Question

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 1

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.