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By Huang X., Yin W.

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Indeed, this is equivalent to the fact that the origin is inside the hyperbolic polygon S(u) with vertices τ0 (u), . . , τs−1 (u) in ∆. 44), one concludes easily that the boundary of S(u) can be deformed, in ∆ \ {0}, to the circle centered at the origin 1−s s−1 with radius s · (s − 1) s u 2s . ) Now, by the Cauchy formula, it holds that α! ∂α f ∗ (0, u) = √ α ∂z 2π −1 ∂S(u) f ∗ (ζ, u) dζ ζ α+1 and ∂ α f ( N+1) α! (0, u) = √ α ∂z 2π −1 ∂S(u) f ( N+1) dζ. ζ α+1 Hence, it follows that ∂ α f ( N+1) ∂α f ∗ 2 (0, u) − (0, u) ≤ Cu 3 N−α−3 .

9). Here F = ( f , g ) = (z + f, w + g) is assumed to be a holomorphic map with f = O(|w| + |z|2 ), g(z, w) = g(w) = O(w2 ), g(w) = g(w) and u = u + g(u). 3. 2, since it is not assumed that F(M) ⊂ M , the reality of g is not automatic from the property that F(M) approximates M to a high order. 2. 11) k=1 j=2 N N with Mnor defined by and let Φ2 be a biholomorphic map from M to Mnor N s−1 s w = z z + 2Re z + a ks+ j z ks+ j + R (z , z ). 12) k=1 j=2 Here R(z, z) = R(z, z) = o(|z|sN+s−1 ) and R (z, z) = R (z, z) = o(|z|sN+s−1 ).

Moser asked in his paper ([21, p. 399]). 3, we next show that a generic Bishop surface with the Bishop invariant vanishing at 0 and with s < ∞ is not even formally equivalent to any algebraic surface in C2 . For this purpose, we borrow the idea used in the CR setting based on the Baire category argument. For the consideration in the CR setting by using the Baire category theorem, the reader is referred to the paper of Forstneric [8]. 16): aαβ z α zβ . 20) A Bishop surface with a vanishing Bishop invariant Write F := {a = (a1 , .

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


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