Asymptotically flat three-manifolds contain minimal planes

TL;DR

This paper proves that in asymptotically flat three-manifolds without closed minimal surfaces, every point lies on a complete properly embedded minimal plane, revealing a rich structure of minimal surfaces in such manifolds.

Contribution

It establishes the existence of minimal planes through every point in asymptotically flat 3-manifolds lacking closed embedded minimal surfaces, a new result in geometric analysis.

Findings

Existence of minimal planes through every point in the manifold

Minimal planes are complete and properly embedded

No closed embedded minimal surfaces are present in the manifold

Abstract

Let $(M,g)$ be an asymptotically flat $3$-manifold containing no closed embedded minimal surfaces. We prove that for every point $p\in M$ there exists a complete properly embedded minimal plane in $M$ containing $p$.