Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

We study rotationally symmetric translators for fully nonlinear extrinsic geometric flows driven by a curvature function, and we establish the fine asymptotics of bowl-type evolutions and, when admissible, the construction and classification of catenoidal-type solutions, together with their asymptotic behavior. Under natural structural and convexity assumptions, we also prove rigidity and uniqueness results within appropriate classes of graphical translators of such curvature flows.

💡 Research Summary

The paper investigates translating solitons—self‑similar solutions moving by translation—of a broad class of fully nonlinear extrinsic geometric flows, denoted γ‑flows, where the normal velocity is given by a symmetric, positive (or signed) curvature function γ(λ₁,…,λₙ). The authors assume that γ is α‑homogeneous (γ(cλ)=c^{α}γ(λ) for c>0), strictly increasing in each principal curvature, and defined on an open symmetric cone Γ containing the positive cone. This framework includes the mean curvature flow, Gauss‑Kronecker flow, various Hessian‑type flows, and many other curvature‑driven evolutions.

The study focuses on rotationally symmetric translators. By parametrising a rotational hypersurface through a planar generating curve (r(s),u(s)) and introducing the angle θ(s) between the tangent and the axis, the authors reduce the translator equation to a system involving the curvature κ of the generating curve and the function ˜γ(x,y)=γ(x,y,…,y). Using the implicit function theorem they write the level set ˜γ(x,y)=z as x=g(y,z), distinguishing two cases: 1‑nondegenerate (γ(0,…,0,1)>0) and 1‑degenerate (γ(0,…,0,1)=0). This implicit representation is the cornerstone for constructing barriers and performing precise asymptotic analysis.

Two families of rotational translators are treated:

1. Bowl‑type translators – entire or bounded graphs u(r) satisfying γ(λ)=⟨ν,e_{n+1}⟩. For 1‑nondegenerate γ the authors prove existence of a global convex graph and obtain a sharp asymptotic expansion u(r)=c r^{α+1}+o(r^{α+1}) as r→∞, where the constant c depends only on γ and α. When γ is 1‑degenerate the same expansion holds but the leading coefficient may vanish, leading to slower growth. The proof relies on the implicit functions g₊ and a careful barrier construction that controls the angle θ and the radial derivative.

2. Catenoidal‑type translators – non‑convex rotational hypersurfaces with a single neck of radius R>0. For signed curvature functions satisfying a mild continuity condition at the origin, the authors construct a unique translator W_R. After removing a large ball, W_R splits into two graphical branches: an upper branch with the same bowl‑type asymptotics, and a lower branch whose behavior depends on the first‑order expansion of g₋ near the origin. If g₋(0,−1)=0 and ∂_y g₋(0,−1)<0, the lower branch exhibits a linear (or logarithmic) decay, whereas in the 1‑nondegenerate case it ends in a bowl‑type cap. The construction uses a barrier method adapted to the signed setting and exploits the even symmetry of the ODE induced by the rational homogeneity of γ.

Beyond existence and classification, the paper establishes rigidity results. Theorem 6.1 shows that any smooth, strictly convex entire γ‑translator asymptotic to the bowl‑type solution must coincide with it up to vertical translation. The proof combines a comparison principle for the fully nonlinear elliptic translator equation with the precise asymptotics derived earlier. Corollary 6.8 proves that no strictly convex translator can be contained inside a round vertical cylinder S^{n−1}×ℝ when γ(0,1,…,1)>0 and α>1/3, ruling out a class of potential singular behaviors.

Methodologically, the work blends several powerful tools: (i) the implicit function theorem to recast the highly nonlinear curvature condition into a tractable scalar relation; (ii) barrier functions tailored to the geometry of the ODE, providing sub‑ and supersolutions that capture the neck and asymptotic regimes; (iii) a fully nonlinear maximum principle adapted to the γ‑translator equation, which yields uniqueness and rigidity. The authors also discuss how their framework recovers known results for the mean curvature flow (e.g., the classical bowl soliton, grim‑reaper cylinders, and translating catenoids) and extends them to a much larger family of curvature flows.

In summary, the paper delivers a comprehensive theory of rotationally symmetric translating solitons for a wide class of fully nonlinear extrinsic flows. It establishes existence, precise asymptotics, classification into bowl‑type and catenoidal‑type families, and strong rigidity/uniqueness theorems, thereby generalizing many classical results from mean curvature flow to the much richer setting of γ‑flows. The techniques introduced are likely to be applicable to further problems involving non‑convex translators, higher codimension flows, and flows with weaker structural assumptions.