Rotationally Symmetric Translators

• Rotationally symmetric translators are hypersurfaces that translate steadily in a fixed direction while exhibiting SO(n)-invariance and strict convexity.
• They are characterized by a reduced ODE formulation based on principal curvatures, with canonical examples including the bowl soliton, catenoidal, and Grim Reaper-type solutions.
• These surfaces are fundamental in singularity formation and rigidity analyses, offering deep insights into the asymptotic behavior of solutions in various curvature flows.

A rotationally symmetric translator is a hypersurface moving by pure translation in a fixed direction under a curvature-driven flow, and which exhibits invariance under the group of rotations (SO(n)) about its axis of translation. These geometric objects arise as soliton solutions for a wide class of fully nonlinear flows—including the mean curvature flow, powers of Gauss curvature flow, and higher-order mean curvature flows—and are central in the analysis of singularity formation, blow-up limits, and rigidity phenomena in geometric PDE. The prototypical example is the "bowl soliton," an entire strictly convex graph, but the analytical framework extends to catenoidal-type, Grim Reaper-type, and more complex solutions depending on the flow.

1. General Framework and Governing Equations

Rotationally symmetric translators are defined as hypersurfaces $\Sigma \subset \mathbb{R}^{n+1}$ whose evolution under a curvature flow is given by translation in the direction $v \in \mathbb{R}^{n+1}$, with the governing equation

$$F(\kappa_1,\ldots,\kappa_n) = \langle \nu, v \rangle,$$

where $F$ is a symmetric, strictly increasing, often $\alpha$-homogeneous function of the principal curvatures $\kappa_i$, and $\nu$ is the unit normal to $\Sigma$ (Santaella, 29 Dec 2025, Rengaswami, 2021, Santaella, 2023). For purely mean curvature flow, $F=H$ (the mean curvature), while for powers of Gauss curvature, $F=K^{\alpha}$, and for higher order flows $F=H_r$.

Assuming SO(n)-symmetry about the $x_{n+1}$-axis, $\Sigma$ can be written as the graph of a radial function $u(r)$ with $r = |x|$: $\Sigma = \{ (x, u(|x|)) : x \in \mathbb{R}^n \}.$ The curvature quantities reduce to: $$\kappa_1 = \frac{u''}{(1 + u'^2)^{3/2}}, \qquad \kappa_2 = \cdots = \kappa_n = \frac{u'}{r(1+u'^2)^{1/2}}$$ yielding an ODE for $u$ parametrized by $F$ and its structure (Rengaswami, 2021, Bourni et al., 2016).

2. Existence, Uniqueness, and Rigidity Results

Under natural structural assumptions (symmetry, monotonicity, $\alpha$-homogeneity, nondegeneracy), the translator ODE admits a unique, strictly convex, entire rotationally symmetric solution (the bowl soliton) for each admissible curvature flow (Santaella, 29 Dec 2025, Bourni et al., 2016, Raji et al., 22 May 2025): $$u''(r) = \text{(nonlinear function of } r,\, u',\, u'').$$ Proofs rely on ODE techniques—fixed-point theorems in weighted spaces, shooting methods, compactness arguments, or power series expansions—that ensure the singular initial value problem at $r = 0$ can be resolved and global existence is achieved (Raji et al., 22 May 2025). The rigidity and uniqueness are established via Alexandrov-type moving plane techniques and strong maximum principles: any strictly convex $\gamma$-translator defined as a graph over a ball with a single asymptotic cylindrical end must be rotationally symmetric and coincide (up to translation) with the bowl soliton (Santaella, 2023, Santaella, 29 Dec 2025).

For flows admitting signed speed functions or lower homogeneity, non-entire solutions with catenoidal geometry, facets, or compact support can occur; these are classified via phase space analysis and ODE integrability (Santaella, 2020, Santaella, 29 Dec 2025).

3. Classification Across Curvature Flows

The theory exhibits robustness across wide classes of flows:

• Mean Curvature Flow (MCF): The rotationally symmetric translator solves

$\frac{u''}{1 + u'^2} + \frac{n-1}{r} u' = 1,$

yielding the unique bowl soliton with quadratic growth at infinity $u(r) \sim \frac{r^2}{2(n-1)}$ (Bourni et al., 2016, Raji et al., 22 May 2025).

• Fully Nonlinear Flows: For $\alpha$-homogeneous functions ($F(\kappa) = \gamma(\kappa)$ with $\gamma(c\kappa) = c^{\alpha} \gamma(\kappa)$), strict convexity, existence, and fine asymptotics of the bowl soliton are proven. Blow-up rates, existence domains, and asymptotic classifications depend on the behavior of $\gamma$ at the boundary of its domain (Santaella, 29 Dec 2025, Rengaswami, 2021).
• Gauss Curvature Flow ($K^{\alpha}$): For K-translators and powers thereof, the profile ODEs involve Gauss curvature combinations and normal height components; explicit formulas are available in special cases ($\alpha=1$, $\alpha=1/2$), and general completeness or blow-up thresholds are driven by the parameter $\alpha$ (Demirci et al., 2024, Aydin et al., 2022, Aydin et al., 2022).
• Anisotropic and Crystalline Flows: When anisotropy and mobility are rotationally symmetric, the translator equation reduces to a singular ODE similar to the bowl soliton, but can admit polyhedral/faceted surfaces or variants with linear growth (Cesaroni et al., 2021).

4. Asymptotic Behavior and Geometric Properties

Entire rotationally symmetric translators exhibit strict convexity, completeness, and prescribed growth rates at infinity reflecting the homogeneity of their velocity function:

• Quadratic or Power-Law Growth: For mean curvature and nondegenerate fully nonlinear flows, the bowl soliton grows as $u(r) \sim r^{1+\alpha}$, where $\alpha$ is the homogeneity (Rengaswami, 2021, Santaella, 29 Dec 2025, Santaella, 2023).
• Cylindrical/Catenoidal Asymptotics: For degenerate or signed curvature speeds, solutions may be confined to finite radius cylinders or present two-ended catenoidal profiles, with each end conforming to distinct asymptotic growth (Santaella, 29 Dec 2025, Lima et al., 2022).
• Facets and Conical Ends: In crystalline/anisotropic flows, profiles may exhibit piecewise linear or polyhedral facets, with flat regions arising from the degeneracy or lack of smoothness in the speed function (Cesaroni et al., 2021).

5. Classification in Product and Curved Geometries

The methodology and results generalize naturally:

• Product Spaces: In $\mathbb{R}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$, SO(n)-invariant translators for higher-order (r-th mean curvature) flows are classified: bowl-type, catenoidal-type, Grim Reaper-type, and cylinders. Explicit ODE reductions and uniqueness hold in both Euclidean and hyperbolic ambient geometries (Lima et al., 2022, Ortega et al., 2024).
• Lorentzian and Minkowski Settings: The causal character of the rotation axis (timelike, spacelike, lightlike) dictates the profile ODE and existence domains for rotational $K^{\alpha}$–translators. Spacelike or timelike solutions exhibit conical, bowl-type, or cylindrical behavior depending on parameter values (Aydin et al., 2022, Bueno et al., 2024).
• Rotating Translators in $\mathbb{H}^2 \times \mathbb{R}$: Helicoidal translators under mean curvature flow are realized as complete families generated by curves in hyperbolic planes, with each wing spiraling to infinity and vertical translation speed matching angular pitch (Lima et al., 2024).

6. Analytical Techniques and Maximum Principles

The reduction to ODEs under rotational symmetry is universal and key to existence and classification. Regularity at the axis (flat-tangent condition), monotonicity, and comparison principles guarantee both uniqueness and the absence of closed (compact) or exotic rotational translators in strict settings. Strong tangential and maximum principles exclude other entire convex solutions beyond the rotational bowl, given convexity and appropriate asymptotics (Santaella, 2023, Santaella, 2020).

Flow Type	ODE Structure	Unique Bowl?
Mean curvature ($H$)	$u''/(1+u'^2) + \frac{n-1}{r}u' = 1$	Yes
Fully nonlinear $\gamma$ (homog.)	$u'' = (1+u'^2)^{\beta+1}g(u'/r,\pm1)$	Yes, if nondeg.
$K^\alpha$ (Gauss curvature)	$(f'f''/(r(1+f'^2)^2))^\alpha = (1+f'^2)^{-1/2}$	Yes, with caveats
$Q_{n-1}$ (elementary quotient)	$s' = (1+s^2)/(-(n-1)r) s$	No (never entire)
r-mean curvature ($H_r$)	$\tau' = C\rho^{1-r}\sqrt{...} - ...$	Yes, plus catenoids
Anisotropic/Crystalline MCF	Mixed piecewise ODEs	Yes/Polyhedral

7. Broader Significance and Open Directions

Rotationally symmetric translators serve as canonical singularity models for blow-up analysis in geometric flows, underpin rigidity theorems and uniqueness classifications in fully nonlinear, mean curvature, and higher-order settings. Their explicit construction via ODE reduction allows for complete asymptotic and geometric characterization, delimiting the possible shapes and global behaviors attainable by soliton solutions. Extensions to curved, product, or Lorentzian spaces show robustness of these classification results. Absence of closed or nontrivial entire solutions (outside the rotational bowl/catenoid/Grim Reaper/cylinder classes) is guaranteed under convexity, homoegeneity, and monotonicity constraints (Santaella, 2023, Bourni et al., 2016, Lima et al., 2022).