Neckpinch — Kähler–Ricci solitons

Steady conformally Kähler Ricci solitons with isoparametric potential

Thu, 29 Oct 2026 00:00:00 GMT

Ernani Ribeiro, Hung Tran — We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are U(2)-invariant nuts on ℝ⁴, including the Bryant and Cao solitons, and bolts on S² × ℝ² and on 𝓞(−m)→ℂP}¹. For each m≥3 there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.

The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-

Wed, 28 Oct 2026 00:00:00 GMT

Tomoyuki 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.

Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

Thu, 15 Oct 2026 00:00:00 GMT

Lei Zhang, Zhenlei Zhang — In this paper, we study heat kernel estimates along the Kähler-Ricci flow with finite time noncollapsing singularities over a compact Kähler manifold. The starting point is the type-II scalar-curvature estimate, which gives the natural parabolic scale r≲ T-t near the singular time. By combining the distance distortion and space-time cut-off estimates of Bamler–Zhang with entropy, mean-value inequality, and conjugate heat-kernel estimates, we obtain two-sided Gaussian bounds and gradient bounds that are uniform on this scale. Under the global type-I scalar-curvature estimate, the same estimates hold on every scale. We then use these estimates and the monotonicity of the W₁-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances dₜ (induced by ω(t)) converge uniformly to this pseudo-distance. Its metric quotient is the unique Gromov–Hausdorff limit. Finally, we apply Bamler's theory and the recent work of Hallgren–Zhang to obtain the following. Every fixed-base-point type-I blow-up admits a subsequential metric-flow limit whose time slices are singular shrinking Kähler–Ricci soliton spaces with normal complex-analytic and log-terminal structure, unique tangent cones, and a polarized Fano fibration. For moving base points, the limiting metric flow, equipped with its limiting conjugate heat flow, is a metric soliton precisely under an entropy-pinching condition.

Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature

Thu, 08 Oct 2026 00:00:00 GMT

Matheus 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.

Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

Sun, 04 Oct 2026 00:00:00 GMT

Liding 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.

Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

Thu, 01 Oct 2026 00:00:00 GMT

Ronan J. Conlon, Alix Deruelle — We establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler-Ricci soliton up to pushforward by biholomorphism.

Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

Thu, 24 Sep 2026 00:00:00 GMT

Matheus Andrade Ribeiro de Moura Horácio — We prove that every complete nonsteady gradient Kähler-Ricci soliton with constant scalar curvature is rigid. This establishes Cao's rigidity conjecture for Kähler-Ricci solitons in arbitrary dimension and gives the corresponding result for expanding solitons, without additional curvature assumptions. The proof uses a rigidity criterion for gradient Ricci solitons with constant scalar curvature, expressed by the vanishing of the Lie derivative of the Ricci tensor along the soliton vector field. In the Kähler case, this vanishing follows from constant scalar curvature, the closedness of the Ricci form, and the classical real holomorphicity of the soliton vector field.

Shrinking Kähler-Ricci solitons on toric fano fibrations

Mon, 21 Sep 2026 00:00:00 GMT

Tristan C. Collins, Genggeng Huang, Freid Tong, Yulun Xu — We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

Sun, 20 Sep 2026 00:00:00 GMT

Yasufumi 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.

Linear Stability of Steady and Expanding Kähler-Ricci Solitons

Fri, 18 Sep 2026 00:00:00 GMT

Lucas Lavoyer, Adam Thompson — We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

Tue, 01 Sep 2026 00:00:00 GMT

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

L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

Thu, 30 Jul 2026 00:00:00 GMT

Guangwen Zhao — We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton (M,g,J,f) and a real-valued pluriharmonic function u, we investigate conditions under which u must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that u is constant whenever ∫_M|∇u|^pdv<∞ for some 0

Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

Tue, 28 Jul 2026 00:00:00 GMT

Ronan J. Conlon, Alix Deruelle — We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.

The Miyaoka-Yau inequality and the delta invariant for Fano varieties

Tue, 28 Jul 2026 00:00:00 GMT

Tomoyuki 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.

The Invariant Szegő metric on strongly pseudoconvex domains

Mon, 25 May 2026 00:00:00 GMT

Anjali Bhatnagar, Jiliang Fan — The Fefferman–Szegő metric g_FS^Ω on a C^∞-smooth bounded strongly pseudoconvex domain Ω⊂ℂⁿ is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its L²-Dolbeault cohomology outside the middle degree: dim H^(p,q)₂(Ω)=0 if p+q≠ n, while dim H^(p,q)₂(Ω)=∞ if p+q=n. We also prove that the metric has C^∞-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman–Szegő metric is a gradient Kahler–Ricci soliton, then Ω is biholomorphic to the unit ball Bⁿ. Moreover, if the metric has constant scalar curvature, then it is Einstein, and again Ω is biholomorphic to Bⁿ. We also give a Ramadanov-type criterion in terms of the Fefferman–Szegő invariant function. Finally, in dimension n=2, assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman–Szegő kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, Ω is simply connected, then Ω is biholomorphic to B².

On weighted extremal Kähler metrics

Sat, 09 May 2026 00:00:00 GMT

Akito 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.

Kähler-Ricci solitons with almost maximal symmetry

Sat, 02 May 2026 00:00:00 GMT

Ha Tuan Dung, Catherine Searle, Hung Tran — This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension n, with an isometry group of dimension at least n²−1. We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension 2². In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily Kähler), the potential function is invariant by the action.

A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

Thu, 26 Mar 2026 00:00:00 GMT

Richard H. Bamler, Eric Chen, Ronan J. Conlon — We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler–Ricci soliton.

An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for t=0

Fri, 19 Dec 2025 00:00:00 GMT

Ivin Babu, Ronan J. Conlon, Alix Deruelle — We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.

Linear stability and instability of Kähler Ricci solitons

Wed, 19 Nov 2025 00:00:00 GMT

Keaton Naff, Tristan Ozuch — We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted L²-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension 4. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

Stability of asymptotically conical gradient Kähler-Ricci expanders

Wed, 08 Oct 2025 00:00:00 GMT

Longteng Chen — In this work, we consider a perturbation of an asymptotically conical gradient expanding Kähler-Ricci soliton metric g in the same Kähler class. We demonstrate that, under suitable assumptions, the normalized Kähler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding Kähler-Ricci soliton metric g_∞. Moreover, if the perturbed initial metric is asymptotic to g at spatial infinity, then the limiting metric coincides with the original soliton, that is, g_∞ = g.

On the Bergman metric of Cartan-Hartogs domains

Tue, 07 Oct 2025 00:00:00 GMT

Andrea Loi, Roberto Mossa, Fabio ZUddas — We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_Omega, mu we prove the equivalence of: (i) M_Omega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_Omega, mu and hat g_Omega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.

Toric geometry of generalized Kähler-Ricci solitons

Mon, 01 Sep 2025 00:00:00 GMT

Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets, Yury Ustinovskiy — We establish a local equivalence between toric steady Kähler-Ricci solitons and A-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are A-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.

Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

Tue, 19 Aug 2025 00:00:00 GMT

Shu-Cheng Chang, Yingbo Han, Chin-Tung Wu — In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

On a general class of free boundary Monge-Ampère equations

Thu, 07 Aug 2025 00:00:00 GMT

Tristan C. Collins, Benjy Firester — We solve a general class of free boundary Monge-Ampère equations given by det D²u = λ(f(−u))/(g(u^⋆)h(∇u))χ_(u<0\) in ℝⁿ, ∇u (ℝⁿ) = P where P is a bounded convex set containing the origin, and h>0 on P. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.

Universal embeddings of flag manifolds and rigidity phenomena

Thu, 31 Jul 2025 00:00:00 GMT

Andrea Loi, Roberto Mossa, Fabio Zuddas — We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous Kähler manifolds. As a first immediate consequence we show the triviality of a Kähler-Ricci soliton submanifod of C × Ω, where C is a flag manifold and Ω is a homogeneous bounded domain. Secondly, we show that no weak-relative relationship can occur among the fundamental classes of homogeneous Kähler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two Kähler manifolds are said to be weak relatives if they share, up to local isometry, a common Kähler submanifold of complex dimension at least two. Our main result precisely shows that if E is (possibly indefinite) flat, C is a flag manifold, and Ω is a homogeneous bounded domain, then: E is not weak relative to C×Ω; C is not weak relative to E×Ω; Ω is not weak relative to E× C. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from relatives to the more flexible notion of weak relatives and dispense with the earlier "special" restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].

On Sun-Zhang's theory of Fano fibrations – weighted volumes, moduli and bubbling Fano fibrations

Tue, 17 Jun 2025 00:00:00 GMT

Yuji Odaka — We revisit the recent theory of Sun-Zhang on general Fano fibration (germs) which emerged from the study of non-compact Kahler-Ricci soliton metrics, primarily from an algebro-geometric perspective. In addition to reviewing the existing framework, we present new results, conjectures, and remarks. These include methods for computing weighted volumes via (restricted) volumes, Laplace transforms, and incomplete Gamma-functions, and a conjectural algebro-geometric construction ("bubbling") of Fano fibration with asymptotically conical base from degenerating Fano fibration.

Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker

Tue, 20 May 2025 00:00:00 GMT

Jiangtao Li — Let (X, g, J, f ) be a non-compact gradient shrinking Kahler-Ricci soliton. We prove that if the scalar curvature of X satisfies a mild assumption, then OP (X), the ring of holomorphic functions with polynomial growth on X, is finitely generated. This gives a partial confirmation to a conjecture of Munteanu and Wang (cf.[MW14]).

Uniqueness of asymptotically conical Kähler-Ricci flow

Wed, 30 Apr 2025 00:00:00 GMT

Longteng Chen — We study the uniqueness problem for the Kähler-Ricci flow with a conical initial condition. Given a complete gradient expanding Kähler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity–appearing as a Kähler cone–must coincide with the forward self-similar Kähler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its Kähler form remains in the same cohomology class as that of the soliton's self-similar Kähler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.

Shrinking gradient Kähler-Ricci solitons are simply-connected

Thu, 06 Mar 2025 00:00:00 GMT

Carlos Esparza — We show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking Kähler-Ricci solitons are simply-connected.