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. ˆR34| ∇ϕ| 2 dμ ≥ ˆR3 r −2 ϕ 2 dμ.We explicitly construct such an example and check its stability in Section A.1.1 Note that a Schrodinger operator with non-negative (but not identically zero) potential cannot be stable on R 2 , thanks to the log-cutoff trick.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

“…We note that similar effective regularity estimates for perimeter minimizing hypersurfaces and stable minimal or constant mean curvature hypersurfaces have been considered in the previous literature under different sets of assumptions. We address the reader for instance to [58,20,25], without the aim of being exhaustive in this list.…”

Liouville-type theorems on manifolds with nonnegative curvature and strictly convex boundary

“…n . Using µ-bubbles, that is, studying stable hypersurfaces with prescribed mean curvature in a manifold with positive scalar curvature (PSC) has given fruitful results in recent years (see for example [Li19] [Zhu21][CL20] [CLS22]). In these work, the fact that the scalar curvature of the manifold has to obtain a strictly positive lower bound is crucial.…”