§ 3.2 The rotational bowl translator (Clutterbuck–Schnürer–Schulze, 2005; Brendle–Choi, 2017; Hoffman–Ilmanen–Martín–White, 2019; Sun–Zhihan Wang, 2023)38B2DD284E09

For each $n\geq 2$, the bowl is the entire graph $z=f(r)$, $r=|x|$ in $\mathbb R^ n$, moving by $z\mapsto z+t$. Its radial profile is specified by the regular ODE

This is an ODE characterization, not an elementary closed formula.

• The bowl is complete, properly embedded, strictly convex, rotationally invariant, and diffeomorphic to $\mathbb R^ n$. For $n\geq 2$ it has one topological end and gives an eternal flow.

• Up to a vertical additive constant, its height has asymptotic expansion

  \[ f(r)=\frac{r^2}{2(n-1)}-\log r+C+O(r^{-1})\quad (r\to \infty ). \]

  A paraboloid is only a leading approximation, not the exact translator.

• It is unique among entire rotationally invariant translating graphs, up to translation. In $\mathbb R^3$ it is the only complete entire translating graph, up to the natural rigid motions and normalization.

• In $\mathbb R^3$, Brendle–Choi's theorem characterizes its flow among noncompact, strictly convex, noncollapsed ancient flows, up to scaling and ambient isometries. These are the hypotheses of the cited classification.

• Graphical weighted stability holds. The stronger dynamical stability theorem of Clutterbuck–Schnürer–Schulze concerns continuous graphical initial data whose height difference from the bowl tends to zero at spatial infinity; the resulting height difference tends uniformly to zero as time tends to infinity.

The supplied notes [HIMWNotes19], Section 4 and Theorem 4.1, PDF pages 10–11, describe the construction and classification. The original ODE and asymptotic analysis is [CSS07], Section 2, Lemmas 2.1–2.2; its Theorem 1.1 gives the stated dynamical stability. The ancient classification is [BrendleChoi19], Theorem 1.1, printed page 36 (PDF page 2). The entropy is $\lambda (\mathrm{Bowl}^ n)=\lambda (S^{n-1}(\sqrt{2(n-1)}))$, in particular $\sqrt{2\pi /e}$ in $\mathbb R^3$. This follows from [SunWang2025], Example 3.6 and Corollary 2.4 with Lemma 2.5, PDF pp. 8–11; it is not an identity for the fixed-center Gaussian area of the bowl.