This is a lemma of the paper “Mean curvature flow singularities for mean convex surfaces” by Gerhard Huisken and Carlo Sinestrari (the paper is available here):

$ \textbf{Lemma 3.2.}$ Suppose $ (1 + \eta) H^2 \leq |A|^2 \leq c_0 H^2$ for some $ \eta, c_0 > 0$ at some point of $ \mathscr{M}_t$ , then we also have

(i) $ -2Z \geq \eta H^2|A|^2$ ;

(ii) $ |H \nabla_i h_{kl} – \nabla_i H h_{kl}|^2 \geq \frac{\eta^2}{4n(n-1)^2c_0} H^2 |\nabla H^2|$

My doubt is concerning to item $ (ii)$ and below is the argument given by the authors

We have (see [10, Lemma $ 2.3$ (ii)], reference [10] is available here)

$ |H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 \geq \frac{1}{4} |\nabla_i H \ h_{kl} – \nabla_k H \ h_{il}|^2 = \frac{1}{2} (|A|^2 |\nabla H|^2 – |\nabla^i H h_{il}|^2).$

Let us denote with $ \lambda_1, \cdots, \lambda_n$ the eigenvalues of $ A$ in such a way that $ \lambda_n$ is an eigenvalue with the largest modulus. Then we have $ |\nabla^i H \ h_{il}|^2 \leq \lambda_n^2 |\nabla H|^2$ and

\begin{align*} |H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 &\geq \frac{1}{2} \sum\limits_{i=1}^{n-1} \lambda_i^2 |\nabla H|^2 = \sum\limits_{i=1}^{n-1} \lambda_i^2 \lambda_n^2 \frac{|\nabla H|^2}{2\lambda_n^2}\ &\geq \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \frac{|\nabla H|^2}{2(n-1)|A|^2}\ &\geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2 \frac{|\nabla H|^2}{n(n-1)|A|^2}\ &= \frac{(|A|^2 – H^2)^2}{4n(n-1)|A|^2} |\nabla H|^2 \geq \frac{\eta^2 H^2}{4n(n-1)c_0} |\nabla H|^2. \square \end{align*}

I would like to understand the following equality and inequalities:

a) $ \frac{1}{4} |\nabla_i H \ h_{kl} – \nabla_k H \ h_{il}|^2 = \frac{1}{2} (|A|^2 |\nabla H|^2 – |\nabla^i H h_{il}|^2)$ ;

b) $ |H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 \geq \frac{1}{2} \sum\limits_{i=1}^{n-1} \lambda_i^2 |\nabla H|^2$ ;

c) $ \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \frac{|\nabla H|^2}{2(n-1)|A|^2} \geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2 \frac{|\nabla H|^2}{n(n-1)|A|^2}$ .

My thoughts:

$ a$ and $ b$ ) I just consider use normal coordinates, but this doesn’t helps me because the right hand side in $ a$ and $ b$ would be zero.

$ c$ ) I just try to prove that $ \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2$ , but I can’t prove this because I don’t know if all eigenvalues are non-negative. Indeed, I even don’t know if the $ H > 0$ because I didn’t see the hypothesis that the hypersurfaces is mean convex on the paper until this lemma.

Thanks in advance!