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

TL;DR

This paper introduces and classifies a new class of transverse $J$-holomorphic curves in nearly Kähler $ ext{CP}^3$, linking them to minimal surfaces in $S^4$ and associative submanifolds, and provides a Bonnet-type theorem.

Contribution

It defines transverse $J$-holomorphic curves, establishes a Bonnet-type theorem for them, and connects these curves to minimal surfaces and flat tori in $S^4$ through moment maps.

Findings

Classification of flat tori in $S^4$.

Introduction of moment-type maps from $ ext{CP}^3$.

Establishment of a Bonnet-type theorem for transverse $J$-holomorphic curves.

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 $S^4$.