Minimal hypersurfaces of $\textbf {R}^ {2m}$ invariant by $\textrm {SO}(m)\times \textrm {SO}(m)$

Abstract:

Let $G = {\text {SO}}(m) \times {\text {SO}}(m)$ act in the standard way on ${{\mathbf {R}}^m} \times {{\mathbf {R}}^m}$. We describe all complete minimal hypersurfaces of ${{\mathbf {R}}^m}\backslash \{ 0\}$ which are invariant under $G$ for $m = 2$, $3$ . We also show that the unique minimal hypersurface of ${{\mathbf {R}}^{2m}}$ which is invariant under $G$ and passes through the origin of ${{\mathbf {R}}^{2m}}$ is the minimal quadratic cone.