(of Theorem 1.3) Let $(M^{n},g,p)$ satisfy $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$. We will first show that for every $q<2$ there exists $C=C(n,{\rm v},q)>0$ such that

Simultaneously, we will show that if $M^{n}$ is Einstein, then this can be improved to

where $r_{x}$ denotes the regularity scale at $x$.

Let $q<2$ and set $\eta=4-2q$. Consider Theorem 7.3 with $\epsilon=\epsilon(n)>0$ chosen from Theorem 6.1 and $\eta$ as above. Thus, there exists $C(n,{\rm v},q)$ such that

Note that by rescaling, we may regard the $\epsilon$-regularity theorem (Theorem 6.1) as stating that if $x$ is $(n-3,\epsilon,2r)$-symmetric then $r_{h}>r$, and if $M^{n}$ is Einstein then $r_{x}>r$. In fact, we have that if $x$ is $(n-3,\epsilon,s)$-symmetric for any $s\geq 2r$, then $r_{h}>r$. This is to say that if $x\not\in\mathcal{S}^{n-4}_{\epsilon,2r}$, then $r_{h}>\frac{s}{2}>r$. The contrapositive gives the inclusion

which by (7.5) gives us the desired estimate

If $M^{n}$ is Einstein, then Theorem 6.1 allows us to replace $r_{h}$ with $r_{x}$, as claimed.

Now, for $q<2$, let us prove the $L^{q}$ bound on the curvature from Theorem 1.3. For this note that if $r_{h}(x)>r$ then by definition there exists harmonic coordinates $\Phi:B_{r}(0^{n})\to M$ with $\phi(0)=x$ and such that

where $g_{ij}=\Phi^{*}g$ is the pullback metric. Since the Ricci curvature satisfies the bound $|{\rm Ric}_{M^{n}}|\leq n-1$, this implies that

where $\Delta_{x}$ denotes the Laplacian written in coordinates. In particular, for every $\alpha<1$ and $s<\infty$, we have the scale invariant estimates

In particular, applying this to $s=q$ we get

Let $\eta=2-q$ be chosen so that $q+\frac{\eta}{2}<2$. Then we have already shown that

for $C(n,{\rm v},q)>0$. Consider the covering $\{B_{r_{h}(x)}(x)\}$ of $B_{1}(p)$, and a subcovering $\{B_{r_{j}}(x_{j})\}$ by mutually disjoint balls, such that

1. (1)

   $B_{1}(p)\subseteq\bigcup B_{r_{j}}(x_{j})$ with $r_{j}=\frac{1}{2}r_{h}(x)$.

2. (2)

   $\{B_{r_{j}/4}(x_{j})\}$ are disjoint.

By using (7.12), we see for each $\alpha\in\mathds{N}$ that

Summing over $\alpha$ this gives

Finally, combining this with (7.11) we get

which finishes the proof of Theorem 1.3. ∎