用Weierstrass表示公式证明Fraser-Li猜想
A new approach to the Fraser-Li conjecture with the Weierstrass representation formula
摘要：
本文给出了$\mathbb{R}^3$中曲面上的曲线与球面正交相交的一个充分条件。这一结果使得在处理球$\mathbb{B}^3$中的自由边界极小曲面时，完全用Weierstrass数据表示边界条件而不需要积分成为可能。此外，我们还证明了嵌入的自由边界极小环的高斯映射是一对一的。利用这一点，Fraser-Li猜想可以转化为确定Gauss映射的问题。另一方面，我们表明，刘维尔型边值问题在一个环的结构，浸入最小环正交领域提供了一些新的见解。这也为Fraser-Li猜想提供了一种新的偏微分方程理论方法。
英文摘要：
In this paper, we provide a sufficient condition for a curve on a surface in $\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.
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