Minimal immersions of some circle bundles over holomorphic curves in complex quadratics into spheres (Q5937562)

A point \(q\) of the complex quadric \(Q^{n-1}\subset\mathbb{C}\mathbb{P}^n\) is represented by \(q = {\mathbf x }+ i {\mathbf y}\), where \({\mathbf{ x,y}} \in \mathbb{R}^{n+1}\) is an orthonormal pair of vectors determined up to a rotation in the circle group \(\text{ SO}(2)\). This defines a circle bundle \(\pi: P\to Q\), where the fibre of \(P\) over \(q\) is the set of such oriented orthonormal pairs. A map \(f:\Sigma\to Q\) of a Riemann surface \(\Sigma\) induces a map \(F: M^3=f^*P\to S^n\subset\mathbb{R}^{n+1}\), given by \(F({\mathbf x }+ i {\mathbf y},\theta)=\cos\theta {\mathbf x}+ \sin\theta {\mathbf y}\). The author finds conditions under which this induced map is a minimal immersion. The main results of this paper are the following: Theorem 1. Let \(f: \Sigma\to Q\) be a holomorphic immersion. If the induced map \(F: f^*P \to S^n\) is an immersion, then \(F\) is minimal if and only if either \(F\) is totally geodesic or \(f\) is first-order isotropic., i.e., the line through \(f(m)\) in \(\mathbb{C}\mathbb{P}^n\) tangent to \(f(\Sigma)\) lies in \(Q\) for each \(m\in \Sigma\). Theorem 2. If \(f: \Sigma\to Q^{2n-1}\) is the directrix curve of a full pseudo-holomorphic map \(\chi:\Sigma\to S^{2n}\), then the induced map \(F: f^*P\to S^{2n}\) is a full, minimal immersion.