IUMJ

Title: Contact geometry of one-dimensional complex foliations

Authors: Giuseppe Tomassini and Sergio Venturini

Issue: Volume 60 (2011), Issue 2, 661-676

Abstract:

Let $V$ be a real hypersurface of class $\mathrm{C}^k$, $k \ge 3$, in a complex manifold $M$ of complex dimension $n+1$ and $\mathrm{HT}(V)$ be the holomorphic tangent bundle to $V$ giving the induced $\mathrm{CR}$ structure on $V$. Let $\theta$ be a contact form for $(V,\mathrm{HT}(V))$ and $\xi_0$ be the Reeb vector field determined by $\theta$, and assume that $\xi_0$ is of class $\mathrm{C}^k$. In this paper we prove the following theorem (cf. Theorem 4.2): if the integral curves of $\xi_0$ are real analytic, then there exist an open neighbourhood $M_0 \subset M$ of $V$ and a solution $u \in C^k(M_0)$ of the complex Monge-Amp\`ere equation $(\mathrm{d}\mathrm{d}^cu)^{n+1} = 0$ on $M_0$ which is a defining equation for $V$. Moreover, the Monge-Amp\`ere foliation associated to $u$ induces on $V$ that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of $\mathrm{CR}$ distributions of codimension one which is of independent interest (cf. Theorem 3.2} below).