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

By Huang X., Yin W.

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12), respectively. Write (z nor, wnor ) = Φ1 (z, w) and write (z ∗nor, w∗nor ) = Φ2 (z , w ). 46) N . A Bishop surface with a vanishing Bishop invariant N N Define Dnor(u) and D∗nor (u), associated with Mnor and Mnor , respectively, in a similar way as for D(u). ) Then τ nor at the origin as that for σ(τ, r). (Notice that u nor = rnor j (u nor ) is nor nor defined such that Aj (u nor ) = rnor · σnor (τ j (u nor ), rnor ). Similarly, we can ∗ ∗ (τ, rnor ), τ ∗nor . 11) up to order N. 34), we obtain N−1 (u ∗nor ) = Anor A∗nor j j (u nor ) + O u nor (u ∗nor ) τ ∗nor j − τ nor j (u nor ) = O (u nor ) N−2 and , as u nor → 0+ .

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). Moreover, the angle between the hyperbolic geodesic (in ∆) connecting τ0∗ (u) to τ1∗ (u) and the hyperbolic geodesic (in ∆) connecting τ0∗ to τ j at their intersection τ0∗ (u) equals, first, to Θj (u) and thus also equals to Θ∗j (u). Hence, we see that τ j (u) = τ ∗j (u). 13. Ψ(τ j (u), r) = τ ∗j (u) for j = 0, . . , s − 1.

21) by the procedure described above with the degree of B s bounded by d, that further satisfy the following properties: Cond (1) H(z, ξ) s are holomorphic over |z|2 + |ξ|2 < 1/m 2 ; Cond (2) max(|z|2 +|ξ|2 )<1/m 2 |H(z, ξ)| ≤ n and |cαβγτ | ≤ n. ∞ d d Write AdB = ∞ n,m=1 A B (n, m) and A B = d=1 A B . 16), is formally equivalent to an algebraic surface if and only if J(M) ∈ J(A B ). (Therefore, M defined X. Huang, W. ) Now, for any sequence {M j } ⊂ AdB (n, m) with M j : w = h j (z, z) = zz + z s + zs + o(|z|s ), by a normal family argument and by passing to a subsequence, we can assume that h j (z, ξ) → H0 (z, ξ) over any compact subset of {|z|2 + |ξ|2 < 1/m 2 }.

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