Holomorphic tubes in Cauchy–Riemann manifolds

Abstract

We consider pseudoholomorphic curves in loop spaces of Cauchy–Riemann manifolds and relate them to certain geometric and topological constructions. Specifically, we discuss pseudoholomorphic functions of one complex variable taking values in the space of transverse loops in a Cauchy–Riemann manifold, with an emphasis on the geometry of their images. In this context, we introduce a special class of foliated solid tori, called holomorphic tubes, which appear connected with a number of interesting geometric topics. In particular, we prove that each isolated singularity of algebraic plane curve defines a holomorphic tube in a small 3-sphere around the singularity endowed with the standard CR-structure. We also show that there exist a plenty of holomorphic tubes containing a given real analytic transverse loop. In particular, such a loop can be holomorphically deformed along certain vector fields on its image.

Keywords:

• Almost complex structure
• Pseudoholomorphic mapping
• Analytic disc
• Cauchy–Riemann manifold
• Transverse loop
• Solid torus
• Hopf fibration
• Villarceau circle
• Plane curve singularity
• AMS 2000 Mathematics Subject Classifications: 32Q60
• 35Q15
• 58D15

Notes

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday.

Email: [email protected]

Email: [email protected]

Additional information

Notes on contributors

G. Khimshiashvili *

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday. Email: [email protected]