In this subsection we recall some basic properties of pointed Gromov-Hausdorff limit spaces

$\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)\,,$

(2.1)

where the ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$ and the noncollapsing assumption ${\rm Vol}(B_{1}(p_{j}))\geq{\rm v}>0$ holds. In particular, we recall the stratification of a noncollapsed limit space, which was first introduced in [ChCo1], and which will play an important role in the proof of Theorem 1.1. The effective version, called the quantitative stratification, which was first introduced in [ChNa13], will be recalled in Section 7. It will play an important role in the estimates of Theorem 1.3.

Given $x\in X$, we call a metric space $X_{x}$ a tangent cone at $x$ if there exists a sequence $r_{i}\to 0$ such that

$\displaystyle(X,r_{i}^{-1}d,x)\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}X_{x}\,.$

(2.2)

That tangent cones exist at every point is a consequence of Gromov’s compactness theorem; see for instance the book [P]. A point is called regular if every tangent cone is isometric to $\mathds{R}^{n}$ and otherwise singular. The set of singular points is denoted by $\mathcal{S}$. As explained below, for noncollapsed limit spaces with a uniform lower Ricci bound, the singular set has codimension $\geq 2$. At singular points, tangent cones may be highly nonunique with ill-defined dimension of the singular set, and even homeomorphism type, see for instance [CoNa2]. Easy examples show that the singular set need not be closed if one just assumes a uniform a lower bound ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$. However, under the assumption of a $2$-sided bound $|{\rm Ric}_{M^{n}_{j}}|\leq(n-1)$, the singular set is indeed closed; see [A90], [ChCo2].

For noncollapsed limit spaces, as shown in [ChCo1], every tangent cone is a metric cone, i.e.

$\displaystyle X_{x}=C(Z)\,,$

(2.3)

for some compact metric space $Z$, with ${\rm diam}(Z)\leq\pi$. With this as our starting point, we introduce the following notion of symmetry.

Thus, in the noncollapsed case, every tangent cone is $0$-symmetric.

The key result of [ChCo1] is the following:

$\displaystyle\dim\mathcal{S}^{k}\leq k\,,$

(2.5)

where dimension is in the Hausdorff sense. Thus, away from a set of dimension $k$, every point has some tangent cone with $(k+1)$ degrees of symmetry. For an effective refinement of this theorem see [ChNa13] and Section 7.