On the Bishop Invariants of Embeddings of S³ into ℂ³

2015/06/26 by Ali M. Elgindi, Elgindi, Ali M.
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV

paper · pdf · doi:10.48550/arxiv.1506.07988

Formal version published at: New York J. of Math. Vol. 20 (2014), 275-292

arxiv created 2015/06/26 · arxiv updated 2015/06/29

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 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 the different possible outcomes from the perturbation of a degenerate complex tangent.

Related

• Explicit Holomorphic Structures for embeddings of closed 3-manifolds into ℂ³
• Totally Real Perturbations and Non-Degenerate Embeddings of S³
• On the Topological Structure Of Complex Tangencies to Embeddings of S³ into ℂ³
• Topological invariants and Milnor fibre for A-finite germs ℂ²→ℂ³
• Cross-caps, triple points and a linking invariant for finitely determined germs