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Suppose that N is the CMC hypersurface which is \epsilon $\epsilon$ close to S. Take the tube of radius \epsilon $\epsilon$ around S. One can check that this has (typically variable) mean curvature on the order of 1/\epsilon. $1/\epsilon$. 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.$1/\epsilon$.

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 $N_t$ 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 $N_t$ 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. $+\infty$.

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.

Rafe

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.

Rafe