Neckpinch — Ricci flow

Conjugate radius and parabolicity along the Ricci flow

Mon, 02 Nov 2026 00:00:00 GMT

Chandan Kumar Mondal — We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if (Mⁿ,g(t))_(t∈[0,T]) is a complete Ricci flow with nonnegative scalar curvature at t=0, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor Ric-1/2 Rg is bounded above, then parabolicity is preserved along the flow and the conjugate radius of g(t) is finite for every t∈(0,T]. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that liminf_(r→∞)Vol(B(p,r))/r²=0, has finite conjugate radius.

Ricci flows and bracket flows on twisted principal bundles

Mon, 02 Nov 2026 00:00:00 GMT

Henry 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.

On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

Fri, 30 Oct 2026 00:00:00 GMT

Yu Li, Longhuan Yan, Wenjia Zhang — We prove strong uniqueness of tangent flows at the first singular time of a closed Ricci flow when one tangent flow is generated by a complete smooth asymptotically conical Ricci shrinker. The modified flow converges to this shrinker in a fixed gauge, with quantitative estimates on expanding domains. The main ingredients are a Łojasiewicz-Simon inequality for Perelman's entropy, based on the global weighted divergence-free gauge constructed in the preceding paper, and estimates controlling the geometric radius under Ricci flow.

Blowdowns along Ricci flow in dimension four

Thu, 29 Oct 2026 00:00:00 GMT

Dain Kim, Tristan Ozuch — We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C¹⁴-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C¹⁴-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂP}²\#ℓℂP}², through finitely many blowdown singularities. (2) On the standard 2ℂP}²\#9ℂP}², there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.

Backward Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the G₂-Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local μ-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the G₂-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

The Yang-Mills flow in vector bundles over higher dimensional Riemannian manifolds

Wed, 28 Oct 2026 00:00:00 GMT

Jared Casey, Chak Hoi Chan, Min-Chun Hong — We develop a parabolic formulation of the Uhlenbeck gauge fixing theorem and use it to prove the maximal existence of the Yang-Mills flow in vector bundles over compact n-dimensional Riemannian manifolds for n ≥ 4. We apply Moser's iteration technique to give a new proof of the bound of the curvature. Using an idea of Hamilton on the Ricci flow, we also provide a new proof on uniform estimates of higher order derivatives. Furthermore, we investigate the blow-up of the Yang-Mills flow at the maximal existence time and improve an asymptotical result of Hong-Tian.

An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

Tue, 27 Oct 2026 00:00:00 GMT

Hongxin Guo, Guozhi Lin — For a rotationally symmetric Ricci flow g(t)=ds²+f(s,t)²g_(Sⁿ⁻¹), n≥3, we study the scale-invariant curvature anisotropy A=1-fₛ²+ffₛₛ introduced by Angenent–Knopf. At the critical power p_*=(n-1)/(n-2), with Φ=sgn(A)|A|^(p_*/2), we prove the distributional identity (∂ₜ−Δ)|A|^(p_*) =−4/n-1 |Φₛ+(n-1)(fₛ)/fΦ|². Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on Sⁿ: if the entropy associated with a singular-time kernel is at most 1 at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the 0/1 quantization of the singular-time anisotropic density.

Variations of the harmonic entropy and linear drift on negatively curved surfaces

Mon, 26 Oct 2026 00:00:00 GMT

Karen Butt, Alena Erchenko, Tristan Humbert — For closed negatively curved surfaces, we obtain formulas for the derivatives of the harmonic entropy of its geodesic flow and the linear drift of its Brownian motion along conformal deformations. We deduce that the harmonic entropy of the geodesic flow of a closed surface of non-constant negative curvature (resp. linear drift of the Brownian motion on its universal cover) is strictly decreasing (resp. increasing) along the normalized Ricci flow. We also prove that any hyperbolic metric on a surface admits a neighborhood in the space of conformal negatively curved metrics in which the linear drift is greater or equal to the Liouville entropy, with equality if and only if the metric is hyperbolic.

Chern holonomy under the first and second Chern–Ricci flows

Mon, 26 Oct 2026 00:00:00 GMT

Yuan Liao — We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)²; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.

Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

Thu, 22 Oct 2026 00:00:00 GMT

G. Bonelli, S. Lacroix, T. Pedroni, A. Tanzini — We construct a family of integrable 2d sigma models through 4d Chern-Simons theory and show that their classical point-particle reduction corresponds to a Hitchin system. The integrable Hitchin quantisation then provides a natural candidate Hamiltonian for the one-loop reduced dynamics of the sigma models, which coincides with the Perelman operator in all cases examined. This allows us to study the corresponding spectral problems using 4d 𝓝=2 gauge theory and 2d conformal field theory methods. For the sausage model, we derive exact quantisation conditions and compute exponentially small corrections essential for the agreement with numerical finite-volume results.

Parametrization of ancient Ricci flows with smooth compact tangent shrinkers

Thu, 22 Oct 2026 00:00:00 GMT

Yu Li — We study ancient Ricci flows whose tangent flow at backward infinity is a smooth compact Ricci shrinker. After normalization, every nontrivial solution either converges exponentially or has polynomial decay governed by neutral directions. We recover the finite-dimensional parametrization of the exponentially convergent solutions and establish finite-order asymptotic expansions for all slowly convergent solutions. For principal slow solutions, we obtain a locally bijective finite-dimensional parametrization compatible with diffeomorphisms, parabolic rescaling, and time translation.

Using Ricci Flow in the Construction of Homogeneous Cosmological Models

Mon, 19 Oct 2026 00:00:00 GMT

Fan 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.

Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

Mon, 19 Oct 2026 00:00:00 GMT

Tsz-Kiu Aaron Chow — We develop a pinching method for mean curvature flow in higher codimension. Motivated by pinching constructions for Ricci flow, the method uses a nonlinear deformation of the second fundamental form. We introduce two tensorial curvature conditions for closed immersions in Euclidean space which extend beyond the scope of classical scalar pinching results. In dimensions at least ten, we apply the method to prove convergence to a round point under the first condition and a canonical neighborhood theorem under the second.

Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

Sat, 17 Oct 2026 00:00:00 GMT

Randall R. Van Why — We demonstrate that Kähler-Ricci flow on a Fano surface decomposes as a symplectic deflation along a plurianticanonical divisor plus an exact symplectic deformation in the divisor complement. We apply this decomposition to the study of certain projective bundles. We establish the optimal decay rate in the setting of Song, Székelyhidi, and Weinkove for a family of toric metrics which match the cohomological profile of Kähler-Ricci flow. In doing so, we identify a C²-obstruction to proving the optimal decay rate in the setting of ruled toric surfaces. We present these ideas as a new symplectic-topological approach to the study of Kähler-Ricci flow.

Type I estimates for the Kähler-Ricci flow II

Fri, 16 Oct 2026 00:00:00 GMT

Wangjian Jian, Jian Song — In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial C⁰ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

Thu, 15 Oct 2026 00:00:00 GMT

Frederick 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.

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.

Long-Time Behavior of the α-Ricci Flow on the 2-Torus

Tue, 13 Oct 2026 00:00:00 GMT

Yifan Guo, Xiaokang Wang — We study the α-Ricci flow (g(t),α(t)), which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus M. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the α-Ricci flow depends on the cohomology class [α] ∈ H¹(M,ℝ), which is invariant under the flow. If [α] is rational, then as t→∞, (M,t⁻¹g(t)) Gromov–Hausdorff converges to S¹ with length ℓ([α])/sqrt 2 where ℓ([α]) is the primitive length of [α]. On the universal cover, the rescaled flow (s⁻¹tilde g(st),tildeα(st)) smoothly converges to an expander (g^*(t),α^*(t)) on ℝ² as s→ ∞ where α^*(t) is nonzero and constant in spacetime and g^*(t) is flat and expanding in the α^* direction. If [α] is irrational, then there is a sequence sᵢ→∞ such that (M,sᵢ⁻¹g(sᵢt)) Gromov–Hausdorff converges to a point. Moreover, on the universal cover, (sᵢ⁻¹tilde g(sᵢt),tildeα(sᵢt)) smoothly converges to (g^*(t),α^*(t)).

Pseudolocality and strong uniqueness for complete Ricci flows

Tue, 13 Oct 2026 00:00:00 GMT

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

The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers

Mon, 12 Oct 2026 00:00:00 GMT

Shaosai 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.

A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

Fri, 09 Oct 2026 00:00:00 GMT

Yuan Liao, Ziyang Qin, Jui-Hui Chung, Ayush Khaitan, Zhengyang Zhang, Hongzhou Lin, Zijian Zhang, Ligeng Zhu, Peng Lu, Bennett Chow — We formalize the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture. The public Lean declaration is reproduced in this paper. Its geometric interpretation follows, and verification is recorded in 4.

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.

Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

Tue, 06 Oct 2026 00:00:00 GMT

Shaochuang Huang, Zhuo Peng — In this paper, we study strong uniqueness for smooth instantaneously complete Chern-Ricci flows on noncompact complex manifolds. Under suitable reference metric and positivity assumptions, we establish strong uniqueness for initial Hermitian metrics which are Kähler outside a compact set. The initial metric may be incomplete, and no curvature, metric comparison, or growth bounds are imposed on the evolving solutions. In particular, we obtain strong uniqueness from complete initial metrics in this class with bounded Chern-Ricci curvature. We also prove stationarity of flows from the Euclidean metric on ℂⁿ and uniqueness of the instantaneously complete flow from the Euclidean metric on the complex unit ball for all positive time. Finally, on complex surfaces, we show that a Ricci flow remaining Hermitian for a fixed complex structure is Kähler at all times if it is Kähler at one time, and derive a corresponding strong uniqueness result for Ricci flows in this class.

HKT geometry on locally conformally hyperkähler manifolds

Tue, 06 Oct 2026 00:00:00 GMT

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

Scale-Invariant Non-Collapse from Raychaudhuri Comparison

Tue, 06 Oct 2026 00:00:00 GMT

Rohit Dhormare — We establish a scale-invariant spatial non-collapse estimate for hypersurface-orthogonal timelike geodesic congruences. The result extends a previous Raychaudhuri-based construction in which a comparison estimate for the expansion was used to control the volume of domains transported along the congruence. The additional ingredient here is a quantitative estimate of the spatial distance distortion induced by the expansion and shear. Assuming that the initial spatial geometry is non-collapsed up to a scale r₀, and that the expansion, shear, and timelike Ricci curvature satisfy suitable uniform bounds, we obtain, before the first blow-up time of the Raychaudhuri comparison solution, \beginequation* \operatornameVol_h_τ\bigl(B_h_τ(p_τ,r)\bigr) \geq κ_L(τ)\, r^3, \endequation* where \beginequation* κ_L(τ) = κ_0 e^-3Kτ \exp\!\left(\int_0^τΘ(s)\,ds\right), \qquad K = \fracΘ_03 + Σ_0 . \endequation* Here Θ denotes the solution of the scalar Raychaudhuri comparison equation and K controls the spatial metric distortion. For every finite time strictly before the comparison barrier, κ_L(τ) is positive. When the comparison solution has a finite blow-up time, the explicit comparison coefficient approaches zero as that time is approached from below. The comparison barrier is a boundary of the estimate and is not, by itself, a statement that the physical congruence develops a focal point or spacetime singularity. The result is structurally analogous to scale-invariant non-collapse estimates in geometric flow, but is mathematically distinct from Perelman's κ-noncollapsing theorem for Ricci flow.

L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

Mon, 05 Oct 2026 00:00:00 GMT

Xianbin Huang, Wancheng Zhang — In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform L² Riemannian curvature bounds and L⁴ Ricci curvature bounds up to dimension 7. These estimates improves L^(2-ε) curvature bounds established by Bamler and L² curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform L² curvature bounds on n-manifolds satisfying an integral L^p Ricci curvature bound for p> n/2, a uniform volume lower bound, and a strong ε-regularity assumption. This result extends the celebrated L² curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an L^p adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori L^(4-ε) curvature radius bound of Ricci flow established by Bamler, we obtain uniform L² curvature bounds on Ricci flows with bounded scalar curvature of dimension n ≤ 7.

Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions

Mon, 05 Oct 2026 00:00:00 GMT

Papa Badiane, Chinh H. Lu, Ahmed Zeriahi — We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.

Characterization of Sasakian Manifolds by Curvature Conditions

Mon, 05 Oct 2026 00:00:00 GMT

Sourav Nayak, Vladimir Rovenski, Dhriti Sundar Patra — Sasakian manifolds, a subclass of contact metric manifolds, play a central role in modern differential geometry and mathematical physics, where they arise naturally in geometric mechanics, CR geometry, and supersymmetric field theories. The weak contact metric (w.c.m.) structure generalizes the contact metric structure and provides a broader framework for studying contact geometry and its applications. A fundamental problem is to understand how curvature conditions constrain the underlying contact metric structure and, in particular, how they distinguish Sasakian geometry from its weaker variants. Using the partial Ricci flow, we characterize Sasakian structure (among w.c.m. manifolds) under the (κ,μ)-nullity condition related to curvature. For κ<1 we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for κ=μ=0 establish splitting and classification results.

On the tangent flows of noncollapsed Ricci flow limit spaces

Sun, 04 Oct 2026 00:00:00 GMT

Yu Li, Bing Wang — We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.