Relating diameter and mean curvature for submanifolds of Euclidean space

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Adiabatic limits of Ricci-flat Kähler metrics (2009, methodology): “metrically into RN and we get that the length of the mean curvature vector of the composite isometric embedding Xy → X→ RN is then uniformly bounded independent of y. We can then apply Theorem 1.1 of [Tp] and get the required diameter bound. Alternatively, ﬁrst one observes that (3.3) implies that t”
On the Shape of Compact Hypersurfaces with Almost‐Constant Mean Curvature (2015, unknown): “ (L+1−a)|B|. (2.5) We shall work under the assumption that δ(Ω) ≤ c(n,L,a) for a suitably small positive constant c(n,L,a) ≤ 1/2: in particular, n 2 ≤ H(x) ≤ 2n, ∀x∈ ∂Ω. (2.6) By Topping’s inequality [Top08], one has diam(Ω) ≤ C(n) Z ∂Ω |H|n−1 , so that (2.4) and (2.6) imply diam(Ω) ≤ C(n)L. (2.7) A”