Counter-examples to the "Jacobian Conjecture at Infinity"

 by   Stepan Orevkov


In the paper [S.Yu.Orevkov, An example in connection to the Jacobian Conjecture, Math. Notes, 47(1990) no. 1-2,  82-88], we constructed an example of an open complex surface  U, a compact rational curve  L  on  of the self-intersection  L.L=+1  and an immersion  $f:U\to C^2$  defined by two functions which are holomorphic on  U-L and meromorphic on  U, such that f is not an embedding (if  were isomorphic to a domain in  $C^2$  such an immersion would provide automatically a counter-example to the well-known Jacobian Conjecture). In this paper we construct a similar example which satisfies the additional property that the restriction of  f  to the boundary of  U  is an immersion of the 3-sphere into  $C^2$  which is regularly homotipic to the standard embedding.  We prove that the constructed immersion can not be extended to a counter-example to the Jacobian Conjecture, analysing coefficients of polynomials  P(x,y),  Q(x,y)  with constant jacobian which might realize such an immersion.  As a corollary, we obtain an example of a pseudo-convex immersion of a 3-sphere into the complex plane which is regularly homotopic to the standard embedding but which is not homotopic through pseudo-convex immersions.