I'm trying to read the paper (Schoen-Simon-Yau '74) and can't figure out one of the bounds they use on the derivative of the second fundamental form.
In detail: Let $M$ a minimal hypersurface embedded in an $n$+1 dimensional Riemannian manifold $N$. It follows $$h_{ijk} = \nabla_k h_{ij} = -\frac 12 \, K_{n+1,ijk},$$ where $h_{ij}$ is the second fundamental form of $M$, $\nabla$ is the induced connection on $M$ and $K$ is the Riemann tensor on $N$. Let the sectional curvatures of $N$ be bounded between $K_1$ and $K_2$.
In the paper, before eq. (1.28) they introduce the inequality $$|K_{n+1,iji}| \leq \frac 12 (K_1 - K_2). \qquad (*)$$
How do I see this result?
I tried to use that $\sum_i h_{ii} = 0$, $\sum_i h_{iik} = 0$, and $(*)$ but that lead to nowhere.
Maybe someone has a tip? I would be very grateful!
It seems that they are off by a factor of $2$.
Let $j\neq i$ (or the inequality is trivial), write $$e = \frac{1}{\sqrt 2} (e_{n+1} + e_j).$$ Note $e$ is a unit vector and is orthogonal to $e_i$. Then
\begin{align*} 2K_{n+1, iji} &= 2K(e_{n+1}, e_i, e_j, e_i) \\ &= K(e_{n+1}+e_j, e_i, e_{n+1} + e_j, e_i) - K(e_{n+1}, e_i, e_{n+1}, e_i) - K(e_j, e_i, e_j, e_i) \\ &= 2K(e, e_i, e, e_i) - K(e_{n+1}, e_i, e_{n+1}, e_i) - K(e_j, e_i, e_j, e_i) \\ &\le 2K_1 - 2K_2. \end{align*}
From (1.24) in the paper, it seems that they assume $K_2 \le K_{ijij} \le K_1$. Since the inequality depends solely on $N$ but not on the minimal submanifolds $M$, I don't think the equation on $h$ helpes. On the other hand, the constant $2$ is not crucial in their calculations. You can absorb it into the $\epsilon$ for example.