Proposition 3.20 . [050F] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Proposition 3.20 .
There is some 3 3 Λ 3 ( 4 ) \Lambda_{3}^{(4)} 3.9
𝔅 0 \displaystyle\mathfrak{B}_{0} ≡ H α y α 4 r d y 1 ∧ d y 2 ∧ d y 3 \displaystyle\equiv\frac{H^{\alpha}y_{\alpha}}{4r}dy_{1}\wedge dy_{2}\wedge dy_{3}
(3.189) + ( Ω i j α β ( 1 16 y α β ^ d r − 3 16 r d y α β ^ ) − 1 16 A i j α β ( y α β ^ d r + r d y α β ^ ) ) ∧ d x i ∧ d x j + r − 3 Λ 3 ( 4 ) , \displaystyle+\Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16}rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}}dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_{i}\wedge dx_{j}+r^{-3}\Lambda_{3}^{(4)},
then the corrected 3 3 ϕ 1 \phi_{1}
(3.190) ϕ 2 ≡ \displaystyle\phi_{2}\equiv ϕ 1 + 𝔅 0 \displaystyle\phi_{1}+\mathfrak{B}_{0}
satisfies
(3.191) Δ ϕ 2 = O ′ ( 1 ) on 𝒰 ∖ P , \Delta\phi_{2}=O^{\prime}(1)\ \text{on}\ \mathcal{U}\setminus P,
and has the expansion
ϕ 2 = \displaystyle\phi_{2}= 1 2 r ( 1 − H α y α 2 ) d y 1 ∧ d y 2 ∧ d y 3 + 1 2 r y β A i α β d x i ∧ d y α ^ − 1 4 A i j α β r ⋅ d y α β ^ ∧ d x i ∧ d x j \displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}
(3.192) + 3 16 ( A i α , α + 1 A j α , α + 2 − A i α , α + 2 A j α , α + 1 ) d ( r y α ) ∧ d x i ∧ d x j + r − 3 Π 3 ( 4 ) + O ′ ( r 2 ) . \displaystyle+\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}).