Flat analytic discs attached to real hypersurfaces of finite type (Q1901662)

The authors construct small analytic discs flat at 1 attached to a large class of hypersurfaces. A smooth analytic disc \(f : \overline \Delta \to \mathbb{C}^n\) with \(f\) being analytic on the unit disc \(\Delta \subset \mathbb{C}\) and smooth up to \(\partial \Delta\), is said to be attached to a manifold \(M \subset \mathbb{C}^n\), if \(f (\partial \Delta) \subset M\). It is flat at 1 if \(f^{(j)} (1) = 0\), \(j\) any positive integer. A main theorem is the following. Consider the hypersurface \(M \subset \mathbb{C}^2\) given by \(\text{Im} z_2 = p(z_1, \overline {z_2})\), where \(p(u, \overline u) = \sum_{2 \leq j + k \leq m} a_{jk} u^j \overline u^k\), with \(a_{jk} = \overline {a_{kj}}\), and \(a_{jk} = 0\) if \(j = k\). Then the following (Theorem 1) is proved. Given \(\varepsilon > 0\), there are nontrivial smooth analytic discs \(f\) attached to \(M\), flat at 1 and satisfying \(|f (\zeta) |< \varepsilon\) for all \(\zeta \in \overline \Delta\). The idea of the proof uses a characterization of the flat condition via the Fourier transform techniques applied to the pullbacks to the real line.