6. The ϵ -regularity Theorem [01YR]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

6. The $\epsilon$ -regularity Theorem

In Section 5, we showed that limit spaces satisfying our assumptions must be smooth away from a closed subset of codimension $4$ . However, the strongest applications come from a more effective version of this statement. In particular, the curvature estimates of Theorem 1.3 and the Minkowski estimates of Theorem 1.1 will require a more rigid statement. Namely, we will prove the following in this section:

Theorem 6.1.

There exists $\epsilon(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq\epsilon$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ . and

\displaystyle d_{GH}\big(B_{2}(p),B_{2}(0)\big)<\epsilon\,,

(6.1)

where $0$ is a vertex of the cone $\mathds{R}^{n-3}\times C(Y)$ , for some metric space $Y$ , then we have

\displaystyle r_{h}(p)\geq 1\,.

(6.2)

Consequently, if $M^{n}$ is Einstein, we have the bound

\displaystyle\sup_{B_{1}(p)}|{\rm Rm}|\leq 1\,.

(6.3)

Proof.

Given $n$ and ${\rm v}>0$ , assume no such $\epsilon$ exists. Then there exists a sequence of spaces $(M^{n}_{j},g_{j},p_{j})$ such that $|{\rm Ric}_{M^{n}_{i}}|\leq\epsilon_{j}\to 0$ , ${\rm Vol}(B_{1}(p_{j}))>{\rm v}>0$ and

\displaystyle d_{GH}\big(B_{2}(p_{j}),B_{2}(0_{j})\big)<\epsilon_{j}\to 0\,,

(6.4)

where $0_{j}\in\mathds{R}^{n-3}\times C(Y_{j})$ is a vertex but $r_{h}(p)<1$ . After possibly passing to a subsequence,we have

\displaystyle B_{2}(p_{j})\to B_{2}(0)\,,

(6.5)

where $0\in\mathds{R}^{n-3}\times C(Y)\equiv X$ is a vertex. But if $C(Y)$ has any point with $r_{h}(x)=0$ , then there is a set of Hausdorff codimension $3$ in $X$ which is not smooth. By the Hausdorff estimate of Theorem 1.1 this is not possible, so we must have that $C(Y)$ is smooth. Thus, $Y$ is a smooth manifold, and in fact, $C(Y)$ is itself be smooth if and only if $Y$ is the unit $2$ -sphere. Thus,

\displaystyle B_{2}(p_{j})\to B_{2}(0^{n})\subseteq\mathds{R}^{n}\,.

(6.6)

But now, we can apply the standard $\epsilon$ -regularity theorem, to conclude $r_{h}(p)\leq 1$ , which is a contradiction. ∎

6. The ϵ -regularity Theorem [01YR]

6. The ϵ\epsilon-regularity Theorem

Theorem 6.1.

Proof.

6. The $\epsilon$ -regularity Theorem