Equivariant Lagrangian minimal \(S^3\) in \(\mathbb CP^3\) (Q2508645)

The author considers the equivariant Lagrangian minimal immersion of \(S^3\) into \(\mathbb CP^3\). The classification and an analytic expression for this immersion are presented in: Theorem A. Set \(S^3=\{(u,w)\in \mathbb C^2/ z \overline z+\omega\overline\omega\}\). Let \(\pi:\mathbb C^4\setminus \{0\}\to \mathbb CP^3:z\to\pi(z)=[z]\) be the natural projection. If \(\varphi: S^3 \to \mathbb CP^3\) is an equivariant Lagrangian minimal immersion with induced metric \(ds^2\), then, up to a holomorphic isometry \(A:\mathbb CP^3\to \mathbb CP^3\) of \(\mathbb CP^3\) and an isometry of \((S^3,ds^3)\), \(\varphi\) is one of the following immersions: (a) \(\varphi\) is totally geodesic, \(\varphi= [z, \omega,\overline z,\overline \omega]\); (b) \(\varphi=\pi\circ f=[f]\), where \(f:S^3\to \mathbb C^4\) is defined by \[ f=\left(z^3+3z\overline \omega^2,\sqrt 3\left(z^2\omega+\omega\overline\omega^2 -2z \overline z\overline\omega\right),\sqrt 3\left(z\omega^2+z\overline z^2-2 \omega\overline z\overline\omega\right),\omega^3+3\omega\overline z^2\right). \]