After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was wondering about the analogous statement on other compact symmetric spaces. In particular,

# Question:

What is the least area minimal hypersurface of $\mathbb C P^{n+1}$? Here, $\mathbb C P^{n+1}$ is endowed with the Fubini-Study metric and I am interested in (real) codimension 1 submanifolds, i.e., manifolds of real dimension $2n+1$.

# A few comments:

I understand there is a well-developed theory for

*complex*codimension 1 (where I suppose the equatorial $\mathbb CP^n$ is the least area one), but I am interested in*real*codimension 1, motivated mainly by the question of finding isoperimetric regions (whose boundary is necessarily a stable constant mean curvature submanifold).My first intuitive guess was that it should be a minimal sphere $S^{2n+1}$, given by the geodesic sphere of radius $r_{min}=\arctan\sqrt{2n+1}$ around any given point $p\in \mathbb C P^{n+1}$. This guess is sort of motivated by the fact that, in this situation, geodesic balls of very small volume should be (up to small perturbations) isoperimetric regions. This sphere of radius $r_{min}$ happens to be minimal, see this post, and one can explicitly compute lots of things about it, e.g., the first (positive) eigenvalue of the Laplacian is $\lambda_1=4n+2$ and $(Ric(\vec n)+\|A\|^2)=2n+4+(2\cot2r_{min})^2+2n(\cot r_{min})^2$ $=4n+4$. [Everything needed for this computation can be found in the 2 links in this post; and I'm 99% sure I computed it correctly... Note the curious coincidence that $(Ric(\vec n)+\|A\|^2)=4n+4$ above is precisely the first (positive) eigenvalue of $\mathbb CP^n$!] Consequently, $\lambda_1=(Ric(\vec n)+\|A\|^2)-2$, which, by the second variation of area, means that the minimal geodesic sphere $S^{2n+1}$ is

**not stable**as solutions of the CMC problem, i.e., it has positive Morse index (which I think is equal to the multiplicity of $\lambda_1$, in this case, $n(n+2)$, see Prop 5.3 here). [**Edit**(see Ian Agol's comment): The fact that this minimal hypersurface is not stable as a critical point of the CMC problem indeed doesn't seem to contradict the possibility of it being a minimizer for area, so maybe the above guess could be correct...]

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