AbstractJ-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$
CP
3
are related to minimal surfaces in $$S^4$$
S
4
as well as associative submanifolds in $$\Lambda ^2_-(S^4)$$
Λ
-
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 $$S^4$$
S
4
and construct moment-type maps from $$\mathbb {CP}^3$$
CP
3
to relate them to the theory of $$\mathrm {U}(1)$$
U
(
1
)
-invariant minimal surfaces on $$S^4$$
S
4
.
In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold [Formula: see text] by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in [Formula: see text] classified by the second author.
Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets