Minimal graphs with micro-oscillations (Q1651937)

The paper is concerned with the graph of a function \(u\), \(\Sigma_u=\{(x,u(x)),\,\,x\in\mathbb{B}^n\}\), where \(\mathbb{B}^n\) is the unit ball in \(\mathbb{R}^n\) and \(u\) is the solution of the equation \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=0\text{ in }\mathbb{B}^n. \] The graph \(\Sigma_u\) is a minimal hypersurface of \(\mathbb{R}^{n+1}\). The authors prove that there are such minimal graphs in \(\mathbb{R}^{n+1}\) for which the intersection with the set \(\Pi:=\{x\in\mathbb{R}^{n+1},\,\,x_{n+1}=0,\,\,|x|<1\}\) has any prescribed geometry, up to a small deformation.