Proposition 3.15 . [0505] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Proposition 3.15 .
Let ϕ 1 \phi_{1} 3 3 3.112 Δ ϕ 1 \Delta\phi_{1}
Δ ϕ 1 = \displaystyle\Delta\phi_{1}= − H α y α 2 r 3 d y 1 ∧ d y 2 ∧ d y 3 − Ω i j α β ( 1 4 r d y α β ^ + 1 4 r 2 y α β ^ d r ) ∧ d x i ∧ d x j \displaystyle-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}-\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr)\wedge dx_{i}\wedge dx_{j}
(3.119) + A i j α β ( 1 4 r 2 y α β ^ d r − 1 4 r d y α β ^ ) ∧ d x i ∧ d x j + r − 5 Π 3 ( 4 ) + O ′ ( 1 ) . \displaystyle+A_{ij\alpha\beta}(\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).