On a minimal hypersurface in \(\mathbb{R}^4 \) (Q1985350)

For a nonparametric surface, with prescribed mean curvature, in a vertical cylinder in three-dimensional (3d) Euclidean space, the existence and behavior of radial limits at a nonconvex corner are well understood. However, the same problem in higher dimensions is largely unknown. To shed light on this quest, the authors consider the specific example of a nonparametric minimal hypersurface \((x,y,z,f(x,y,z))\) in 4d and investigate its behavior at a nonconvex conical point of the boundary of its domain. They consider \(f\) over a 3d domain obtained by rotating a symmetric nonconvex quadrilateral about the \(x\)-axis while the function prescribed on the boundary is piecewise linear. The rotational symmetry of the domain allows the authors to employ results and techniques from prescribed mean curvature surfaces in 3d whose boundary has a nonconvex corner. The two main theorems establish, in particular, the existence of the limit of \(f\) at the nonconvex corner by respectively taking paths along the symmetry axis and along the boundary of the 3d domain. As a corollary of the first theorem, it follows that the Dirichlet problem for the minimal surface equation in 4d may fail to have classical solutions over nonconvex domains, as also happens in 3d. Finally, by comparing to what is known for nonparametric minimal surfaces, the authors list some open questions, such as deciding on the equality or not of the two limits along the paths considered in the main theorems and establishing the existence of radial limits at the nonconvex corner along distinct paths.