Picture below is from Huisken's The volume preserving mean curvature flow.
$h_{ij}$ is second fundamental form. $H=g^{ij}h_{ij}$, and $|A|^2=g^{ij}g^{kl}h_{ik}h_{jl}$. How 2.1.Theorem equal to the red line ? Besides, what is come close together ? It means $h_{ij}-h_{kl}\rightarrow 0$ ?
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If we let $\kappa_i$ denote the eigenvalues of the second fundamental form, then $|A|^2 = \sum_i \kappa_i^2$ and $H = \sum_i \kappa_i$. Some algebraic manipulation should then convince you that $$Q := |A|^2 - \frac 1n H^2 = C_n \left(\sum_{i \ne j} (\kappa_i - \kappa_j)^2\right),$$
so $Q$ is some kind of total measurement of the distance between the eigenvalues. Dividing it by $H^2$ we get a scale-invariant measurement, and the theorem then states that
$$ \frac Q {H^2} \lesssim H^{-\delta};$$
so if $H \to \infty$ then $Q / H^2 \to 0$, i.e. the relative difference between the eigenvalues approaches zero.
As an aside, maybe compare this to Hamilton's original paper on Ricci Flow for Ricci-positive 3-manifolds, where an almost identical estimate is shown for the eigenvalues of $\rm Rc$.