(Higher order estimates) For every $\epsilon>0$ there exists $\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta$ and $u:B_{2}(p)\to\mathds{R}^{k}$ is a $\delta$-splitting map, then the following hold:

1. (1)

   There exists $\alpha(n)>0$ such that for each $1\leq a\leq k$,

   $\displaystyle\fint_{B_{1}(p)}\frac{|\nabla^{2}u^{a}|^{2}}{|\nabla u^{a}|^{1+\alpha}}<\epsilon\,.$

   (1.14)

2. (2)

   Let $\omega^{\ell}\equiv du^{1}\wedge\cdots\wedge du^{\ell}$, $1\leq\ell\leq k$. Then

   $\displaystyle\fint_{B_{1}(p)}\big|\Delta|\omega^{\ell}|\big|<\epsilon\,.$

   (1.15)