Minimal immersions of 3-dimensional sphere into spheres
Let \(S^ n_ c\) be the n-dimensional sphere with constant curvature c and let \(f: S^ n_ c\to S^ N_ 1\) be an isometric minimal immersion. It is known then that \(c=k(d)\), for some integer d, and that for \(n\geq 3\) and \(d\geq 4\) such minimal immersions are not uniquely determined [the reviewer and \textit{N. R. Wallach}, Ann. Math., II. Ser. 93, 43-62 (1971; Zbl 0218.53069)]. In the case \(n=3\), \(S^ 3\) has a Lie group structure, namely, \(S^ 3=SU(2)\). By studying the orbits of the linear representations of SU(2), the author obtains interesting examples of minimal isometric immersions of \(S^ 3_{3/d(d+2)}\) into \(S_ 1^{2d+1}\), for \(d\geq 4\), and of \(S^ 3_{3/d(d+2)}\) into \(S^ d_ 1\), for \(d\geq 6\). In particular, this provides an example [cf. also \textit{N. Ejiri}, Proc. Am. Math. Soc. 83, 759-763 (1981; Zbl 0474.53051)] of an isometric minimal immersion \(f: S^ 3_{1/16}\to S^ 6_ 1\) with the additional property that the image \(f(S^ 3_{1/16})\) is neither an embedded sphere nor an embedded projective space in \(S^ 6_ 1\).
scientific article; zbMATH DE number 3890300