Contact geometry of one-dimensional complex foliations (Q2898369)

The authors give a non-trivial necessary and sufficient condition for the existence of a defining function \(u\) for a contact CR hypersurface \(V\), with \(u\) satisfying the complex Monge-Ampère equation in a neighborhood of \(V\). More precisely, let \(M\) be an \((n+1)\)-dimensional complex manifold, and \(V\subset M\) a closed real hypersurface of class \(C^k\) with \(k\geq 3\). Assume that the Levi form of \(V\) is non-degenerate; then \(V\) necessarily is a contact manifold. Let \(\theta\) denote the contact form of \(V\), and \(\xi_0\) the associated Reeb vector field. Assume that \(\xi_0\) is of class \(C^k\) too. Then the authors prove that the following conditions are equivalent: NEWLINENEWLINENEWLINENEWLINE (i) the integral curves of \(\xi_0\) are real analytic as curves in \(M\); NEWLINENEWLINENEWLINENEWLINE (ii) there exists an open neighborhood \(M_0\subseteq M\) of \(V\) and a function \(u\in C^k(M_0)\) defining \(V\) (that is, \(u|_{V}\equiv 0\) and \(du|_V\neq 0\)), satisfying the complex Monge-Ampère equation \((dd^c u)^{n+1}\equiv 0\) and inducing on \(V\) the given contact structure, that is \(d^cu|_{TV}\equiv\theta\) and \(du\wedge d^cu\wedge (dd^c u)^n\neq 0\) on \(M_0\). NEWLINENEWLINENEWLINENEWLINE When \(\theta\) and \(V\) are real analytic (and hence (i) is automatically satisfied), the existence of \(u\) was originally proved by \textit{E. Bedford} and \textit{D. Burns} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 381--414 (1979; Zbl 0422.32021)]. The main tool for dealing with the \(C^k\) situation is the following Cauchy-type theorem proved by the authors: if \(V\subset M\) is a real hypersurface of class \(C^k\), with \(k\geq 2\), and \(\xi_0\) is a \(C^k\)-vector field on \(V\) with real analytic integral curves, then there exists a (locally) unique pair \((\xi,u)\) defined in a neighborhood \(M_0\) of \(V\) with \(\xi\) a \(C^k\)-vector field on~\(M_0\) such that \([\xi,J\xi]\equiv O\) and \(\xi|_V=\xi_0\), and \(u\in C^k(M_0)\) such that \(du(\xi)\equiv 1\), \(d^cu(\xi)\equiv 0\) and \(u|_V\equiv 0\).