https://neckpinch.com/topics/kahler-einstein/New arXiv papers on kähler–einstein & k-stability, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.38137arxiv: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://arxiv.org/abs/2610.34340arxiv:2610.34340Sun, 25 Oct 2026 00:00:00 GMTJunhao Tian, Kai Zheng — In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any Tⁿ-invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.31270arxiv: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://arxiv.org/abs/2610.31068arxiv:2610.31068Thu, 22 Oct 2026 00:00:00 GMTLi Ma, Li Sheng — In this paper, we propose the Chern-Calabi Flow for extremal metrics in conformal Hermitian geometry. After the existence of extremal metric, we consider the existence of global flows to the Chern-Calabi flows for extremal metrics on compact complex surfaces with Gauduchon metrics. To achieve this goal, we use the Adams inequality and the Gronwall inequality method to obtain fourth order derivative norm estimates. We also study the convergence of the global solution to the Chern-Calabi flow and its limit is the solution of the Chern-Calabi extremal metric problem. We propose a fourth order elliptic problem and derive the Bochner formula relevant to the elliptic estimates for the Chern-Calabi heat flow.Other curvature flowsKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.21934arxiv: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://arxiv.org/abs/2610.11935arxiv: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://arxiv.org/abs/2610.05217arxiv:2610.05217Sun, 04 Oct 2026 00:00:00 GMTHaozhao Li, Bing Wang, Kai Zheng — We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.Other curvature flowsSingularitiesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.05201arxiv:2610.05201Sun, 04 Oct 2026 00:00:00 GMTLiding Huang, Xiaohua Zhu — We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold (M,J,T), and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial T-invariant generalized Kähler metric with its symplectic form in 2πc₁(M,J), we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2610.03592arxiv:2610.03592Fri, 02 Oct 2026 00:00:00 GMTCarlos Esparza, Junsheng Zhang — For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compactKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.36418arxiv: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://arxiv.org/abs/2609.23630arxiv: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://arxiv.org/abs/2609.23627arxiv: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://arxiv.org/abs/2609.20513arxiv: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://arxiv.org/abs/2609.07124arxiv: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://arxiv.org/abs/2609.06451arxiv:2609.06451Sun, 06 Sep 2026 00:00:00 GMTAlexander Bednarek — We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is n-1, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.Ricci flowKähler–Ricci flowKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2609.03332arxiv: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://arxiv.org/abs/2607.25181arxiv:2607.25181Tue, 28 Jul 2026 00:00:00 GMTTomoyuki Hisamoto, Masataka Iwai — We establish the following Miyaoka-Yau inequality for any n-dimensional klt Fano variety X, possibly K-unstable, in terms of its delta invariant: (2(n+1)c₂(X)−n c₁(X)²)· c₁(X)ⁿ⁻² ≥ −n (1-min\1,δ(X)\)² · c₁(X)ⁿ. Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kähler-Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessKähler–Einstein & K-stabilityAnalytic minimal model program https://arxiv.org/abs/2606.09235arxiv:2606.09235Mon, 08 Jun 2026 00:00:00 GMTShing Tak Lam — We prove that the asymptotics of the Hermitian-Yang-Mills flow on a slope semistable holomorphic vector bundle over a compact Kähler manifold are determined algebro-geometrically, via the iterated filtration defined by Haiden-Katzarkov-Kontsevich-Pandit. This proves a conjecture of Haiden-Katzarkov-Kontsevich-Pandit in this setting. Moreover, we prove a non-linear analogue, relating the asymptotics of the Calabi flow near a cscK manifold to the iterated balancing filtration of the deformation space. In both settings, we reduce the infinite-dimensional flow to a finite-dimensional flow, following the foundational work of Chen-Sun. In finite dimensions, we prove that the asymptotics of the moment map flow are determined by the iterated balancing filtration, proving a conjecture of Ibáñez Núñez.Other curvature flowsKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2606.01153arxiv: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://arxiv.org/abs/2605.08796arxiv: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://arxiv.org/abs/2604.17954arxiv: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://arxiv.org/abs/2604.02632arxiv:2604.02632Fri, 03 Apr 2026 00:00:00 GMTYi Li, Jie Wang, Pingsan Yuan, Chao Zheng — In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.Other curvature flowsKähler geometryKähler–Einstein & K-stabilityDiscrete & graph Ricci https://arxiv.org/abs/2603.19405arxiv:2603.19405Thu, 19 Mar 2026 00:00:00 GMTJingrui Cheng, Junhao Tian — In this paper, we observe that if the initial data of pseudo Calabi flow has volume form C⁰ close to a smooth one, then the flow is immediately smooth for t>0. As an application, we show that if the initial data has volume form C⁰ close to that of a cscK metric, then the pseudo Calabi flow exists for t∈ (0,+∞). We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to c₁(M).Other curvature flowsKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2602.19913arxiv: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://arxiv.org/abs/2601.22934arxiv:2601.22934Fri, 30 Jan 2026 00:00:00 GMTPak 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.Other curvature flows4-manifoldsLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2601.18931arxiv: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://arxiv.org/abs/2512.21910arxiv: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://arxiv.org/abs/2512.21904arxiv: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 https://arxiv.org/abs/2512.04572arxiv:2512.04572Thu, 04 Dec 2025 00:00:00 GMTJie He, Haozhao Li — In this paper, we study a family of twisted Calabi flows connecting the J-flow and Calabi flow on a compact Kähler manifold with a constant scalar curvature (cscK) metric. We show that for any initial data the twisted Calabi flow near the J-flow has long time existence and converges smoothly to the cscK metric. Moreover, we show that if a twisted Calabi flow has long time existence and converges, then the nearby twisted Calabi flow with the same initial data also has long time existence and converges. These results imply the openness of the continuity method to study Chen's long time existence conjecture on (twisted) Calabi flow on cscK manifolds.Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2512.03323arxiv:2512.03323Wed, 03 Dec 2025 00:00:00 GMTCharles Cifarelli, Carlos Esparza — Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration (π: X → Y, ξ). In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric g implies K-polystability of (π: X → Y, ξ), in the case that the Ricci curvature of g decays at infinity. As an application, we give a non-existence result: if M is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space X of the cube root of K_M is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.Ricci solitonsSingularitiesSurfacesKähler geometryStability & uniquenessBoundaries & non-compactKähler–Einstein & K-stabilityAnalytic minimal model program