Neckpinch — Limit spaces & RCD

Calabi flow on polarized Kähler surfaces

Sun, 01 Nov 2026 00:00:00 GMT

Haozhao Li, Bing Wang, Kai Zheng — We study global smooth Calabi flows on polarized Kähler surfaces (M,L), assuming uniformly bounded Sobolev constants and the algebraic conditions χ(𝓞_M)=1 and c₁(M)· c₁(L)>0. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and |S|≤1. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ℙ¹×ℙ¹ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.

Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Sun, 01 Nov 2026 00:00:00 GMT

Haozhao Li, Bing Wang, Kai Zheng — We study global Calabi flows g(t) on compact Kähler surfaces with a uniform Sobolev constant. For any sequence tⱼ→∞, set gⱼ(t):=g(tⱼ+t). After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime hat C^∞-Cheeger–Gromov topology: it is uniform in the Gromov–Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.

On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis

Thu, 29 Oct 2026 00:00:00 GMT

Yu Li, Longhuan Yan, Wenjia Zhang — We develop the weighted Sobolev and Hölder analysis needed to study perturbations of complete asymptotically conical (AC) Ricci shrinkers. Weighted elliptic estimates yield a global weighted divergence-free slice in spaces that retain the asymptotic data at infinity. As a geometric application, we prove pointed Gromov-Hausdorff rigidity of nontrivial AC shrinkers with vanishing infinitesimal solitonic deformation space.

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.

Fixed geodesics of involutions on 3-spheres

Wed, 21 Oct 2026 00:00:00 GMT

Bin Shen, Hui Zhang — Let an isometric involution act on a smooth Riemannian 3-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most 2π, and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop–Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis–Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.

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.

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)).

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.

Radial Powered Mean Curvature Flow with Robin Boundary Conditions

Tue, 06 Oct 2026 00:00:00 GMT

Tianhong Pu, Xiaoqian Xin, Lixia Yuan — We study a positive radial graph evolving by V=H^α+b, with α>0 and b<0, in the unit cylinder and with uᵣ(1,t)=u(1,t). We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for H, uᵣ/r, and uᵣᵣ. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

Thu, 01 Oct 2026 00:00:00 GMT

Yeyun Xu, Linfeng Zhou — We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

Thu, 01 Oct 2026 00:00:00 GMT

Jiyuan Han — We prove a uniform L^∞-estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

Smoothing polyhedral spaces via Ricci flow

Tue, 29 Sep 2026 00:00:00 GMT

Richard H. Bamler, Esther Cabezas-Rivas — We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.

Heat kernel on Ricci shrinker metric measure spaces

Wed, 16 Sep 2026 00:00:00 GMT

Bing Wang, Jie Wang — As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function f admits a heat kernel H_f under the weighted volume measure e^(−f)dv. In this paper, we study H_f systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between H_f and the spacetime heat kernel H(x,t;y,s) under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that |∇R|=o(f^(3/2)) implies R=o(f).

Chern-Ricci flow on Kato surfaces

Wed, 16 Sep 2026 00:00:00 GMT

Daniele Angella, Mauricio Corrêa — Let S be a Kato surface and D its maximal reduced divisor of rational curves. On S∖ D we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption 0<μ<2, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is 2πb₂(S); in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

Sat, 12 Sep 2026 00:00:00 GMT

Shaochuang Huang, Zhuo Peng — In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

Under Ricci flow, a 3-torus goes flat

Sun, 30 Aug 2026 00:00:00 GMT

John Lott — We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ℝ³, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric g(t) converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of (M,t⁻¹g(t)) is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, s⁻¹widetilde g(sτ), converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.

Topology of low-dimensional generalized Ricci solitons and string backgrounds

Wed, 26 Aug 2026 00:00:00 GMT

Jeffrey Streets — Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to S³ / Γ or S³ × S¹ / Γ. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion G₂, strong torsion Spin(7)-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered 5-manifolds over complex orbifolds.

Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

Wed, 26 Aug 2026 00:00:00 GMT

JeongHyeong Park, Wooseok Shin — We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation |R|²∇f=0 for the soliton potential f. Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.

A Lean Formalization of Hamilton's Three-Manifold Theorem

Fri, 21 Aug 2026 00:00:00 GMT

Bennett Chow, Yuan Liao, Ziyang Qin — We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow

Fri, 14 Aug 2026 00:00:00 GMT

Dongwoo Gang — For a closed connected Riemannian manifold (M,g), the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding x↦ pₜ(x,·) dvol_g. The resulting family gₜ agrees with Ricci flow to first order in t, but in general not to second order. We prove that gₜ =g-2tRic_g +t²(−ΔRic_g +2Ric_g²−2/3𝓠_g) +O_(C⁰)(t³), where 𝓠_g is quadratic in the full curvature tensor. The term −ΔRic_g also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while gₜ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is O(t²), and round spheres show that this estimate is sharp.

Compactness and Rigidity of Complete Kähler Ricci Shrinkers

Tue, 11 Aug 2026 00:00:00 GMT

Tongxin Xu, Zhenlei Zhang — In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.

A note on generalized BV mean curvature flow with a critical forcing term

Thu, 06 Aug 2026 00:00:00 GMT

Kiichi Tashiro — In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

Tue, 28 Jul 2026 00:00:00 GMT

Yu Li, Junsheng Zhang — We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an S¹-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

Spacelike Brakke flows with boundary in pseudo-Euclidean space

Sat, 11 Jul 2026 00:00:00 GMT

Javier Hervás — We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

Thu, 11 Jun 2026 00:00:00 GMT

Yu Li — Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any t₀>0 and any connected component Z_(t₀)' of the time-slice Z_(t₀) of the completion, we prove that (𝓡ₜ',d_(gₜ)) xrightarrow[t↗ t₀]mathrmGromov-Hausdorff (Z_(t₀)',d_(t₀)^Z), where 𝓡ₜ' is the corresponding connected component of the regular part and 𝓡ₜ' denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic 1-rectifiable, and its time image has vanishing 1/2-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most 1. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.

On the structure of complete G₂-solitons

Thu, 04 Jun 2026 00:00:00 GMT

Haozhao Li, Yuanqing Ma, Kai Zheng — In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G₂-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C^(1,α) convergence. Our principal result shows that this C^(1,α) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G₂-solitons. Since the available C^(1,α) control does not directly close an elliptic bootstrap for the G₂-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C^(1,α) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G₂-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L^(7/2) -curvature bound then yields pointed C^∞ compactness.

Spectral Obstructions to Contracting Transport Maps on Curved Spaces

Sat, 23 May 2026 00:00:00 GMT

Shrey Aryan — Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). Recently, Beck and Jerison [BJ21] raised related questions on the round hemisphere. The existence of a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. In this work, we construct counterexamples to the corresponding spectral comparisons on spheres and on weighted manifolds satisfying CD(1,∞) in dimensions d≥4, yielding obstructions to contracting transport maps. In dimensions d≥5, the weighted counterexamples can be chosen to satisfy Ric_g≥ 0 and ∇_g² V ≥ g separately. In dimension two, we use inverse mean curvature flow to construct a contracting transport map from the suitably rescaled round sphere to every closed connected Riemannian surface satisfying the same positive Ricci curvature lower bound. This implies the spectral comparison in dimension two. Together with the recent counterexample in dimension three by Lin, Wang, and Xu [LWX26], this settles the spherical spectral comparison in every dimension d≥2. Using the same method, we also construct a contracting transport from the uniform probability measure on a hemisphere onto the normalized uniform measure on any geodesically convex subset of positive volume, thereby answering affirmatively the remaining case of a conjecture by Beck and Jerison [BJ21] following the work of Fathi, Fradelizi, Gozlan, and Zugmeyer [FFGZ26].

Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

Fri, 22 May 2026 00:00:00 GMT

Xu Cheng, Franciele Conrado, Neilha Pinheiro, Detang Zhou — Let (M, g, f, τ) be a complete Ricci shrinker satisfying Ric+∇²f=g/2τ and let R denote its scalar curvature. For a confined function V on M, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator −Δ+R/4+V, expressed in terms of an integral quantity involving V and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's μ-functional. We also study the drifted Schrödinger operator −Δ_f+V on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if V is affine.

Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

Thu, 21 May 2026 00:00:00 GMT

Haoyuan Sun — We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.