On the Simon conjecture for minimal immersions with \(S^1\)-symmetry (Q1100747)

Let \(\psi: S^ 2\to S^ n(1)\) be a minimal immersion of the 2-sphere \(S^ 2\) into the \(n\)-sphere \(S^ n(1)\) of constant sectional curvature 1. It has been conjectured by U. Simon that if the Gaussian curvature \(K\) of the metric \(g\) induced on \(S^ 2\) by \(\psi\) is pinched between \(2/s(s+1)\) and \(2/(s+1)(s+2)\) for some positive integer \(s\), then \(K\) is constant. In this paper the conjecture is proved under the additional assumption that the isometry group of \((S^ 2,g)\) contains a non-trivial 1-parameter subgroup \(G\). The result is proved using projective and metric invariants of a holomorphic curve (the directrix curve) associated to \(\psi\). The crucial step is the proof of a formula involving these invariants and the values of \(K\) at the two fixed points of \(G\).