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

Shin Kato·Masaaki Umehara·Kotaro Yamada
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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.

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Shin Kato, Masaaki Umehara, Kotaro Yamada. 1997-09-02. General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux. https://arxiv.org/abs/dg-ga/9709007

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