ScalingStacks

Proof. As above we embed $(X,\omega_{X})$ isometrically into $\mathbb{R}^{N}$ and we get that the length of the mean curvature vector of the composite isometric embedding $X_{y}\to X\to\mathbb{R}^{N}$ 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 there is a uniform constant $\kappa$ so that that geodesic balls in $X_{y}$ of radius $r<1$ have volume at least $\kappa r^{2(n-m)}$ (Lemma 3.2 in [H]). Since the total volume of $X_{y}$ is constant equal to $1$, an elementary argument gives the required diameter bound. ∎