Parabolic hypersurfaces with constant mean curvature in Euclidean space (Q2078103)

In the present paper the authors consider parabolic hypersurfaces with constant mean curvature \(H\) in Euclidean space \(\mathbb R^{n+m}\) \((m,n\geq 2)\). They give classification results in the case when these hypersurfaces are invariant under a group \(\mathrm{O}(m)\times \mathrm{O}(n)\) and its Gauss-Kronecker curvature \(K\) does not change sign. Case \(H\ne 0\) leads to generalizations of circular cylinders, namely \(\mathbb R^m \times \mathbb S^{n-1}(\rho_1)\) and \(\mathbb R^n \times \mathbb S^{m-1}(\rho_2)\) and hyperspheres \(\mathbb S^{m+n-1}(\rho_3)\), where the positive reals \(\rho_1, \rho_2, \rho_3\) are determined by \(H, m\) and \(n\). In case of \(H=0\) the (non-extendable) solutions are cones \(\mathcal{C}_{m,n}\) generated by the profile curve \(y =\sqrt{\frac{n-1}{m-1}} x\) under the action of \(\mathrm{O}(m)\times \mathrm{O}(n)\).