By A. Libgober, P. Wagreich
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Extra resources for Algebraic Geometry. Proc. conf. Chicago, 1980
O(Γi + |vi |H01 )+ |w|H01 . We are left with the projection terms associated to v i , hi and with the contribution of v i to g. 4 The contribution of v i to g comes from αj ωj j=i 0(|v i |2 ) and thus yields o(|v i |H01 )|w|H01 . − α4i ωi4 v i + January 17, 2007 11:55 WSPC/Book Trim Size for 9in x 6in ﬁnalBB Recent Progress in Conformal Geometry 50 For the projection terms, the contribution of v i and hi is gathered. For = i, 5/2 Ωi e w ≤ Cεi |w|H01 . For = i, we use Lemmas 20–24 and derive that these projection terms contribute (use also Lemmas 25–27) εi |v |H01 + o(|v i |H01 + Γi ) |w|H01 .
Hence, we derive the estimate |ω2∞ |εij +O(εij ). This is a direct estimate. 5/2 ω14 ∂ω1 ∂ω2 . (4) We may assume that ω2 is the new concentrated one. We split between ω14 ∂ω1 ∂ω2 + Ωc1 Ω1 5/2 ω14 ∂ω1 ∂ω2 = Ω1 ω14 ∂ω1 ∂ω2 +O ε12 (I) ω1 is completely deconcentrated. On Ω1 , λr = λ|x−a2 | ≥ (due to rescaling). Then ω14 ∂ω1 ω2∞ 0(δ2 ) +C |(I)| ≤ Ω1 δ15 Ω1 0(|ω2∞ |ε12 ) O(|ω2∞ |ε12 ) + C r≥ ε 1 1 ε12 where λ = 1 ε12 δ2 ≤ 1 + λ2 |x − a2 |2 1/6 r2 dr (1 + r2 )9 = O(|ω2∞ |ε12 + ε12 ). 12).
6 6 − xi | λi |x − xi | (λi Max εim )3 January 17, 2007 11:55 WSPC/Book Trim Size for 9in x 6in ﬁnalBB Sign-Changing Yamabe-Type Problems Proof of 2. 2]. Next, if conclude. δ 4 δi = 0 εi √ λ 24/5 |ωk |6/5 5/6 5/6 . 1 10 |x √ − xi |. Then, δi (x) ≤ c λ εi and where D is the domain where |x − xi | ≥ Or |x−x | ≥ 12 |x −xi | and |x−xi | ≤ and Dc ω or k = i. The claim then follows from [Bahri 2001, and k are diﬀerent from i, we use 1. and H¯ older to Proof of 3. Either |x − xi | ≥ D Ωi 37 δ 4 δi ≤ Cλ2i ε4i 1 |x−xi |≤ 10 |x √ λ2i λi 4 =C εi λ3i 1 10 |x |x −xi | .
Algebraic Geometry. Proc. conf. Chicago, 1980 by A. Libgober, P. Wagreich