ScalingStacks

Lemma 4.5 . [023L]

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Lemma 4.5.

The function f0=Δ​u0f_{0}=\Delta u_{0} satisfies the fast decay near infinity

|∇T​Ykf0|=O⁡(e−c​rD1n−1n)|\nabla_{TY}^{k}f_{0}|=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}})

and the total integral

12​π​(n−1)​∫D1∖D2f0​ωT​Yn−1=∫D1∖D2d​dc​u0∧ωT​Yn−2=d2n−2n−1​∫Yc1​(L0)n−2≠0.\frac{1}{2\pi(n-1)}\int_{D_{1}\setminus D_{2}}f_{0}\omega_{TY}^{n-1}=\int_{D_{1}\setminus D_{2}}dd^{c}u_{0}\wedge\omega_{TY}^{n-2}=\frac{d_{2}^{n-2}}{n-1}\int_{Y}c_{1}(L_{0})^{n-2}\neq 0.

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