The Bishop invariant is a powerful tool in the analysis of real submanifolds of complex space that associates to every (nondegenerate) complex tangent of the embedding a nonnegative 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 C³. We then exhibit some examples over S³ 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 some of the different possible outcomes of the perturbation of a degenerate complex tangent.

Supported in part by IRG grant No. 4010501011418 from King Faisal Foundation