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

TL;DR

This paper constructs smooth embeddings of closed 3-manifolds into complex 3-space with prescribed complex tangent sets and tangent spaces, linking geometric structures with complex analysis invariants.

Contribution

It provides a method to realize any null-homologous link with a specified 2-plane field as the set of complex tangents in an embedding into 3^3, extending previous work.

Findings

Existence of embeddings with prescribed complex tangent sets and tangent spaces.

Explicit relation between Bishop invariant and the angle of complex tangents.

Extension of prior constructions to more general links and plane fields.

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.