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I have 3 questions:

1) Let $(M_j^n,p_j)_{j\in \mathbb{N}}$ be a sequence of pointed mean-convex embedded hypersurfaces in $\mathbb{R}^{n+1}$. Does a subsequence converge in the Gromov-Hausdorff topology to some pointed metric space $(X,d,p)$?

2) More generally, given a constant $k\in \mathbb{R}$ and a sequence $(M_j^n,p_j)_{j\in \mathbb{N}}$ of pointed embedded hypersurfaces in $\mathbb{R}^{n+1}$ with mean curvature $\geq k$, does a subsequence converge in the Gromov-Hausdorff topology to some pointed metric space $(X,d,p)$?

According to Gromov's precompactness theorem, 1) and 2) are true iff there exists a function $\mathcal{N}:(0,\infty)^2\to (0,\infty)$ such that, for every $j,\epsilon,r>0$, the maximum number of disjoint balls of radius $\epsilon$ contained inside of $B_r(p_j)\subset M_j^n$ is bounded above by $\mathcal{N}(\epsilon,r)$. (See for example Petersen's book Riemannian Geometry.)

However, I am not sure whether such a function exists.

The analogous statement for Ricci curvature bounded below is proven using Bishop-Gromov volume monotonicity, which leads to the last question:

3) Is there an analogue of volume monotonicity for hypersurfaces with mean curvature bounded below? My guess is that the comparison would be against constant curvature hypersurfaces like the round sphere.

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