论文标题
在$ \ widetilde {e（2）} $中的螺旋形最小表面的结构和最小环
Constructions of helicoidal minimal surfaces and minimal annuli in $\widetilde{E(2)}$
论文作者
论文摘要
在本文中，我们在三维谎言组$ \ widetilde {e（2）} $中构建了两个单参数的最小表面的单参数家族，这是欧几里得平面的一组刚性运动的普遍覆盖。第一个可以看作是一个螺旋内的家族，而第二个是一个偶然的最小表面家族。我们用于构建这些表面的主要工具是由Meeks，Mira，Pérez和Ros引入的Weierstrass型表示，以最小的表面为第三个尺寸的谎言组。最后，我们研究了顶齿最小表面的极限。作为此限制情况的应用，我们获得了$ \ widetilde {e（2）} $中最小表面的半空间定理的新证明。
In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in $\widetilde{E(2)}$.
