A central result of this paper is the $\epsilon$-regularity theorem, Theorem 6.1. The original $\epsilon$-regularity theorems for Einstein manifolds were given in [A90], [T90] , [BKN89]. They state that if $M^{n}$ is an Einstein manifold, ${\rm Ric}_{M^{n}}=\lambda g$, with $|\lambda|\leq n-1$, and if for $B_{2}(p)\subset M^{n}$,

then $\sup_{B_{1}(p)}|{\rm Rm}|\leq 1$.

In [CCT02], [Ch2], [CD13], $\epsilon$-regularity theorems were proved under the assumption of $L^{q}$ curvature bounds, $1\leq q<n/2$, provided $B_{2}(p)$ is assumed sufficiently close to a ball in a cone which splits off an isometric factor $\mathds{R}^{n-2q}$.

On the other hand, the regularity theory of [ChNa13] for Einstein manifolds depends on $\epsilon$-regularity theorems which do not assume any $L^{q}$ curvature bounds. In particular, it follows from the work of [A90] that there exists $\epsilon(n)>0$ such that if $|{\rm Ric}_{M^{n}}|\leq\epsilon(n)$ and if

where $B_{2}(0^{n})\subseteq\mathbb{R}^{n}$, then $|{\rm Rm}|\leq 1$ on $B_{1}(x)$.

This result can be extended in several directions. In order to state the extension in full generality, we first recall the notion of the harmonic radius:

We call a mapping $\Phi:B_{r}(0^{n})\to X$ as above a harmonic coordinate system. Harmonic coordinates have an abundance of good properties when it comes to regularity issues; see the book [P] for a nice introduction. In particular, if the Ricci curvature is uniformly bounded then in harmonic coordinates, the metric, $g_{ij}$ has a priori $C^{1,\alpha}\cap W^{2,q}$ bounds, for all $\alpha<1$ and $q<\infty$. If in addition, there is a bound on $|\nabla{\rm Ric}_{M^{n}}|$, then in harmony coordinates, $g_{ij}$ has $C^{2,\alpha}$ bounds, for all $\alpha<1$.

The primary theorem we wish to review in this subsection is the following:

By the results of the previous subsection, it is possible to find balls satisfying the above constraint off a subset of Hausdorff codimension $2$. Moreover, when combined with the quantitative stratification of [ChNa13], see also Section 7, this $\epsilon$-regularity theorem leads to a priori $L^{p}$ bounds on the curvature. The primary result of the present paper can be viewed as Theorem 6.1, which states that the conclusions of Theorem 2.3 continue to hold if $\mathds{R}^{n-1}$ is replaced by $0\in\mathds{R}^{n-3}\times C(Z)$.