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** Additional info for A Bishop surface with a vanishing Bishop invariant**

Some earlier studies related to non-algebraicity for CR manifolds can be found, for instance, in [2, 14, 17]. However, by a result of the first author with Krantz [16] and a result of the first author in [15], a Bishop surface with an elliptic complex tangent can always be holomoirphically transformed into the algebraic Levi-flat hypersurface C × R and also into the Heisenberg hypersurface in C2 . 18), the condition that λks+ j = 0 for j = 0, 1, k = 1, 2, . . can be compared with the Cartan–Chern–Moser chain condition in the case of strongly pseudoconvex hypersurfaces (see [6]).

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 . 54) Here, we used the obvious fact that the Euclidean length of ∂S(u) is bounded by a constant. 53). Assume that z is on the hyperbolic geodesic segment in D(u) connecting Aj (u) to Aj+1 (u) for a certain j ∈ [0, s − 1]. ) X. Huang, W. 55) j+1 for a certain t ∈ [0, 1]. 22), we get |z(u, t)| = s · (s − 1) 1−s s |1 + t(e √ 2π −1 s − 1)|u s−1 s + o(u s−1 s ). 56) for 0 < u 1. 14.

Then f (Aj (u), u) − A∗j (u ) |u| N−1 , for j = 0, . . 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 ).