AN INVERSE PROBLEM OF THE FLUX FOR MINIMAL-SURFACES

For a complete minimal surface in the Euclidean 3-space, the so-called flux vector corresponds to each end. The flux vectors are balanced, i .e., the sum of those over all ends are zero. Consider the following i nverse problem: For each balanced n vectors, find an n-end catenoid wh ich attains the given vectors as flux. Here, an n-end catenoid is a co mplete minimal surface of genus 0 with ends asymptotic to the catenoid s. In this paper, the problem is reduced to solving algebraic equation . Using this reduction, it is shown that, when n = 4, the inverse prob lem for the 4-end catenoids has solutions for almost all balanced 4 ve ctors. Further obstructions for n-end catenoids with parallel flux vec tors are also discussed.