Minimal planes in asymptotically flat three-manifolds (Q2135664)

The authors improve a previous result of \textit{O. Chodosh} and \textit{D. Ketover} [Adv. Math. 337, 171--192 (2018; Zbl 1397.53074)] by showing that a Riemannian manifold \(M\), which is diffeomorphic to \({\mathbb R}^3\), asymptotically flat, and such that it does not contain closed embedded minimal surfaces, has the following properties: \begin{itemize} \item For every \(p\in M\) and every \(2\)-plane \(V\subset T_pM\), there is a properly embedded minimal plane \(\Sigma\) in \(M\) such that \(p\in\Sigma\) and \(T_p\Sigma =V\); \item For every choice of three points in \(M\) there exists a properly embedded minimal plane that contains these points. \end{itemize} Concerning uniqueness, the authors consider \({\mathbb R}^3\) with an asymptotically Euclidean metric and construct as a counterexample two distinct minimal planes tangent at the origin.