Nitsche猜想
The Nitsche conjecture
摘要：
该猜想涉及圆环A（r，R）与A（r*，R*）之间调和同胚的存在性，部分是由具有给定边界的双连通极小曲面的存在性问题所激发的. 1962年，J.C.C. Nitsche观察到图像环不能太薄，但可以任意厚（甚至是穿孔的磁盘）。然后，他解释说，这样一个映射存在，我们必须有以下不等式，现在被称为Nitsche界：R*/r* 是大于或等于（R/r+r/R）/2。本文对他的猜想给出了肯定的回答。作为推论，我们发现在给定环上的所有极小图中，悬链面的上板具有最大的共形模。
英文摘要：
The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In 1962 J.C.C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). Then he conjectured that for such a mapping to exist we must have the following inequality, now known as the Nitsche bound: R*/r* is greater than or equal to (R/r+r/R)/2. In this paper we give an affirmative answer to his conjecture. As a corollary, we find that among all minimal graphs over given annulus the upper slab of catenoid has the greatest conformal modulus.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
