On the Bishop Invariants of Embeddings of $S^3$ into $\mathbb{C}^3$

TL;DR

This paper provides a practical formula for computing the Bishop invariant for embeddings of 3-spheres into complex 3-space, explores its behavior in various examples, and analyzes perturbations of degenerate tangents.

Contribution

It introduces a new, easily applicable formula for the Bishop invariant in graphical embeddings of 3-manifolds into ^3 and examines its behavior in different scenarios.

Findings

Different configurations of the Bishop invariant along complex tangents are demonstrated.

The behavior of the Bishop invariant under perturbations of degenerate tangents is analyzed.

Examples illustrate the range of possible Bishop invariant values for embeddings of S^3.

Abstract

The Bishop invariant is a powerful tool in the analysis of real submanifolds of complex space that associates to every (non-degenerate) complex tangent of the embedding a non-negative real number (or infinity). It is a biholomorphism invariant that gives information regarding the local hull of holomorphy of the manifold near the complex tangent. In this paper, we derive a readily applicable formula for the computation of the Bishop invariant for graphical embeddings of 3-manifolds into $\mathbb{C}^3$. We then exhibit some examples over $S^3$ demonstrating the different possible configurations of the Bishop invariant along complex tangents to such embeddings. We will also generate a few more results regarding the behavior of the Bishop invariant in certain situations. We end our paper by analyzing the different possible outcomes from the perturbation of a degenerate complex tangent.