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 $n\ge 2$ 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 $b_2(\Sigma ;{\mathbb{Z}}_2)<\infty$.

9 Pages