The Height of Choe's Higher-Dimensional Enneper Surface

We prove that Choe’s higher-dimensional Enneper hypersurfaces in ${\mathbb{R}}^{n+1}$, $3\le n\le 6$, cannot be confined between two parallel hyperplanes, disproving his slab confinement conjecture. We further strengthen this conclusion by determining their precise height growth exponent. If $H(R)$ denotes the maximal absolute height above the asymptotic hyperplane within the horizontal ball of radius $R$, then $H(R)=R^{{\alpha }_1+o(1)}$ as $R\to \mathrm{\infty }$, where ${\alpha }_1=\frac{\sqrt{(n-2{)}^2+4{\lambda }_1(\mathrm{\Omega })}-(n-2)}{2}$, $\mathrm{\Omega }=S^{n-1}\setminus [0,\mathrm{\infty }{)}^n$, and ${\lambda }_1(\mathrm{\Omega })$ is the first Dirichlet eigenvalue of the spherical Laplacian. In particular, $0<{\alpha }_1<1$, so the height is unbounded but grows sublinearly. The proof combines the blow-down behavior of a fundamental piece of Choe’s hypersurface with the rigidity of positive harmonic functions on cones. In dimension two, the formula gives $2\mathrm{/}3$, recovering the classical Enneper height-growth exponent.