圆环边界上的规定次数调和映射与悬链面的分支
Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids
摘要：
设$A \subset \mathbb{R} ^2 $是光滑双连通区域.我们考虑狄利克雷能量$E(u)=\int_{A} |\nabla u|^2$，其中$u:A \rightarrow \mathbb{C}$，并寻找这个能量的临界点与规定的模$|u|=1$上$\partial A$和规定的程度上的两个连接组件的$\partial A$。这个变分问题是一个缺乏紧性的问题，因此我们不能使用变分法的直接方法。我们的分析依赖于所谓的霍普夫微分和这个问题和问题之间的强有力的联系，找到所有极小曲面有界的两个$p$覆盖的圆在平行的平面。然后，我们构造新的浸入极小曲面在$\mathbb{R} ^3$与此属性。这些曲面是由悬链线的$p$ -覆盖族分支得到的。
英文摘要：
Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components of $\partial A$. This variational problem is a problem with lack of compactness hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf differential and on a strong link between this problem and the problem of finding all minimal surfaces bounded by two $p$ covering of circles in parallel planes. We then construct new immersed minimal surfaces in $\mathbb{R} ^3$ with this property. These surfaces are obtained by bifurcation from a family of $p$-coverings of catenoids.
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