  For each end of complete minimal surface in the Euclidean 3-space, the flux
vector is defined. It is well-known that the sum of the flux vector over all
ends are zero. Consider the following inverse problem: For each balanced
n-vectors, find an n-end catenoid which realizes these vectors as flux. Here,
an n-end catenoid is a complete minimal surface of genus zero with ends
asymptotic to the catenoids. In this paper, we show that the inverse problem
can be solved for almost all balanced n vectors for arbitrary n, which is
grater than 4. The assumption "almost all" is needed because nonexistence is
known for special balanced vectors. We remark that in the case of n=4, the same
result has been obtained by the authors (dg-ga/9709006). And the case n=3 is
treated by Lopez and Berbanel.
