Minimal planes in asymptotically flat three-manifolds

Abstract

In this paper, we improve a result by Chodosh and Ketover. We prove that, in an asymptotically flat $3$-manifold $M$ that contains no closed minimal surfaces, fixing $q\in M$ and a $2$-plane $V$ in $T_{q}M$ there is a properly embedded minimal plane $\Sigma$ in $M$ such that $q\in \Sigma$ and $T_q\Sigma =V$. We also prove that fixing three points in $M$ there is a properly embedded minimal plane passing through these three points.

22 pages, second version a counterexample is added