Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

Mon, 26 Oct 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Ruijia Zhang, Rong Zhou — We study quermassintegral-preserving curvature flows in ℍⁿ⁺¹, n≥3, driven by the curvature quotient (Eₖ/E_ℓ)^(1/(k-ℓ)). For 1≤ℓ

Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow

Wed, 14 Oct 2026 00:00:00 GMT

Esther Cabezas-Rivas, Carlo Sinestrari — We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

Ricci flow on manifolds with positive isotropic curvature in all dimensions

Thu, 01 Oct 2026 00:00:00 GMT

Simon Brendle, Raphael Tsiamis — We extend the classification of manifolds with positive isotropic curvature to all dimensions n ≥ 5. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact κ-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

Wed, 16 Sep 2026 00:00:00 GMT

Yuyue Chen, Shiying Wu — We establish the eventual monotonicity of the Willmore deficit ∫_ΣH² dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ³, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

Gradient Ricci Solitons on PTR-Submanifolds of Complex Space Forms

Sat, 05 Sep 2026 00:00:00 GMT

Jae Won Lee — We study gradient Ricci solitons on partially totally real (PTR) submanifolds of complex space forms. In nonzero ambient holomorphic sectional curvature, the Gauss–soliton equation reconstructs the square P² of the tangential Kähler operator from the Hessian endomorphism and an extrinsic Ricci operator. Consequently, the PTR tangency spectrum is encoded by intrinsic soliton data together with the second fundamental form. For compact minimal PTR solitons with positive ambient curvature we obtain λ≤ c(n-1)/4, with a strict version under a relative-nullity condition. We also prove compact triviality criteria, an exact trace-free anisotropy identity, a parallel-data rigidity result, and a nonexistence theorem for compact real hypersurfaces. A flat multi-level Gaussian model and a nonflat Segre CR-product provide explicit consistency checks and identify the limiting cases of the reconstruction formulas.

Pinching cones for positive isotropic curvature in dimensions seven and eight

Thu, 27 Aug 2026 00:00:00 GMT

Jae Ho Cho — We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE d/dtR=Q(R) in dimensions n=7,8, thereby extending the pinching estimate established by Brendle for n≥ 12 and by Chen for 9≤ n≤ 11. In dimension n=8, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at n=8 by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension n=7, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension n=8. The n=8 pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible (n-1)-dimensional space forms, extending a theorem of Brendle from n≥ 12. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact κ-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from n=4 or n≥ 12.

Small Normal Curvature and Three-Manifold Topology

Tue, 18 Aug 2026 00:00:00 GMT

Tsz-Kiu Aaron Chow, Jingbo Wan — For m=2,3, we prove that every smooth immersion F:ℝℙ^mlooparrowrightoverline{B^(,N)(1) satisfies κ(F)²≥ 2m/(m+1), with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with κ(F)≤√(3/2) is diffeomorphic to S³, ℝℙ³, or S²× S¹. All three possibilities occur, while κ(F)<√(3/2) forces X≅ S³. These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality (min_Y R_g)sys(g)²<6π² for every spherical three-space form Y with |π₁(Y)|>2. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for ℝℙ³ of Bray–Brendle–Eichmair–Neves.

Non-compact Ricci Solitons of Finite Volume with Potential Field of Constant Length

Fri, 24 Jul 2026 00:00:00 GMT

Hemangi Madhusudan Shah, Sharief Deshmukh, Mohammad Aqib — We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise 3-dimensional Ricci solitons of constant length and of finite volume on almost α-cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension 3 admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow

Fri, 03 Jul 2026 00:00:00 GMT

Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum, Ziyi Zhao — We obtain the unique asymptotics of SO(k)× SO(n-k+1)-invariant, compact, simply-connected, factorwisely non-self-similar n-dimensional κ-solutions of the Ricci flow (Mⁿ, g(t)), where n≥ 4 and 2≤ k≤ n-2. More precisely, these κ-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere Sⁿ, having a positive curvature operator metric g(t) and a cylindrical tangent flow at -∞, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric g(t) of every SO(k)× SO(n-k+1)-invariant ancient oval is represented in the form g(t)=dz⊗ dz + F²(z,t) g_(S^(k-1)) + G²(z,t)g_(S^(n-k)) (up to flipping k-1 and n-k). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function G(z, t), and prove that the uniqueness of G(z, t) implies the uniqueness of F(z, t). In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional κ-solutions of the Ricci flow.

Hyperbolic solitons on trans-Sasakian space forms and its submanifolds

Mon, 22 Jun 2026 00:00:00 GMT

Bidhan Mondal, Sibsankar Panda, Nirabhra Basu — Kong and Liu introduced the concept of hyperbolic Ricci flow in 2007 and used it to study the wave character of metrics. After that, many mathematicians have used this new geometric flow to study the evolution of manifolds and their structures. Ricci solitons and hyperbolic Ricci solitons are self-similar solitons of the Ricci flow and hyperbolic Ricci flow respectively. In this paper, we introduce the concept of hyperbolic *-Ricci solitons and hyperbolic Ricci-Yamabe solitons on a trans-Sasakian space forms and characterized the nature of some hyperbolic solitons. Additionally, we deduce the Ricci tensors of submanifolds of trans-Sasakian space forms and conformal trans-Sasakian space form and found the nature of solitons on submanifolds. Finally, we have included an example which will justify our result.

Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

Fri, 19 Jun 2026 00:00:00 GMT

Jason DeVito, David González-Álvaro, Masoumeh Zarei — We prove that the Ricci flow preserves positive sectional curvature on homogeneous spheres and complex projective spaces. In conjunction with prior results, this completes the classification of which homogeneous spaces have positively curved metrics flowing outside the set of positively curved metrics and which do not.

κ-solutions with the round cylinder as an asymptotic shrinker

Thu, 14 May 2026 00:00:00 GMT

Aprameya Girish Hebbar — We show that κ-solutions to the Ricci flow in dimensions n≥ 4 whose asymptotic shrinking Ricci soliton is the round cylinder Sⁿ⁻¹×ℝ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.

Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

Tue, 28 Apr 2026 00:00:00 GMT

Tianci Luo, Yong Wei, Rong Zhou — We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space ℍⁿ⁺¹. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain conditions. This class of flows includes the kth mean curvature flow as a special case. We show that if the initial hypersurface has positive sectional curvature, then this property is preserved along the flow, and the evolving hypersurface contracts to a round point in finite time.

PIC1 pinched manifolds are flat or compact

Mon, 23 Mar 2026 00:00:00 GMT

Alix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping — Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

Curvature pinching of asymptotically conical gradient expanding Ricci solitons

Mon, 06 Oct 2025 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry

Mon, 29 Sep 2025 00:00:00 GMT

Lucas Lavoyer, Luke T. Peachey — We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with O(p) × O(q) symmetry in dimension p+q for any pair of integers p,q ≥ 2 and discuss their asymptotic geometry.

Curvature tensors and hyperbolic solitons on Lorentzian trans-Sasakian space form

Mon, 22 Sep 2025 00:00:00 GMT

Bidhan Mondal, Nirabhra Basu, Arindam Bhattacharyya — Lorantzian trans-Sasakian space form is a special type of space form in which the nature of even and odd dimensional space form both exist. Various curvature tensors with respect to Levi-Civita connection on the space form are derived in this paper. We have shown that if an odd-dimensional Lorentzian trans-Sasakian space form admits a hyperbolic Ricci soliton and hyperbolic conformal Ricci soliton then they will be η-Einstein. We also obtained the conditions for the solitons to be expanding, steady or shrinking. Finally, an example has been constructed which justifies the results obtained.

On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Tue, 16 Sep 2025 00:00:00 GMT

Zhengnan Chen — For all dimensions n≥5, let (M,g,f) be a n-dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that ∇²f is 2-nonnegative and the curvature tensor is WPIC1 at some point x∈ M. Then (M,g) must be a quotient of either Sⁿ or Sⁿ⁻¹×ℝ. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions n≥9. Namely that a complete ancient solution to the Ricci flow of dimension n≥9 with uniformly PIC must be weakly PIC2.

Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

Fri, 29 Aug 2025 00:00:00 GMT

Ruojing Jiang, Franco Vargas Pallete — This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic 3-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by −1 and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by −6 and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly C⁰-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.

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.

Almost Ricci Solitons on Class 𝓐 Hypersurfaces of Product Spaces

Sat, 16 Aug 2025 00:00:00 GMT

Ahmet Umut Çoraplı, Burcu Bektaş Demirci, Nurettin Cenk Turgay — In this paper, we study hypersurfaces in the product spaces ℚ_ε³ × ℝ for which the tangential component T of the vector field ∂/∂t is a principal direction, where ℚ_ε³ denotes the three-dimensional non-flat Riemannian space form with sectional curvature ε= ± 1, and ∂/∂t is the unit vector field tangent to the ℝ-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field T. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Sat, 28 Jun 2025 00:00:00 GMT

Jie Wu — In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space ℍⁿ and for convex hypersurfaces in the sphere Sⁿ. The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.

Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

Sun, 20 Apr 2025 00:00:00 GMT

Yuxing Deng — In this paper, we study gradient Ricci soitons on smooth orbifolds. We prove that the scalar curvature of a complete shrinking or steady gradient Ricci soliton on an orbifold is nonnegative. We also show that a complete κ-noncollapsed steady gradient Ricci soliton on a Riemannian orbifold with positive curvature operator, compact singularity and linear curvature decay must be a finite quotient of the Bryant soliton. Finally, we show that a complete steady gradient Ricci soliton on a Riemannian orbifold with positive sectional curvature must be a finite quotient of the Bryant soliton if it is asymptotically quotient cylindrical.

Orbifold singularity formation along ancient and immortal Ricci flows

Mon, 21 Oct 2024 00:00:00 GMT

Alix Deruelle, Tristan Ozuch — In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension 4 with Einstein orbifolds as tangent flows at infinity. For instance, for any k∈ℕ₀, we obtain continuous families of non-isometric ancient Ricci flows on \#k(S²× S²) depending on a number of parameters growing linearly in k, and a family of half-PIC ancient Ricci flows on ℂP}²\#ℂP}². The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics.

Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

Thu, 19 Sep 2024 00:00:00 GMT

Yaoting Gui, Yuqiao Li, Jun Sun — We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau. And also a uniqueness result follows

Expanding Ricci solitons coming out of weakly PIC1 metric cones

Fri, 19 Apr 2024 00:00:00 GMT

Pak-Yeung Chan, Man-Chun Lee, Luke T. Peachey — Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.

Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

Thu, 28 Mar 2024 00:00:00 GMT

Huai-Dong Cao, Junming Xie — This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by |Rm|\leq R. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kaehler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.

On the inverse mean curvature flow by parallel hypersurfaces in space forms

Thu, 21 Mar 2024 00:00:00 GMT

Alancoc dos Santos Alencar, Keti Tenenblat — We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by the distance function of the parallel hypersurfaces. The solutions to the flow are obtained explicitly when the distinct principal curvatures have the same multiplicity. This is an additional assumption only for isoparametric hypersurfaces of the hyperbolic space or of the sphere with two or four distinct principal curvatures. The boundaries of the maximal interval of definition, when finite, are determined in terms of the number g of distinct principal curvatures, their multiplicities m and the mean curvature H of the initial hypersurface. We describe the collapsing submanifolds of the flow at the boundaries of the interval. In particular, we show in the Euclidean space the solutions are eternal, while in the hyperbolic space there are eternal and immortal solutions. Starting with a connected isoparametric submanifold of the sphere, we show that the flow is an ancient solution, that collapses into a minimal hypersurface whose square length of its second fundamental form and its scalar curvature are constants given in terms of g and n. The minimal hypersurface is totally geodesic when g=1, it is a Clifford minimal hypersurface of the sphere when g=2 and it is a Cartan type minimal submanifold when g∈\3,4,6\.

Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

Wed, 20 Mar 2024 00:00:00 GMT

David González-Álvaro, Masoumeh Zarei — We prove that there exist SU₃-invariant metrics on Aloff-Wallach spaces W⁷_(k₁, k₂), as well as SU₅-invariant metrics on the Berger space B¹³, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

Neckpinch — Sphere theorems

O(3)-symmetry of four-dimensional steady Ricci solitons