Minimal Legendrian surfaces in the tangent sphere bundle of \(\mathbb{R}^3\) (Q6630746)

The authors study minimal Legendrian surfaces in the tangent sphere bundle \(T_1 \mathbb{R}^3\) of \(\mathbb{R}^3\). They show that a totally geodesic Legendrian surface in \(T_1 \mathbb{R}^3\) is locally congruent to one of the following:\N\begin{itemize}\N\item \(M_1 :=\{ (0, \nu )\in \mathbb{R}^3 \times S^2 \mid \nu \in S^2 \}\);\N\item \(M_2 :=\{ (p, 0, \nu_0 )\in \mathbb{R}^2 \times \mathbb{R} \times S^2 \mid p \in \mathbb{R}^2 \}\) (\(\nu_0 =(0, 0, 1)\));\N\item \(M_3 :=\{ (x, 0, 0, 0, \cos \theta , \sin \theta ) \in \mathbb{R}^3 \times S^2 \mid x\in \mathbb{R} , \theta \in [0, 2\pi )\}\).\N\end{itemize}\NMoreover, the authors show that a complete non-negatively curved minimal Legendrian surface in \(T_1 \mathbb{R}^3\) is congruent to one of the above three surfaces \(M_1\), \(M_2\), \(M_3\), and that a minimal Legendrian sphere is congruent to \(M_1\), while they also show that there is no closed minimal Legendrian torus in \(T_1 \mathbb{R}^3\). They give a stability result, which states that a complete minimal Legendrian surface in \(T_1 \mathbb{R}^3\) is stable if and only if it is congruent to either \(M_1\) or \(M_2\).