TITLE: Complete stable minimal hypersurface in positively curved manifolds
QUESTION [5 upvotes]: Let $(M^n,g)$ be a complete noncompact orientable Riemannian manifold with positive sectional curvature. Can we find an orientable stable minimal hypersurface $N$ in $M$?
It follows from R. Schoen's work that if $n=3$, no such hypersurface $N$ exists. Moreover, if $N$ is compact, this is also impossible by the stability inequality. Are there any result for the general case?

REPLY [2 votes]: You can construct a positively curved $(M^n,g)$ for any $n\geq 4$ that admits a stable minimal hypersurface. This is described in Example 1.2 here. (That paper also contains some non-existence results for $n=4$ under additional curvature assumptions.)