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By Huang X., Yin W.

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63) α,β =0 β where β = γ −α + 2s ∈ {0, 1, 2, . . } and bαβ = β bαβγ . 60), we conclude that, for each fixed α and for any β, γ with β + 2s , it holds that |bαβγ | ≤ C0 · R2sα+2sβ ≤ C0 · (R2s )α+β . Thus, β = γ −α 2 |bαβ | ≤ C0 · 2s(α + β ) + 1 R2sα+2sβ ≤ C0 (1 + R)4s(α+β ) . 63). 62) holds for each α ≥ 0, we see that f (z, u) = f ∗ (z, u) in the formal sense. Hence, f (z, u) is also given by a convergent power series. 11 is finally complete. 2. 11 have the same content. 5. 16. Let (M, 0) be a real analytic elliptic Bishop surface with the Bishop invariant vanishing and the Moser invariant s < ∞ at 0.

41) first in the formal sense and thus also hold as functions of u. Now, we can use an automorphism of ∆ to map τ0 to the origin and τ1 to a point in the positive real line. Then we easily see that Θj and L 1( j+1) uniquely determine τ j (u). Recall Ψ(·, τ) is an automorphism of ∆ and thus is an isometry with respect to the Poincaré metric, that maps τ j (u) to τ ∗j (u) for j = 0, 1. Write τ j (u) = Ψ(τ j (u), r) ∈ ∆ for each j ∈ [2, s − 1]. Then the hyperbolic distance between τ j (u) and τ0∗ (u) = Ψ(τ0 (u), r) equals to that between τ j (u) and τ0 (u), that is L 1( j+1)(u) and thus is also the same as L ∗1( j+1)(u).

From this, it follows that F(B) = B , where B is the branching locus of π near the origin. We next give the precise defining equation of B near 0. 5) where h 1 (ζ) is holomorphic near 0. 3), we get w = h 2 (ζ) = (1 − s)ζ s + o(ζ s ). 6), we get − w = (h 3 (ζ))s s−1 with h 3 (ζ) = ζ + o(ζ). 7) Hence, we get ζ= w − s−1 h −1 3 z = h1 ◦ h −1 3 = s(−1)− s w 1 w − s−1 s−1 s 1 s = (−1) 1 s · (s − 1) 1−s s 1 s 1 s−1 = h 1 (−1) + o(w s−1 s ). 8) X. Huang, W. Yin Here, h j s are holomorphic functions near 0.

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A Bishop surface with a vanishing Bishop invariant by Huang X., Yin W.

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