Stability for area in codimension one

Abstract

We present three results investigating aspects of the stability of minimal hypersurfaces. First, in joint work with Otis Chodosh, Chao Li, and Paul Minter, we prove that every complete oriented stable minimal hypersurface in R^5 is a hyperplane, which settles the stable Bernstein problem in dimension 5. Some consequences of this classification are curvature estimates for stable minimal hypersurfaces in 5-dimensional Riemannian manifolds. Second, in joint work with Otis Chodosh and Chao Li, we prove a rigidity result for complete two-sided stable minimal hypersurfaces in 4-dimensional manifolds with bounded and nonnegative sectional curvature and uniformly positive scalar curvature. We use this rigidity result to prove new topological obstructions for complete noncompact 4-dimensional manifolds satisfying these curvature assumptions. Finally, we present a new localization technique to prove the existence of minimal hypersurfaces in some complete noncompact manifolds. The minimal hypersurfaces produced by these methods are not necessarily stable, but they are quantitatively almost stable (index at most one). This result can be viewed as the sharp generalization to complete noncompact manifolds of the successful min-max theory of minimal hypersurfaces in closed manifolds.