Pairs of intersecting real manifolds in complex space
Let \(M_1,M_2\) be a pair of totally real and real analytic \(n\)-manifolds in \(\mathbb C^n (n\geq 2)\) which intersect transversally at \(0\). The author studies the simultaneous normalization of the pair under local biholomorphic maps, and the existence of analytic annuli in \(\mathbb C^n\) with one bounding circle in \(M_1\) and the other in \(M_2\). The linear case is studied by \textit{B. Weinstock} [Math. Ann. 282, No. 1, 131--138 (1988; Zbl 0628.32015)]. The author considers three main cases: the generic case, the real Lagrangian case (\(M_j\subset\mathbb C^{2n}\) with Re\,\(dz_1\wedge dz_{n+1}+\dots+dz_n\wedge d_{2n}| _{M_j}=0\)), and the holomorphic reversible case (\(M_2=\tau M_1\) for some holomorphic involution \(\tau\)). These problems are related to an earlier important work of \textit{J. K. Moser} and the author on real manifolds with a non-vanishing Bishop invariant [Acta Math. 150, 255--296 (1983; Zbl 0519.32015)]. The author's main theorem on pairs of real Lagrangian surfaces is the following: Let \(r_1(z,\overline z)=az^2+bz\overline z+\overline a\overline z^2+O(3)\) and \(r_2(z,\overline z)=z\overline z+O(3)\) be two real analytic functions in \(z\in \mathbb C\). Let \(M_j: p=\partial_zr_j(z,\overline z)\) be the corresponding pair of real Lagrangian surfaces in \(\mathbb C^2\). Assume that \((b^2-4| a| ^2+1)/b>2\). Then there exists a real analytic \(1\)-parameter family of analytic annuli \(A_c\subset \mathbb C^2\), \(0<c<c_0\), bounding in \(M_1\cup M_2\), and shrinking to the origin as \(c\to 0\). These annuli sweep out a three dimensional Levi-flat manifold lying on a real analytic set biholomorphically equivalent to Im\,\((zp)=0\) with Re\((zp)>0\). The author also proves an interesting result on the existence of an invariant holomorphic surface of a holomorphic symplectic map on \(\mathbb C^{2n}\). The map, restricted to the invariant surface, has a normal form \((z,p)\to (\mu(zp)z,\nu(zp)p)\). It is crucial that the normal form satisfies both \(\mu\nu=1\) and \(| \mu(0)| \neq1\), which allow the author to show the existence of analytic annuli for higher dimensional \(M_1, M_2\).
scientific article; zbMATH DE number 2113422