Explicit Holomorphic Structures for embeddings of closed 3-manifolds into $\mathbb{C}^3$

Abstract

Expanding on my former work along with the more recent work of Kasuya and Takase, we demonstrate that for a given link $L \subset M$ which is null-homologous in $H_1(M)$ and for any smooth oriented 2-plane field $\eta$ over $L$ there exists a smooth embedding $F:M \hookrightarrow \mathbb{C}^3$ so that the set of complex tangents to the embedding is exactly $L$ and at each $x \in L$ the holomorphic tangent space is exactly $\eta_x$. Furthermore, we demonstrate how the "analyticity" of a complex tangent, as given by the Bishop invariant, may be determined exactly from the angle formed between the holomorphic complex line and the the curve of complex tangents.