ScalingStacks

Theorem 7.3 (Quantitative Stratification, [ ChNa13 ] ) . [01YX]

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Theorem 7.3 (Quantitative Stratification,[ChNa13]).

Let MnM^{n} satisfy Ric≥−(n−1){\rm Ric}\geq-(n-1) with Vol⁡(B1​(p))>v>0{\rm Vol}(B_{1}(p))>v>0. Then for every ϵ,η>0\epsilon,\eta>0 there exists C=C⁡(n,v,ϵ,η)C=C(n,{\rm v},\epsilon,\eta) such that

Vol⁡(Tr​(𝒮ϵ,rk​(M)∩B1​(p)))≤C​rn−k−η.\displaystyle{\rm Vol}\left(T_{r}\left(\mathcal{S}^{k}_{\epsilon,r}(M)\cap B_{1}(p)\right)\right)\leq Cr^{n-k-\eta}\,. (7.2)

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