Rotational symmetry & warped products

Statistical manifolds, introduced by Amari in information geometry, carry a pair of dual torsion-free connections $(\nabla ,{\nabla }^{\ast })$ 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 $\nabla$ and ${\nabla }^{\ast }$, together with the statistical analogue of almost quasi-Yamabe solitons. We discuss two principal settings: (i) statistical warped products $\mathbb{R}{\times }_f{N}_2$ and $N_1{\times }_f\mathbb{R}$ for (pseudo) Riemannian manifolds $N_1$ and $N_2$, where the soliton equation constrains the warping function; and (ii) statistical submanifolds of Kenmotsu statistical manifolds, where the position of the Reeb field $\xi$ relative to the submanifold and the presence of concircular or concurrent vector fields determine the Einstein, $\eta$-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 ${\nabla }^{\theta }$ 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.

For a rotationally symmetric Ricci flow $g(t)=d{s}^2+f(s,t{)}^2{g}_{{\mathbb{S}}^{n-1}}$, $n\ge 3$, we study the scale-invariant curvature anisotropy $A=1-f_s^2+f{f}_{ss}$ introduced by Angenent–Knopf. At the critical power $p_{\ast }=(n-1)/(n-2)$, with $\Phi =sgn(A)|A{|}^{p_{\ast }/2}$, we prove the distributional identity $$({\partial }_t-\Delta )|A{|}^{p_{\ast }}=-\frac{4}{n-1}{\left|{\Phi }_s+(n-1)\frac{f_s}{f}\Phi \right|}^2.$$ Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on ${\mathbb{S}}^n$: 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.

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 $t\to -\infty$, and that these properties hold whenever the length is $o(\sqrt{-t})$. 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 $L^2$ estimate for the height perturbation.

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.

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.

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.

We study entire graphical translating solutions of the mean curvature flow, $$\mathrm{div}\left(\frac{\nabla G}{\sqrt{1+|\nabla G{|}^2}}\right)=\frac{1}{\sqrt{1+|\nabla G{|}^2}}\text{in }{\mathbb{R}}^N.$$ 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 $N\ge 8$ 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 ${\mathbb{R}}^8$ 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.

For every pair of integers $3\le k<n$ with $k$ odd, we construct a compact embedded rotational torus in ${\mathbb{R}}^{n+1}$ whose homothetic dilations satisfy the unnormalised ${\sigma }_k$-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 $1\le p<k/(k-1)$. 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 $⟨X,\nu ⟩=-{\sigma }_k$, where $X$ is the position vector and $\nu$ 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 $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow $(M^2,\partial M^2,g(t))$ 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.

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.

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.

For every $n\ge 3$, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round $n$-sphere ${\text{Ss}}^n$. 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 ${\text{Ss}}^n$. These examples show that the collection of ancient Yamabe flows on ${\text{Ss}}^n$ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on ${\text{Ss}}^2$, all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on ${\mathbb{R}}^n$, 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 ${\mathbb{R}}^n$. 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.

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 ${\mathbb{S}}^1\times {\mathbb{S}}^{n-1}$ 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 $x^2+r^2=2n$, 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.

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 $\Psi ={\Phi }^2$ 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 ${\Delta }_{{\mathbb{S}}^3}+3$, with spectrum \beginequation* σ_\ell=-(\ell-1)(\ell+3), \endequation* and spectral gap $5$ after removal of the scaling mode.

We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with $\mathrm{SO}(2)\times \mathbb{R}$ 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 $\mathcal{C}\ge 0$. For $\mathcal{C}=0$, the asymptotic geometry exhibits power-law decay; for $\mathcal{C}>0$, the metric converges exponentially fast to a flat cylinder of finite radius.

This is a continuation of the research in [16]. Let $(\overline{M},g_{-1})$ be a closed geodesic $r_0$-ball in the hyperbolic space $({\mathbb{H}}^n,g_{-1})$. Let $m\ne 1$ be a positive constant. In this paper, we show that for $n\ge 3$, starting from the metric $m{g}_{-1}$ on $\overline{M}$, with certain prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{{\mathbb{S}}^{n-1}}]$ on the boundary $\partial M$, 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 $\overline{M}$ to a complete hyperbolic metric as $t\to \infty$(see Theorem 1.1 for details). Under some additional conditions, we show the same conclusion holds for $n=2$.

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in ${\mathbb{H}}^n$, for $n\ge 3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{{\mathbb{S}}^{n-1}}]$ 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 $B_{r_0}(0)$ to a complete hyperbolic metric as $t\to \infty$(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 $t\to \infty$.

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 $\mathbb{P}{\times }_{\chi }\mathbb{R}$. Here, $\mathbb{P}$ is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and $\chi$ is a radial function defined in $\mathbb{P}$. In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor $\mathbb{R}$ in the warped product. This setting encompasses the cases of translating solitons in ${\mathbb{R}}^{n+1}$, ${\mathbb{H}}^n\times \mathbb{R}$ and ${\mathbb{H}}^{n+1}$ 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 $\mathbb{P}$. Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.

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 results

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.