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Lemma 7.5 .
On U ∖ D U\setminus D C > 0 C>0
(7.52) C − 1 ( − log | S | 2 ) 1 n − 1 ω c y l ≤ ω T Y ≤ C ( − log | S | 2 ) 1 n ω c y l , C^{-1}(-\log|S|^{2})^{\frac{1}{n}-1}\omega_{cyl}\leq\omega_{TY}\leq C(-\log|S|^{2})^{\frac{1}{n}}\omega_{cyl},
and for all k ≥ 1 k\geq 1 C k , m k > 0 C_{k},m_{k}>0
(7.53) | ∇ ω c y l k ω T Y | ω c y l ≤ C k ( − log | S | 2 ) m k . |\nabla^{k}_{\omega_{cyl}}\omega_{TY}|_{\omega_{cyl}}\leq C_{k}(-\log|S|^{2})^{m_{k}}.