Examples of constant mean curvature immersions of the 3-sphere into euclidean 4-space

Abstract

Mean curvature is one of the simplest and most basic of local differential geometric invariants. Therefore, closed hypersurfaces of constant mean curvature in euclidean spaces of high dimension are basic objects of fundamental importance in global differential geometry. Before the examples of this paper, the only known example was the obvious one of the round sphere. Indeed, the theorems of H. Hopf (for immersion of S(2) into E(3)) and A. D. Alexandrov (for imbedded hypersurfaces of E(n)) have gone a long way toward characterizing the round sphere as the only example of a closed hypersurface of constant mean curvature with some added assumptions. Examples of this paper seem surprising and are constructed in the framework of equivariant differential geometry.