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,


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:

  1. 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).

  2. 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...]

  • $\begingroup$ Are the minimal hypersurfaces of Lawson (ams.org/mathscinet-getitem?mr=267492) and Huang (ams.org/mathscinet-getitem?mr=662381) (Hopf fibration images of $S^p\times S^q$'s in $S^{2n+3}$) stable by any chance? If not, this comment will self-destruct :-) $\endgroup$ – Francois Ziegler Jul 13 '13 at 2:53
  • $\begingroup$ Hi Renato! Am I not mistaken that Fubini-Study metric has positive Ricci curvature (in fact it is positive Einstein, I think), so you can't have stable minimal surfaces... $\endgroup$ – Otis Chodosh Jul 13 '13 at 2:56
  • 2
    $\begingroup$ @Otis: Yes, $Ric=(2n+2)$ is positive so there are no stable minimal hypersurfaces (take $f=const$ on the stability formula). However, if you only consider functions with zero average (corresponding to being stable as critical point for CMC problem), than they could be stable in positive Ricci. Take, e.g., the equator $S^n$ on the round sphere $S^{n+1}$. For any $f$ with zero average $\int_{S^n} |\nabla f|^2\geq\int_{S^n}(Ric+|A|^2)f^2=n\int_{S^n} f^2$ because the first eigenvalue of $S^n$ is $n$. However, if you don't insist $f$ has zero average you can take $f=const$ and get a contradiction. $\endgroup$ – Renato G. Bettiol Jul 13 '13 at 4:45
  • $\begingroup$ @Francois: Thank you for the references, it seems promising (at least so I can learn about these constructions!) I will try to look it up in detail this weekend :) $\endgroup$ – Renato G. Bettiol Jul 13 '13 at 4:50
  • $\begingroup$ I don't get why your argument is a contradiction: a CMC minimizer for hypersurfaces bounding the same volume might not be a minimal surface, only CMC? $\endgroup$ – Ian Agol Jul 13 '13 at 5:18

This is now solved at least for $n+1=2,3$. Ramírez Luna has recently proven that an index $1$ minimal surface in $\mathbb{C}P^{n+1}$ is a Clifford hypersurface (as described in the paper). Because $\mathbb{C}P^{n+1}$ is simply connected and has positive Ricci curvature, the least area-minimal surface has index $1$ by work of Mazet--Rosenberg.

The dimensional restriction here is because of the usual singular dimension of stable minimal hypersurfaces. Presumably, it should be possible to overcome this issue here (but it might not be so easy).


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