Orbits of SU(2)-representations and minimal isometric immersions

Abstract

In contrast to all known examples, we show that in the case of minimal isometric immersions of \(S^3\) into \(S^N\) the smallest target dimension is almost never achieved by an \(SU(2)\)-equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of \(S^3\) into \(S^N\). The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order \(d \ge 1\) play an important role in the study of the moduli space of all minimal isometric immersions of \(S^3\) into \(S^N\). Using a new necessary and sufficient condition for immersions of isotropy order \(d \ge 1\), we derive a general existence theorem of such immersions.