Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Gauss map of minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$   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.

9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int