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 U 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 U 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.