Minimal planes in asymptotically flat three-manifolds

TL;DR

This paper proves the existence of properly embedded minimal planes in asymptotically flat 3-manifolds under certain conditions, extending previous results and providing new existence theorems for minimal surfaces.

Contribution

It improves prior results by establishing the existence of minimal planes passing through specified points and tangent directions in asymptotically flat 3-manifolds without closed minimal surfaces.

Findings

Existence of minimal planes through a point with a given tangent plane.

Existence of minimal planes passing through three specified points.

Extension of previous minimal surface existence results.

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 $T_qM$ 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.