https://neckpinch.com/topics/einstein-metrics/New arXiv papers on einstein metrics, from Neckpinch Wed, 04 Nov 2026 13:15:59 GMT https://neckpinch.com/papers/2610.38137/arxiv:2610.38137Wed, 28 Oct 2026 00:00:00 GMTTomoyuki 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.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/2610.37151/arxiv:2610.37151Tue, 27 Oct 2026 00:00:00 GMTFengjiang 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.Ricci solitonsEinstein metricsScalar curvatureStability & uniqueness https://neckpinch.com/papers/2610.31270/arxiv:2610.31270Thu, 22 Oct 2026 00:00:00 GMTHongyu 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.Other curvature flowsEinstein metricsRicci-flat & Calabi–YauSingularitiesKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://neckpinch.com/papers/2610.21934/arxiv:2610.21934Thu, 15 Oct 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Junbang Liu — Let X be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class [ω], Jian–Shi–Song proved that there exists a unique cscK metric in the class 2πc₁(K_X)+ε[ω] when ε∈ (0,ε₀], using the properness criterion of Chen–Cheng. We prove that the cscK metrics in 2πc₁(K_X)+ε[ω] converge, as εdownarrow0, in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of X as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on X.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program https://neckpinch.com/papers/2610.12930/arxiv:2610.12930Fri, 09 Oct 2026 00:00:00 GMTFengyuan 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.Ricci solitonsEinstein metricsAlmost & η-Ricci solitonsStability & uniquenessBoundaries & non-compact https://neckpinch.com/papers/2610.11935/arxiv:2610.11935Thu, 08 Oct 2026 00:00:00 GMTMatheus 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.Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsCurvature pinchingKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://neckpinch.com/papers/2609.36418/arxiv:2609.36418Tue, 29 Sep 2026 00:00:00 GMTWenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete gradient Kähler–Ricci shrinker satisfying Ric+∇² f=1/2 g. We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer k∈\0,…,m\ such that R≡ k and (M^(2m),g,J)≅ bigl(N^(2k)×ℂ^(m-k), g_N+g_Euc,J_N⊕ J₀bigr), where (N^(2k),g_N,J_N) is Kähler–Einstein and Ric_(g_N)=1/2 g_N. As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition 𝓛_∇fRic=0 implies radial flatness and hence rigidity.Ricci solitonsEinstein metricsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability 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/2609.23630/arxiv:2609.23630Sun, 20 Sep 2026 00:00:00 GMTYasufumi Nitta — We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability 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/2609.20513/arxiv:2609.20513Thu, 17 Sep 2026 00:00:00 GMTWangjian Jian, Jian Song — The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.Ricci flowKähler–Ricci flowEinstein metricsSingularities(Non)collapsingKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://neckpinch.com/papers/2609.07124/arxiv:2609.07124Mon, 07 Sep 2026 00:00:00 GMTYifan Xiao — In this paper, we study the Kähler–Ricci flow on ℂP^m-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type I.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/2609.03332/arxiv:2609.03332Thu, 03 Sep 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Hung Tran — We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds N := N₁ × ⋯ × Nᵣ, with metric constructed using the ansatz considered in, et. al. In the earlier work by the authors, we considered the "two-bolt" case where both ends of the foliation close with the "bolt" N. In this article, we continue our work on the more subtle "nut-bolt" and "two-nut" cases. The former has one end of the interval closes with a nut-type collapse (i.e. N' := N₂ × ⋯ × Nᵣ) and the other with a bolt (i.e. N). The compactification M is then a ℂP}^(m+1)-bundle over N'. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the Nᵢ's must be ℂP}^(m₀) and ℂP}^(m_ℓ), and the compactification M is a ℂP}^(m₀+m_ℓ+1)-bundle over ∏_(k≥ 3)Nₖ. We proved that in all "two-bolt", "nut-bolt" and "two-nut" caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be (Σ^(m+1), g_Σ(t)) × (ℂ^(k), flat) with m, k ≥ 0, where Σ is one of the following: ℂP}^(m+1), Tot(𝓛^(⊕(m+1))), or a projectivization ℙ(𝓞^(⊕(m₀+1)) ⊕ 𝓛^(⊕(m_ℓ+1))) with m₀ + m_ℓ = m, and 𝓛 is a line bundle over the product of some of the N₁, ⋯, Nᵣ factors. The metric g_Σ(t) is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.Ricci flowKähler–Ricci flowRicci solitonsEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/2608.21732/arxiv:2608.21732Sat, 22 Aug 2026 00:00:00 GMTFengyuan Zhang — The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \beginequation D_ijk=\nabla_iR_jk+\nabla_jR_ki+\nabla_kR_ij=0\nonumber \endequation These structures include non-compact gradient shrinking Ricci soliton, compact (m > 1)-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.Ricci solitonsEinstein metricsAlmost & η-Ricci solitonsBoundaries & non-compact https://neckpinch.com/papers/2608.16248/arxiv:2608.16248Mon, 17 Aug 2026 00:00:00 GMTJungwoo Moon — We investigate a compact Einstein-type manifold whose potential vector field generates a Riemannian foliation. In particular, we prove necessary conditions for such a manifold to be taut and to have a splitting property. Additionally, some properties of taut Riemannian foliations on a compact almost Ricci solitons and a compact Einstein manifolds are provided.Ricci solitonsEinstein metricsAlmost & η-Ricci solitons 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/2608.08781/arxiv:2608.08781Sun, 09 Aug 2026 00:00:00 GMTBeatrice Brienza, Anna Fino, Udhav Fowdar, Gueo Grantcharov — Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a ∇-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact ∇-Einstein manifold in dimension 5 and 7. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with S¹.Ricci flowGeneralized & G₂ flowsEinstein metricsKähler geometryBoundaries & non-compactSasakian & contact geometry https://neckpinch.com/papers/2607.06975/arxiv:2607.06975Wed, 08 Jul 2026 00:00:00 GMTFumika Mizoguchi — In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers. Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons. Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.Ricci solitonsEinstein metricsHomogeneous spacesDiscrete & graph Ricci https://neckpinch.com/papers/2607.05002/arxiv:2607.05002Mon, 06 Jul 2026 00:00:00 GMTRasmus Jouttijärvi — We study the stability problem for Einstein manifolds with boundary with respect to the Einstein-Hilbert action. The geometric boundary conditions we are using arise naturally from studying the calculus of variations associated with Ricci flow. Upon the introduction of a boundary, the space of TTg tensors no longer arise naturally as the defining space for the stability condition. Thus we must settle for the larger subspace of tensors preserving the scalar curvature, the total volume and the Bianchi gauge condition, which we call the space of TVg tensors. As a test-case, we shall discuss stability of the Riemannian Schwarzschild anti-deSitter family of metrics; a problem known as "a Black hole in a box".Ricci flowEinstein metricsScalar curvatureStability & uniquenessBoundaries & non-compactPhysics https://neckpinch.com/papers/2606.13066/arxiv:2606.13066Thu, 11 Jun 2026 00:00:00 GMTAtreyee Bhattacharya, Sayoojya Prakash — Rigidity, stability and local minimizing properties of Einstein metrics as critical points of quadratic Riemannian functionals defined by L²-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature have been extensively studied. However, there are non-Einstein critical points of these functionals that are not so well understood. In this paper, we study Ricci solitons, a generalization of Einstein metrics, that are critical points of a special quadratic curvature functional and analyze their rigidity.Ricci solitonsEinstein metricsScalar curvatureStability & uniqueness https://neckpinch.com/papers/2606.01153/arxiv:2606.01153Sun, 31 May 2026 00:00:00 GMTBo-Yong Chen, Yuanpu Xiong, Liyou Zhang — Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both ∫_M |S_C⁻(g)|ⁿdV_g and rm vol_g(M) (normalized by S_C(g)≥ −1) are bounded below by ((nπ)ⁿ)/(n!)CanVol(M) for any Hermitian metric g on a compact complex n-manifold M. Here S_C denotes the Chern scalar curvature, S_C⁻=max\−S_C,0\ and rm CanVol(M) is the canonical volume of M, i.e., the volume of the canonical line bundle K_M. Moreover, if rm vol_g(M)=((nπ)ⁿ)/(n!)CanVol(M) holds for some Kähler metric with S_C≥ −1, then it has to be the Kähler-Einstein metric of negative scalar curvature. The completely new phenomenon is that if M is a compact Kähler manifold such that K_M is nef, then rm MinVol_C(M)=𝓘_C(M)=𝓘_C⁻(M)=((nπ)ⁿ)/(n!)CanVol(M), where rm MinVol_C(M) is the infimum of rm vol_g(M) with S_C(g)≥ −1 and 𝓘_C⁻(M)=inf_g ∫_M |S_C⁻(g)|ⁿdV_g, 𝓘_C(M)=inf_g ∫_M |S_C(g)|ⁿdV_g. It remains unknown whether the nef condition is superfluous. The answer is positive when M is obtained by blowing up a finite number of points from a projective manifold with big and nef canonical line bundle. The arguments are based on the asymptotic behaviour of the Bergman kernel of mK_M as m→ ∞, the theory of Kähler-Ricci flow and singular Kähler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://neckpinch.com/papers/2605.25723/arxiv:2605.25723Mon, 25 May 2026 00:00:00 GMTSergey Stepanov — The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.Ricci flowEinstein metricsCurvature pinchingStability & uniquenessEigenvalues & spectrum https://neckpinch.com/papers/2605.17001/arxiv:2605.17001Sat, 16 May 2026 00:00:00 GMTHanbing Fang, Yu Li — In this paper, we extend the results of to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed 𝓦-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder ℝ^k × N^(n-k), where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset 𝓢^k_qc(N)⊂ 𝓢^k, consisting of points at which some tangent flow is given by ℝ^k × N^(n-k) or its quotient, is horizontally parabolic k-rectifiable.Ricci flowEinstein metricsSingularitiesStability & uniquenessEigenvalues & spectrum https://neckpinch.com/papers/2605.08796/arxiv:2605.08796Sat, 09 May 2026 00:00:00 GMTAkito Futaki — The notion of weighted extremal Kähler metrics extends the classical notion of Calabi's extremal Kähler metrics, but includes many well-studied objects in Kähler geometry such as Kähler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal Kähler metrics.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry https://neckpinch.com/papers/2604.21002/arxiv:2604.21002Wed, 22 Apr 2026 00:00:00 GMTSamuel 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.Ricci solitonsEinstein metricsAlmost & η-Ricci solitons4-manifoldsLimit spaces & RCD https://neckpinch.com/papers/2604.17954/arxiv:2604.17954Mon, 20 Apr 2026 00:00:00 GMTAndrew Gracyk — We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly Kähler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, or more closely a Kähler cross-entropy Hessian. This recovers a Kähler-Ricci flow variation up to a time derivative and expectation, or an average-valued Kähler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived Kähler flow.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/2602.19913/arxiv:2602.19913Mon, 23 Feb 2026 00:00:00 GMTMan-Chun Lee, Valentino Tosatti, Junsheng Zhang — We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stabilityAnalytic minimal model program https://neckpinch.com/papers/2601.18931/arxiv:2601.18931Mon, 26 Jan 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Hung Tran — In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://neckpinch.com/papers/2512.21910/arxiv:2512.21910Fri, 26 Dec 2025 00:00:00 GMTAlexander Bednarek — This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of C⁻¹(T-t)^(n-m) Ω≤ ω(t)ⁿ≤ C(T-t)^(n-m)Ω. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate √(T-t). Furthermore, several precise C⁰-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryScalar curvatureKähler–Einstein & K-stabilityAnalytic minimal model program https://neckpinch.com/papers/2512.21904/arxiv:2512.21904Fri, 26 Dec 2025 00:00:00 GMTAlexander Bednarek — This is the first of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. Given a Fano fibration which is generated by Kawamata's theorem from a compact Kähler manifold X endowed with an ample, rational line bundle L and non-nef canonical line bundle K_X, we construct a (1,1)-form on the regular part of the base analytic variety which is related to the Weil-Petersson metric. It is also proven that the singular Kähler metric constructed by Zhang, Zhang, on the base analytic variety satisfies a twisted Kähler-Einstein equation involving this (1,1)-form and, for a submersion, that the Chern classes of X and the base manifold decompose in terms of this (1,1)-form.Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stabilityAnalytic minimal model program