Complete stable minimal hypersurfaces in positively curved 4-manifolds

Abstract

We show that the combination of non-negative sectional curvature (or $2$-intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a $4$-manifold with bounded curvature. In particular, this implies the nonexistence of complete two-sided stable minimal hypersurface in a closed $4$-manifold with positive sectional curvature. Our work leads to new comparison results. We also construct various examples showing rigidity of stable minimal hypersurfaces can fail under other curvature conditions.

arXiv commentTypos corrected, exposition improved. To appear in JEMS

Cited by in corpus (1)

• Volume growth and positive scalar curvature