Calabi-Yau Conjecture for Minimal Hypersurfaces in $\mathbb{R}^4$ with bounded geometry

Abstract

The Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ for $n\geq 2$ ask whether a complete minimal hypersurface must be unbounded, and more strongly whether it must be proper. In this work, we resolve this conjecture for complete, connected, embedded minimal hypersurfaces $ \Sigma^3 \subset \mathbb{R}^4$ with bounded second fundamental form and finite second Betti number $b_2(\Sigma;\mathbb{Z}_2)<\infty$.