General existence of minimial surfaces of genus zero with catenoidal ends and prescribed flux (Q1971637)

In this paper the authors study the inverse problem of the flux formula: Problem A. For a given set of unit vectors \(v:=\{v_1,v_2,\ldots ,v_n\}\) in \(\mathbb R^3,\) and nonzero real numbers \(a:=\{a_1,\ldots ,a_n\}\) satisfying \(\sum_{j=1}^na_jv_j =0\) (the pair \((v,a)\) is called the flux data), is there a non-branched \(n\)-end catenoid (i.e. every end is of catenoid type) \(X: \mathbb C\cup\{\infty\}\setminus \{q_1,\ldots,q_n\}\rightarrow \mathbb R^3\) such that \(\nu (q_j)=v_j\) (limit unit normal vector) and \(a_j\) is the weight at the end \(q_j\)? The principal result of this paper is the following: For each integer \(n\geq 3\) Problem A is solved for almost all flux data.