Proposition 6.8 (Uniform injectivity estimate on the neck) . [0557] 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 6.8 (Uniform injectivity estimate on the neck).
For any sufficiently large parameter T ≫ 1 T\gg 1 6.30
(6.60) ℒ : 𝔖 1 → 𝔖 2 , − 1 ∂ ∂ ¯ ϕ ↦ Δ ϕ \mathscr{L}:\mathfrak{S}_{1}\rightarrow\mathfrak{S}_{2},\quad\sqrt{-1}\partial\bar{\partial}\phi\mapsto\Delta\phi
is an isomorphism and satisfies the uniform injectivity estimate,
(6.61) ‖ − 1 ∂ ∂ ¯ ϕ ‖ 𝔄 ≤ C L ⋅ ‖ Δ ϕ ‖ 𝔅 . \|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{A}}\leq C_{L}\cdot\|\Delta\phi\|_{\mathfrak{B}}.
Here the constant C L > 0 C_{L}>0 T ≫ 1 T\gg 1