论文标题
$ \ mathbb {s}^2 \ times \ mathbb {r} $中的最小表面的高斯图
The Gauss map of minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$
论文作者
论文摘要
在这项工作中，我们考虑$ \ Mathbb {s}^2 \ times \ Mathbb {r} $等速度为$ \ Mathbb {r}^3 \ setMinus \ {0 \} $，赋予了$ \ mathbb的euuse ausp and wes＆weuuss＆weept and wes and wes＆wes y mathbb {同样，在$ 3- $ Euclidean空间中，该模型的表面也表面。我们将其作为主要结果表明，任何两个最小的保形沉浸在$ \ mathbb {s}^2 \ times \ times \ mathbb {r {r} $中，具有相同的非Constant Gauss映射仅通过两种类型的环境异构体差异：$ f =（$ f =（要么是$ f =（\ mathrm {id}，t）$ transb $ at $ trranse of translab of translab translab， $ f =（\ MATHCAL {a}，t）$，其中$ \ Mathcal {a} $表示$ \ Mathbb {s}^2 $上的对抗地图。此外，如果高斯地图是单数的，我们表明它一定是恒定的，然后仅在$ \ mathbb {s}^2 $的大地测量器上垂直圆柱体，$ \ mathbb {s}^2 \ times \ times \ times \ mathbb {r} $出现此假设。我们还研究了一些特殊情况，其中我们证明，在$ \ mathbb {s}^2 \ times \ mathbb {r} $中，高斯映射是一个非恒定抗恒定的抗模态图。
In this work, we consider the model of $\mathbb{S}^2\times\mathbb{R}$ isometric to $\mathbb{R}^3\setminus \{0\}$, endowed with a metric conformally equivalent to the Euclidean metric of $\mathbb{R}^3$, and we define a Gauss map for surfaces in this model likewise in the $3-$Euclidean space. We show as a main result that any two minimal conformal immersions in $\mathbb{S}^2\times\mathbb{R}$ with the same non-constant Gauss map differ by only two types of ambient isometries: either $f=(\mathrm{id},T)$, where $T$ is a translation on $\mathbb{R}$, or $f=(\mathcal{A},T)$, where $\mathcal{A}$ denotes the antipodal map on $\mathbb{S}^2$. Moreover, if the Gauss map is singular, we show that it is necessarily constant, and then only vertical cylinders over geodesics of $\mathbb{S}^2$ in $\mathbb{S}^2\times\mathbb{R}$ appear with this assumption. We also study some particular cases, among them we prove that there is no minimal conformal immersion in $\mathbb{S}^2\times\mathbb{R}$ which the Gauss map is a non-constant anti-holomorphic map.
