Area maximizing hypersurfaces in Minkowski space having an isolated singularity
We consider area maximizing hypersurfaces in Minkowski space \(L^{n+1}\), i.e. spacelike hypersurfaces given by \(u\in C^{0,1}({\bar \Omega})\), \(\Omega \subset {\mathbb{R}}^ n\), which solve the variational problem \[ \int_{\Omega}\sqrt{1-| Dv|^ 2}\to \max \quad in\quad K=\{v\in C^{0,1}({\bar \Omega})| | Dv| \leq 1\quad a.e.\quad in\quad \Omega,\quad v|_{\partial \Omega}=u\}. \] The corresponding Euler- Lagrange-equation is given by \[ -\sum^{n}_{i=1}D_ i(D_ iu/\sqrt{1-| Du|^ 2})=0,\quad | Du| <1\quad in\quad \Omega; \] solutions of this equation are called maximal hypersurfaces. We prove that area maximizing hypersurfaces are asymptotic to an upper resp. lower light cone near an isolated singularity. Furthermore we show that entire maximal hypersurfaces having an isolated singularity are radially symmetric up to Lorentz transformations.
quasilinear elliptic second order equations
scientific article; zbMATH DE number 3956038