Minimal hypersurfaces foliated by geodesics of 4-dimensional space forms (Q690159)

The author gives a detailed determination of minimal hypersurfaces \(M=\{ \exp_ p (t \mathbb{R}):\;p \in \Sigma,\;t \in \mathbb{R}\}\), where \(\Sigma\) is a minimal surface of constant curvature \(c\) in a 4-dimensional space form \(\widetilde M = \widetilde M^ 4(c)\) and \(\overline \xi\) is a local unit vector field on \(\Sigma\). In fact the author solves the corresponding P.D.E. when \(\Sigma\) is totally geodesic in \(M\), the minimal Clifford torus \(S^ 1 \times S^ 1 \subset S^ 3 \subset S^ 4\), and when \(\Sigma\) is the Veronese surface of \(S^ 4\). That enables the author to find all minimal hypersurfaces \(M\) of \(S^ 4\), satisfying the conditions: 1) \(M\) contains a Veronese surface \(\Sigma\) of \(S^ 4\), 2) \(M\) is foliated by great circles \(S^ 1\) of \(S^ 4\) which intersect \(\Sigma\) orthogonally, 3) the rank of the shape operator of \(M\) is equal to 3 on some open set which intersects \(\Sigma\). The paper ends with some comments about technically related work by Dajczer-Gromoll.