An inverse problem of the flux for minimal surfaces

Details An inverse problem of the flux for minimal surfaces An inverse problem of the flux for minimal surfaces

1997-09-02

arxiv.org/abs/dg-ga/9709006v1

Indiana Univ. Math. J., vol. 46 (1997) 529--560

Mathematics

Differential Geometry

28 pages, AMSLaTeX 1.1, with 8 figures, To appear in Indiana University Mathematics Journal

Scientific paper

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 inverse problem: For each balanced n vectors, find an n-end catenoid which attains given vectors as flux. Here, an n-end catenoid is a complete minimal surface of genus 0 with ends asymptotic to the catenoids. In this paper, the problem is reduced to solving algebraic equation. Using this reduction, it is shown that, when n=4, the inverse problem for 4-end catenoid has solutions for almost all balanced 4 vectors. Further obstructions for n-end catenoids with parallel flux vectors are also discussed.

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