https://neckpinch.com/topics/rotational-symmetry/New arXiv papers on rotational symmetry & warped products, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.38723arxiv:2610.38723Wed, 28 Oct 2026 00:00:00 GMTAliya Naaz Siddiqui, Bang-Yen Chen — Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections (∇,∇^(*)) together with a Riemannian metric. In this chapter we survey the theory of statistical solitons, that is, vector fields and constants that solve the Ricci-soliton equation simultaneously for ∇ and ∇^(*), together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products R×_f N₂ and N₁×_fR for (pseudo) Riemannian manifolds N₁ and N₂, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field ξ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, η-Einstein or quasi-Einstein character of the submanifold. We further review the results of Blaga and Chen on gradient solitons and statistical structures, in particular the connections ∇^θ defined by a 1-form, volume formulas and bounds for the Ricci tensor. We also describe Chen-type inequalities for statistical warped product submanifolds, which give the extrinsic context in which such solitons live. Key steps of the proofs, remarks on the sharpness of hypotheses, worked examples, and open problems are included.Ricci solitonsAlmost & η-Ricci solitonsRotational symmetry & warped productsSasakian & contact geometry https://arxiv.org/abs/2610.37154arxiv:2610.37154Tue, 27 Oct 2026 00:00:00 GMTHongxin 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.Ricci flowSingularitiesHeat kernelRotational symmetry & warped products https://arxiv.org/abs/2610.34274arxiv:2610.34274Sun, 25 Oct 2026 00:00:00 GMTBeomseok Goh, Taehun Lee — We study ancient solutions to curve shortening flow on a class of warped product surfaces. For closed embedded ancient solutions whose curves are not null-homotopic, we prove that graphicality at every time is equivalent to a uniform length bound as t→-∞, and that these properties hold whenever the length is o(√(−t)). Under an additional positivity condition on the linearized height equation, every such solution with bounded length is a family of slices. We derive backward convergence to a limiting circle from the length bound and prove slice rigidity by an L² estimate for the height perturbation.Curve shorteningAncient solutionsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2610.32819arxiv:2610.32819Fri, 23 Oct 2026 00:00:00 GMTBen S. Lambert, Oliver C. Schnürer — We study rotationally symmetric graphical solutions to mean curvature flow that are initially close to a homothetic reference expander, in particular near infinity. Based on a computer algebra aided proof, we show that under appropriate conditions, at large times, the solutions are closer to a different expander than to the reference expander.Mean curvature flowRicci solitonsRotational symmetry & warped products https://arxiv.org/abs/2610.11209arxiv:2610.11209Thu, 08 Oct 2026 00:00:00 GMTMatheus Andrade Ribeiro de Moura Horácio — We study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.Ricci solitonsScalar curvatureRotational symmetry & warped products https://arxiv.org/abs/2609.23424arxiv:2609.23424Sun, 20 Sep 2026 00:00:00 GMTMohammadjavad Habibivostakolaei, Abbas M. Sherif, Yen-Kheng Lim — We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.Ricci solitonsAlmost & η-Ricci solitons4-manifoldsStability & uniquenessRotational symmetry & warped productsEigenvalues & spectrumPhysics https://arxiv.org/abs/2609.04675arxiv:2609.04675Fri, 04 Sep 2026 00:00:00 GMTJuan Davila, Manuel del Pino, Juncheng Wei — We study entire graphical translating solutions of the mean curvature flow, div (frac∇G√(1+|∇G|²)) =frac1√(1+|∇G|²) in ℝ^N. Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every N≥ 8 there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri–De Giorgi–Giusti (BDG) entire minimal graph in ℝ⁸ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2609.01346arxiv:2609.01346Tue, 01 Sep 2026 00:00:00 GMTHaoxuan Cheng, Junqi Lai, Guoxin Wei — For every pair of integers 3≤ k<n with k odd, we construct a compact embedded rotational torus in ℝⁿ⁺¹ whose homothetic dilations satisfy the unnormalised σₖ-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity C^(1,1/k) and Sobolev regularity W^(2,p) for every 1≤ p<k/(k-1). Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation ⟨ X,ν⟩=−σₖ, where X is the position vector and ν is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical C² rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.Other curvature flowsRotational symmetry & warped products https://arxiv.org/abs/2608.26619arxiv:2608.26619Thu, 27 Aug 2026 00:00:00 GMTKyeongho Bang, Eric Chen, Wenkui Du — We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow (M², ∂M², g(t)) on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.Ricci flowAncient solutionsSingularitiesSurfacesStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2608.11128arxiv:2608.11128Tue, 11 Aug 2026 00:00:00 GMTRoman Cherniha, John R. King — The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.Ricci flowSurfacesRotational symmetry & warped products https://arxiv.org/abs/2607.06100arxiv:2607.06100Tue, 07 Jul 2026 00:00:00 GMTJihui Wang, Matteo Lulli, Antonino Marciano — We investigate the stochastic Ricci flow of spherically symmetric perturbations of the Schwarzschild–Anti de Sitter black-hole metric. Elaborating on the Ricci-flow analysis of Headrick and Wiseman, we include a negative cosmological constant through a Ricci-target term and study how the flow is correlated with the thermodynamic heat capacity of the black hole. Numerical simulations show that, in the positive heat-capacity regime, perturbations of the angular sector of the metric relax toward the Schwarzschild–Anti de Sitter fixed point, while in the negative heat-capacity regime they grow under the deterministic Ricci flow. We then introduce a multiplicative stochastic noise and find that sufficiently strong stochasticity can suppress the growth of these perturbations, effectively stabilizing configurations that would otherwise be thermodynamically unstable. Finally, we reformulate the dynamics in terms of an entropy variable evolving on a thermodynamic free-energy landscape, and support the metric-flow results through Monte Carlo simulations and the associated Fokker–Planck equation. These results suggest that stochastic fluctuations can modify the relation between geometric stability under Ricci flow and thermodynamic stability in asymptotically Anti de Sitter black-hole spacetimes.Ricci flowStability & uniquenessRotational symmetry & warped productsPhysicsNumerics & visualization https://arxiv.org/abs/2606.22355arxiv:2606.22355Sun, 21 Jun 2026 00:00:00 GMTHaixia Chen, Seunghyeok Kim, Monica Musso — For every n ≥ 3, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round n-sphere Ssⁿ. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of Ssⁿ. These examples show that the collection of ancient Yamabe flows on Ssⁿ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on Ss², all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on Rⁿ, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner–outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on Rⁿ. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted Hölder estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.Ricci flowYamabe flowAncient solutionsSingularitiesRotational symmetry & warped products https://arxiv.org/abs/2606.20276arxiv:2606.20276Thu, 18 Jun 2026 00:00:00 GMTPeng Peng — Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type S¹× Sⁿ⁻¹ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker x²+r²=2n, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2606.19493arxiv:2606.19493Wed, 17 Jun 2026 00:00:00 GMTAndrew Lesniewski — The Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ=Φ² satisfies the linear forced heat equation \beginequation* D_tΨ=Ψ_ss-2, \endequation* while the fixed-lapse coordinate form contains the associated transport term. Since the Bures–Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \beginequation* g(t)=(1-4t)g_BH, \endequation* with scalar curvature 6/(1-4t) and extinction time T=1/4. Thus the metric remains inside the monotone cone for all t<T and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume–normalized flow, for which the Bures–Helstrom metric is a fixed point. Its linearization is the shifted round–sphere Laplacian Δ_(S³)+3, with spectrum \beginequation* σ_\ell=-(\ell-1)(\ell+3), \endequation* and spectral gap 5 after removal of the scaling mode.Ricci flowRicci solitonsScalar curvatureHarmonic maps & DeTurck trickRotational symmetry & warped productsEigenvalues & spectrum https://arxiv.org/abs/2606.03983arxiv:2606.03983Tue, 02 Jun 2026 00:00:00 GMTMiguel Pino Carmona — We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with SO(2)×ℝ symmetry, providing a cylindrical counterpart to the spherically symmetric solitons recently found by Podestà and Raffero. The family is parametrized by a flux constant k>0 and a conserved quantity 𝓒≥ 0. For 𝓒=0, the asymptotic geometry exhibits power-law decay; for 𝓒>0, the metric converges exponentially fast to a flat cylinder of finite radius.Generalized & G₂ flowsRicci solitons3-manifoldsBoundaries & non-compactLong-time existence & convergenceRotational symmetry & warped products https://arxiv.org/abs/2604.20383arxiv:2604.20383Wed, 22 Apr 2026 00:00:00 GMTGang Li — This is a continuation of the research in [16]. Let (M,g₋₁) be a closed geodesic r₀-ball in the hyperbolic space (ℍⁿ,g₋₁). Let m≠1 be a positive constant. In this paper, we show that for n≥3, starting from the metric m g₋₁ on M, with certain prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class [g_(Sⁿ⁻¹)] on the boundary ∂M, the solution g(t) to the normalized Ricci flow (1.2) which is continuous up to the boundary, exists for all t>0, and converges locally uniformly in the interior M of M to a complete hyperbolic metric as t→∞(see Theorem 1.1 for details). Under some additional conditions, we show the same conclusion holds for n=2.Ricci flowBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2604.20375arxiv:2604.20375Wed, 22 Apr 2026 00:00:00 GMTGang Li — In this paper, we show that starting from a geodesic ball overlineB_(r₀)(0) in ℍⁿ, for n≥3, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class [g_(Sⁿ⁻¹)] on the boundary, the solution g(t) to the normalized Ricci flow (1.2) which is continuous up to the boundary, exists for all t>0 and converges locally uniformly in B_(r₀)(0) to a complete hyperbolic metric as t→∞(see Theorem 1.2 for details). Moreover, the sectional curvature of g(t) maintains less than −1 for t>0. For dimension 2, to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as t→∞.Ricci flowBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2604.18974arxiv:2604.18974Tue, 21 Apr 2026 00:00:00 GMTJorge Herbert Soares de Lira, Rafael Rocha de Farias — In this paper we prove existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows that are invariant by translations in warped product manifolds ℙ×_χℝ. Here, ℙ is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and χ is a radial function defined in ℙ. In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor ℝ in the warped product. This setting encompasses the cases of translating solitons in ℝⁿ⁺¹, ℍⁿ × ℝ and ℍⁿ⁺¹ studied in recent papers. In particular we prove the existence of families of bowl-type and catenoid-type translating solitons under mild assumptions about the curvature of the warped product. We also describe the asymptotic behavior for those solitons in terms of the geometry at infinity of ℙ. Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.Mean curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products https://arxiv.org/abs/2604.14366arxiv:2604.14366Wed, 15 Apr 2026 00:00:00 GMTJosé N. V. Gomes, Willian I. Tokura, Hikaru Yamamoto — We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we develop a method for constructing explicit warped product solutions to the Ricci-Bourguignon flow and present examples that illustrate the scope and geometric relevance of our resultsRicci flowBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2604.13734arxiv:2604.13734Wed, 15 Apr 2026 00:00:00 GMTSara Albert-Niclòs, Esther Cabezas-Rivas — We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle.Other curvature flowsVolume-preserving & constrained flowsLong-time existence & convergenceRotational symmetry & warped products https://arxiv.org/abs/2602.02424arxiv:2602.02424Mon, 02 Feb 2026 00:00:00 GMTRonaldo F. de Lima, Álvaro K. Ramos — We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension n+1≥ 3. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2512.24524arxiv:2512.24524Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel leading mode condition, together with a new "induction over thresholds" argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingRotational symmetry & warped products https://arxiv.org/abs/2512.23623arxiv:2512.23623Mon, 29 Dec 2025 00:00:00 GMTJosé Torres Santaella — We develop a rotational theory for translating solitons of fully nonlinear extrinsic curvature flows in Euclidean space. Furthermore, we obtain fine asymptotic expansions for bowl-type translators in nondegenerate and degenerate regimes. On the other hand, we also introduce a signed-neck framework which yields the construction and classification of catenoidal-type translators, distinguishing complete embedded families from maximal admissible pieces according to the selected signed branch. As applications, we prove uniqueness results for strictly convex entire graphical translators with prescribed bowl-type asymptotics and obtain catenoidal-barrier nonexistence results for bounded graphical translators.Other curvature flowsSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2512.06027arxiv:2512.06027Thu, 04 Dec 2025 00:00:00 GMTMasoumeh Khalili, Ghodratallah Fasihi-Ramandi, Shahroud Azami — This paper introduce the idea of second Ricci solitons. A second Ricci soliton is nothing but a steady hyperbolic Ricci soliton. We study the geometry of closed and compact second Ricci soliton manifolds. Immersed submanifolds as second solitons also will be investigated. Finally, we investigate this structure on warped product manifolds.Ricci solitonsRotational symmetry & warped products https://arxiv.org/abs/2511.12144arxiv:2511.12144Sat, 15 Nov 2025 00:00:00 GMTAtreyee Bhattacharya, Sayoojya Prakash — Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.Ricci flowRicci solitonsEinstein metricsAlmost & η-Ricci solitonsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2510.13057arxiv:2510.13057Wed, 15 Oct 2025 00:00:00 GMTValter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos — In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.Ricci solitonsAlmost & η-Ricci solitons4-manifoldsRotational symmetry & warped productsEigenvalues & spectrum https://arxiv.org/abs/2509.25132arxiv:2509.25132Mon, 29 Sep 2025 00:00:00 GMTAntonio Airton Freitas Filho — We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and n-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as warped products. A modified Ricci soliton appears as part of a special solution of the modified Ricci-harmonic flow, which result a new characterization of n-quasi-Einstein metrics. We also study the modified Ricci almost solitons. In the spirit of the Lichnerowicz and Obata first eigenvalue theorems, we prove that in the class of compact Riemannian manifolds with constant scalar curvature the standard sphere with a structure of gradient modified Ricci almost soliton is rigid under some specific geometric conditions. Moreover, we display an example of modified Ricci-harmonic soliton.Ricci flowRicci solitonsEinstein metricsAlmost & η-Ricci solitonsScalar curvatureRotational symmetry & warped productsEigenvalues & spectrum https://arxiv.org/abs/2507.19428arxiv:2507.19428Fri, 25 Jul 2025 00:00:00 GMTGuanhua Shao, Jiahua Zou — For a general class of cones in ℝ³, we construct self-expanders of positive genus asymptotic to these cones. As a result, we use these self-expanders to construct a mean curvature flow with genus strictly decreasing but not to zero at the first singular time. We also construct a sequence of self-expanders with unbounded genus which are asymptotic to the same rotationally symmetric cone. Moreover, we characterize the asymptotic behavior of the sequence.Mean curvature flowSelf-shrinkers & translatorsWeak & singular flowsRotational symmetry & warped products https://arxiv.org/abs/2505.23157arxiv:2505.23157Thu, 29 May 2025 00:00:00 GMTMing Hsiao — We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on ℝⁿ⁺¹ that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.Ricci flowSingularities(Non)collapsingWeak & singular flowsHarmonic maps & DeTurck trickRotational symmetry & warped products https://arxiv.org/abs/2505.16688arxiv:2505.16688Thu, 22 May 2025 00:00:00 GMTHakar Raji, Oliver C. Schnürer — There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.Mean curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products