Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold

2012-06-20T04:29:25Z

Let $M$ be Riemannian manifold and $S\subset M$ a minimal submanifold, with $\dim S<\dim M-1$. According to a few references (e.g., Mahmoudi, Mazzeo & Pacard), it should not be hard to see that:

''The closer a constant mean curvature (CMC) hypersurface of $M$ is to $S$ (in the Hausdorff metric), the larger its mean curvature must be.''

I was wondering if this claim is indeed not hard to see, given that I am still unable to find a simple/elementary proof. Any suggestions?

Moreover, I was wondering if anything is known about the speed in which the value of the mean curvature $H(t)$ of a family $N_t$ of CMC hypersurfaces of $M$ "condensing" (i.e., collapsing) onto $S$ diverges to $+\infty$. For instance, is it always the case that both $H$ and $H'$ diverge to $+\infty$ as those hypersurfaces collapse onto the minimal limit submanifold? Perhaps, in some special situation (e.g., if the sequence of CMC hypersurfaces collapsing is a solution to the Mean Curvature Flow (MCF)), this is implied by some property of the MCF?

Answer by Rafe Mazzeo for Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold

2012-06-20T06:41:03Z

Suppose that N is the CMC hypersurface which is $\epsilon$ close to S. Take the tube of radius $\epsilon$ around S. One can check that this has (typically variable) mean curvature on the order of $1/\epsilon$. If one chooses this tube correctly, it is tangent to N and this gives a lower bound for the mean curvature of N at the point of contact. Since the mean curvature of N is constant, this means that it is everywhere bigger than $1/\epsilon$.

The situation is now a bit better understood than when that paper was written. First, Rosenberg proved a theorem that if N is a surface in a 3-manifold and has mean curvature H and H is very large, then one of the two regions N bounds in the ambient manifold has inradius less than C/h.

Second, it is only true that the set on which the $N_t$ collapse must be minimal if the norm of the second fundamental form of N is comparable to the mean curvature. If this fails -- and a typical scenario is a sequence of spheres joined by very small necks, then it seems to be the case that the collapsing set is not necessarily minimal.

Finally, regarding your question -- by the same comparison result (using geodesic tubes) it is certainly the case that H is increasing as $N_t$ collapses, so this gives some sort of average increase of H'(t). There maybe could be some exotic examples where the mean curvature wobbles and H' is not necessarily tending to $+\infty$.

There are lots of other good questions here -- Harold Rosenberg had several questions which remain unanswered. For example, if T is a `geodesic triangle' in a 3-manifold, then is there a CMC tube which condenses to T? The problem is whether it is possible for the CMC surfaces to turn the corners at the vertices of T.

Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold - MathOverflow

Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold

Answer by Rafe Mazzeo for Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold