General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux

TL;DR

This paper proves that for most balanced flux vectors, there exist minimal surfaces of genus zero with multiple catenoidal ends in Euclidean 3-space, solving an inverse problem for arbitrary numbers of ends greater than four.

Contribution

It establishes the existence of minimal surfaces with prescribed flux vectors for almost all balanced vectors when the number of ends exceeds four, extending previous results.

Findings

Existence of minimal surfaces for almost all balanced flux vectors with more than four ends.

Nonexistence results for certain special balanced vectors.

Extension of previous work for four ends and new results for higher numbers of ends.

Abstract

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.