On a Generalization of the Catenoid

Abstract

It is a classical result that the only surface of revolution in Euclidean space E3 which is minimal is the catenoid. Of course the surface is conformally flat, but if Mn, n ≧ 4, is a conformally flat hypersurface of Euclidean space En+1, then Mn admits a distinguished direction [2] (“tangent to the meridians“). Thus we seek to characterize conformally flat hypersurfaces of En+1 which are minimal. Specifically we prove the followingTHEOREM. Let Mn, n ≧ 4, be a conformally flat, minimal hypersurface immersed in En+1.