ScalingStacks

Lemma 8.5 . [01ZD]

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Lemma 8.5.

Let RicM4≥−3​δ{\rm Ric}_{M^{4}}\geq-3\delta and Vol⁡(B1​(p))>v>0{\rm Vol}(B_{1}(p))>{\rm v}>0 with δ≤1\delta\leq 1. Then for each δ′>0\delta^{\prime}>0 and x∈B2​(p)x\in B_{2}(p) there exists at most N⁡(v,δ′)N({\rm v},\delta^{\prime}) scales α∈ℕ\alpha\in\mathds{N} such that

|𝒱rα+1δ​(x)−𝒱rαδ​(x)|<δ′.\displaystyle\big|\mathcal{V}^{\delta}_{r_{\alpha+1}}(x)-\mathcal{V}^{\delta}_{r_{\alpha}}(x)\big|<\delta^{\prime}\,. (8.10)

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