solve the dual Orlicz–Minkowski equation with c=1. A uniform strict monotonicity condition under rescaling ensures uniqueness for c=1 and convergence of the unnormalized flow in any C^k norm at an exponential rate. The problem considered here includes the capillary Lₚ-Minkowski and Lₚ dual Minkowski problems as special cases.Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsHomogeneous spacesStability & uniquenessFree boundary & capillary https://neckpinch.com/papers/2611.02477/arxiv:2611.02477Mon, 02 Nov 2026 00:00:00 GMTHenry 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.Ricci flowRicci solitonsHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2611.02062/arxiv:2611.02062Sun, 01 Nov 2026 00:00:00 GMTVicente 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.Generalized & G₂ flowsRicci solitons4-manifoldsHomogeneous spaces https://neckpinch.com/papers/2610.40994/arxiv:2610.40994Thu, 29 Oct 2026 00:00:00 GMTErnani 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.Ricci solitonsKähler–Ricci solitons(Non)collapsing4-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compact https://neckpinch.com/papers/2610.34835/arxiv:2610.34835Mon, 26 Oct 2026 00:00:00 GMTFanqi 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.Ricci solitons3-manifoldsHomogeneous spacesPhysics https://neckpinch.com/papers/2610.27340/arxiv:2610.27340Mon, 19 Oct 2026 00:00:00 GMTFan Zhang, Lee Lindblom — This study develops numerical methods for computing homogeneous and isotropic cosmological models on manifolds with non-trivial spatial topologies. The use of Ricci flow is introduced here to aid in the construction of globally homogeneous initial data on these manifolds. The initial data produced in this way are evolved by solving Einstein's equation numerically to construct homogeneous and isotropic cosmological models on three exemplar manifolds in this study: Seifert-Weber space, the lens space L(5,2), and Sixth-Turn space.Ricci flowHomogeneous spacesPhysicsNumerics & visualization https://neckpinch.com/papers/2610.25883/arxiv:2610.25883Mon, 19 Oct 2026 00:00:00 GMTFumika 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.Ricci solitonsRicci-flat & Calabi–YauHomogeneous spacesDiscrete & graph Ricci https://neckpinch.com/papers/2610.21884/arxiv:2610.21884Thu, 15 Oct 2026 00:00:00 GMTKurando Baba, Naoyuki Koike — Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.Mean curvature flowHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2610.20042/arxiv:2610.20042Wed, 14 Oct 2026 00:00:00 GMTYuxin 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.Ricci solitons3-manifoldsSurfacesHomogeneous spacesSasakian & contact geometryPhysics https://neckpinch.com/papers/2610.17229/arxiv:2610.17229Mon, 12 Oct 2026 00:00:00 GMTShaosai Huang — Let compact RCD(κ,N) spaces converge in the measured Gromov–Hausdorff sense, and suppose that they have (δ,ρ)-Reifenberg local covers for one fixed ρ>0, with |κ|ρ²≤1 and δ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of O(n) whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with O(n), with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition Aₛ it is a closed aspherical manifold. Three appendices treat smooth manifolds with Ric≥−(n-1) and small local rewinding Reifenberg error: a uniform short-time Ricci flow with |Rm|≤α/t, also for complete non-compact data, where it is unique among complete flows with curvature at most A/t; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov–Hausdorff approximations with small error.Ricci flow(Non)collapsingWeak & singular flowsLimit spaces & RCDHomogeneous spacesBoundaries & non-compact https://neckpinch.com/papers/2610.07897/arxiv:2610.07897Tue, 06 Oct 2026 00:00:00 GMTGiovanni 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.Ricci flowRicci-flat & Calabi–YauKähler geometryHomogeneous spaces https://neckpinch.com/papers/2610.08859/arxiv:2610.08859Mon, 05 Oct 2026 00:00:00 GMTHuibin 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.Generalized & G₂ flowsRicci solitonsHomogeneous spaces https://neckpinch.com/papers/2610.04428/arxiv:2610.04428Sat, 03 Oct 2026 00:00:00 GMTAbdelhadi Belkhirat — Let V be an oriented Euclidean vector space and let D(V)=ΛV^*⊗ΛV^* be the algebra of double forms. The two exterior factors carry natural factorwise exterior and interior multiplication operators and therefore two commuting Clifford actions. The central result of the paper is an intrinsic realization of these Clifford actions inside Greub's composition algebra. For a∈ V^* we introduce the inhomogeneous double form χₐ=((1⊗ a)−(a⊗1))e^g, prove that the χₐ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas C_R(a)ω=χₐ∘ω, C_L(a)ω=−ω∘χₐ. For h∈D^(1,1) this yields a canonical four-component decomposition of the coupled left–right Clifford action into exterior multiplication, double contraction and two generalized Bianchi operators. For the metric double form g these four operators generate two commuting sl₂ actions. The construction extends canonically to arbitrary bidegree. In bidegree (2,2) its degree-preserving component admits a direct comparison with the double-form extension of the sharp product: for every R∈D^(2,2), Γ_(1,1)(R)(ω)=1/4R\#ω. The point of this identity is not a new definition of R\#, which is already constructed from Clifford commutators, but the fact that it appears canonically as the central homogeneous component of the higher left–right Clifford transform. As an application, the algebraic quadratic map 𝓠(R)=−1/4R\#R satisfies 𝓠(R)=−Γ_(1,1)(R)R, and its linearization at R is the operator −2Γ_(1,1)(R). For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.Ricci flowHomogeneous spaces https://neckpinch.com/papers/2610.03795/arxiv:2610.03795Thu, 01 Oct 2026 00:00:00 GMTZhiqi 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.Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauScalar curvatureHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2610.01608/arxiv:2610.01608Thu, 01 Oct 2026 00:00:00 GMTLorenzo 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.Generalized & G₂ flowsRicci solitons3-manifolds4-manifoldsHomogeneous spaces https://neckpinch.com/papers/2609.31023/arxiv:2609.31023Fri, 25 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.Other curvature flowsHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2609.29199/arxiv:2609.29199Thu, 24 Sep 2026 00:00:00 GMTHongyi Sheng, Weimin Sheng, Jiazhuo Yang — Let H^n+1 be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^betar; we treat both regimes: beta>1+kalpha and beta=1+kalpha. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_infty respectively in two different cases.Other curvature flowsHomogeneous spaces https://neckpinch.com/papers/2609.28280/arxiv:2609.28280Wed, 23 Sep 2026 00:00:00 GMTLorenzo 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.Ricci flowGeneralized & G₂ flowsRicci solitonsEinstein metricsRicci-flat & Calabi–Yau3-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compact https://neckpinch.com/papers/2609.23627/arxiv:2609.23627Sun, 20 Sep 2026 00:00:00 GMTNaotoshi Fujihara, Toru Kajigaya, Albert Wood — We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies ρ= Cω+ ndd^cf, and which satisfy C>0. With analogous assumptions on the flow, we obtain subconvergence to an f-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that (M,g,f) is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.Mean curvature flowCurve shorteningLagrangian mean curvature flowEinstein metricsSingularitiesSurfacesKähler geometryHomogeneous spacesLong-time existence & convergenceKähler–Einstein & K-stability https://neckpinch.com/papers/2609.13052/arxiv:2609.13052Fri, 11 Sep 2026 00:00:00 GMTTomoya Tatsuno — We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval [−3, −3/2]. The upper bound −3/2 is achieved by the complex Heisenberg group Heis₃(ℂ) with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold N has the pinching constant −3/2, then N admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant −3. This is derived by showing that there is an open neighborhood U of Heis₃(ℝ)× ℝⁿ⁻³ in the space of n-dimensional 2-step nilmanifolds such that the pinching constant is −3 on U, and any 2-step nilpotent Lie group N has a metric g such that (N,g) lies in U. In fact, if N is not isomorphic to Heis₃(ℝ)× ℝⁿ⁻³, then there is a curve gₜ of metrics on N with (N,gₜ)∈ U, showing that there are uncountably many left-invariant metrics on N such that the pinching constant is −3. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant −3/2 is also given, and the pinching constants of various examples are computed.Ricci solitonsCurvature pinchingHomogeneous spacesStability & uniqueness https://neckpinch.com/papers/2609.16029/arxiv:2609.16029Thu, 10 Sep 2026 00:00:00 GMTHuibin Chen, Zhiqi Chen, Fuhai Zhu — In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions 6m+d for m>d≥1. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra mathfrak m₂(n), n≥5, admits a generalized nilsoliton if and only if 5≤ n≤8.Generalized & G₂ flowsRicci solitonsHomogeneous spacesEigenvalues & spectrum https://neckpinch.com/papers/2609.12237/arxiv:2609.12237Thu, 10 Sep 2026 00:00:00 GMTAbdou Bousso, Ameth Ndiaye — In this article, we study Ricci-Yamabe solitons on the Lie group Sol × ℝⁿ equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to Span\∂_(t₁), …, ∂_(tₙ)\, and we explicitly provide the Perelman potential.Ricci solitonsAlmost & η-Ricci solitonsHomogeneous spaces https://neckpinch.com/papers/2608.30027/arxiv:2608.30027Sun, 30 Aug 2026 00:00:00 GMTJohn Lott — We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ℝ³, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric g(t) converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of (M,t⁻¹g(t)) is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, s⁻¹widetilde g(sτ), converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.Ricci flowRicci solitonsGeometrization3-manifoldsLimit spaces & RCDHomogeneous spacesLong-time existence & convergence https://neckpinch.com/papers/2608.25704/arxiv:2608.25704Wed, 26 Aug 2026 00:00:00 GMTJeongHyeong 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 solitons4-manifoldsLimit spaces & RCDHomogeneous spacesStability & uniquenessBoundaries & non-compact https://neckpinch.com/papers/2608.25619/arxiv:2608.25619Wed, 26 Aug 2026 00:00:00 GMTElia 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.Ricci flowGeneralized & G₂ flowsRicci solitonsRicci-flat & Calabi–YauScalar curvatureHomogeneous spacesStability & uniqueness https://neckpinch.com/papers/2608.14358/arxiv:2608.14358Fri, 14 Aug 2026 00:00:00 GMTBeatrice 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.Generalized & G₂ flowsEinstein metrics4-manifoldsKähler geometryScalar curvatureHomogeneous spaces https://neckpinch.com/papers/2608.10745/arxiv:2608.10745Tue, 11 Aug 2026 00:00:00 GMTSara Albert-Niclòs, Esther Cabezas-Rivas, Shujing Pan — We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential C^∞ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.Other curvature flowsCurvature pinchingHomogeneous spacesLong-time existence & convergenceGeometric inequalities https://neckpinch.com/papers/2608.08901/arxiv:2608.08901Sun, 09 Aug 2026 00:00:00 GMTKe Chen, Ruilin Hu, Quoc-Hung Nguyen — We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in W^(1,∞) by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted Hölder estimates whose weighted quantities remain bounded as tdownarrow0. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.Mean curvature flowHomogeneous spacesBoundaries & non-compactLong-time existence & convergence https://neckpinch.com/papers/2608.08370/arxiv:2608.08370Sat, 08 Aug 2026 00:00:00 GMTYoussef 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.Ricci solitons4-manifoldsHomogeneous spacesPhysics https://neckpinch.com/papers/2608.02840/arxiv:2608.02840Mon, 03 Aug 2026 00:00:00 GMTEric Cochran, Arseny Mingajev, Lawrence Mouillé, Nazia Valiyakath — We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.Ricci flowHomogeneous spacesNumerics & visualization