Transverse $J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$

Abstract

$J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$ are related to minimal surfaces in $S^4$ as well as associative submanifolds in $\Lambda^2_-(S^4)$. We introduce the class of transverse $J$-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $S^4$ and construct moment-type maps from $\mathbb{CP}^3$ to relate them to the theory of $\mathrm{U}(1)$-invariant minimal surfaces on $S^4$.