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.