HIGHER DIMENSIONAL MINIMAL SUBMANIFOLDS GENERALIZING THE CATENOID AND HELICOID

Abstract

For each $k$-dimensional complete minimal submanifold $M$ of $\boldsymbol{S}^n$ we construct a $(k+1)$-dimensional complete minimal immersion of $M\times \boldsymbol{R}$ into $\boldsymbol{R}^{n+2}$ and $(k+1)$-dimensional minimal immersions of $M\times \boldsymbol{R}$ into $\boldsymbol{R}^{2n+3}, \boldsymbol{H}^{2n+3}$ and $\boldsymbol{S}^{2n+3}$. Also from the Clifford torus $M=\boldsymbol{S}^k(1/\sqrt{2})\times \boldsymbol{S}^k(1/\sqrt{2})$ we construct a $(2k+2)$-dimensional complete minimal helicoid in $\boldsymbol{R}^{2k+3}$.