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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?

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  • $\begingroup$ This seems like a difficult problem. I suspect it might be open, because the following question is: `If $\Sigma^3 \subset \mathbf{R}^4$ is a complete stable minimal surface with trivial normal bundle, is $\Sigma$ a plane?' It is a conjecture of Schoen that this is true; it's recorded as Conjecture 2.12 in the book of Colding--Minicozzi for example. One of the difficulties is that estimates for the area growth are hard to come by when $n \geq 4$ (when these are available you can use the work of Schoen--Simon--Yau), another that the logarithmic cut-off estimate is not available anymore. $\endgroup$
    – Leo Moos
    Commented Oct 12, 2020 at 21:55
  • $\begingroup$ Be careful also to distinguish between 'orientable' and 'trivial normal bundle'. They are not the same; surfaces with the latter property are also called two-sided. The way it's stated I think your claim is not quite correct: $\mathbf{R} P^{2n-1} \subset \mathbf{R} P^{2n}$ is an orientable, stable minimal hypersurface. $\endgroup$
    – Leo Moos
    Commented Oct 12, 2020 at 21:59
  • $\begingroup$ @LeoMoos Here the ambient manifold is assumed to be orientable. So the "two-sided" condition is equivalent to the orientable hypersurface. $\endgroup$
    – Totoro
    Commented Oct 12, 2020 at 23:14
  • $\begingroup$ My bad, you're right, I missed that part of your question! $\endgroup$
    – Leo Moos
    Commented Oct 13, 2020 at 0:37
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    $\begingroup$ A small comment. The assumption that $M^n$ is orientable is superfluous. A complete noncompact positively curved manifold $M^n$ must be diffeomorphic to $\mathbb R^n$ because its soul is a point. $\endgroup$
    – Vitali Kapovitch
    Commented Mar 9, 2022 at 17:24

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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.)

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