Calabi-Yau Conjecture for Minimal Hypersurfaces in ℝ⁴ with bounded geometry

2026/08/05 by Shrey Aryan, Alexander D. McWeeney, Giuseppe Tinaglia
Mathematics · #math.DG #math.AP

paper · pdf

9 Pages

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

The Calabi-Yau conjectures for complete minimal hypersurfaces Σⁿ⊂ ℝⁿ⁺¹ for n≥ 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 Σ³ ⊂ ℝ⁴ with bounded second fundamental form and finite second Betti number b₂(Σ;ℤ₂)<∞.

Citations

• Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4
• Two rigidity results for stable minimal hypersurfaces
• The embedded Calabi-Yau conjecture for finite genus
• Chord arc properties for constant mean curvature disks
• Minimal hypersurfaces asymptotic to Simons cones
• Density theorems for complete minimal surfaces in R³
• The Calabi-Yau conjectures for embedded surfaces
• Review of geometry and analysis