§ 3.16 Nonrotational entire convex translatorsBD00A99565D1

In every dimension $n\geq 3$, Xu-Jia Wang constructs entire convex solutions $u:\mathbb R^ n\to \mathbb R$ of the unit-speed upward translator equation

whose level sets are not spheres or round cylinders in any orthogonal coordinate system. This is an existence construction, not an explicit formula.

• The graphs are embedded, complete, proper and diffeomorphic to $\mathbb R^ n$; the last three facts also follow directly from being smooth entire graphs. Their flows exist for all $t\in \mathbb R$ by translation.

• They are convex and break the spherical or cylindrical rotational pattern of the bowl and its Euclidean products. The corresponding statement for complete entire graphs in $\mathbb R^3$ is false: there the bowl is the only example.

• Under spatial blow-down $u_ h(x)=h^{-1}u(\sqrt h\, x)$, Wang's theorem gives, after rotations depending on $h$, a quadratic limit of the form

  \[ \frac{x_1^2+\cdots +x_ k^2}{2(k-1)},\qquad 2\leq k\leq n. \]

  The theorem does not rule out slow rotation of the axes as $h\to \infty $. This is an asymptotic limit, not a formula for $u$.

• The same paper also constructs nonrotational complete convex graphical translators over slab regions. No uniqueness theorem for the general higher-dimensional families is claimed here.

For the construction in Section 5, write $x'=(x_1,\ldots ,x_{n-1})$. Wang solves a convex Dirichlet problem on ellipsoids in the Legendre-dual variables, fixes an unequal ratio of the dual level-set widths, and passes to a limit after transforming back. When $n\geq 3$, the choice $\theta >1$ in (5.8) gives an entire solution. This construction retains rotations in $x'$ and reflection in $x_ n$. Wang's term $k$-rotational also requires independence of the other coordinates: partial rotation symmetry alone does not make these examples bowls or bowl products.

For $n=3$, the graph lies in $\mathbb R^4$, with $x_4=u(x_1,x_2,x_3)$. A horizontal level $\{ u=h\} $ can be drawn in the base $\mathbb R^3$; the construction has bounded sublevel sets by (5.13). The figure uses a convex schematic for these levels and also offers the graph slice $x_2=0$. Its rescaling control illustrates how compact levels can approach a cylindrical level locally while their caps escape. The cylinder $\{ x_1^2+x_2^2=2\} $ is the exact level $Q=1$ of the $k=2$ quadratic model. Neither this quadratic nor the finite drawing formula is a translator solution; the picture selects a permitted limit model without asserting $k=2$ for all examples or a rate of convergence.

See [XJWang11], Theorems 1.2–1.3, printed pages 1187–1188 (PDF pages 3–4), and Section 5, printed pages 1225–1234 (PDF pages 41–50), especially (5.7)–(5.13) and Section 5.3. The translator construction uses $\sigma =1$ in (1.2); the $\sigma =0$ case concerns level-set flow.