[Paper Review] Contact geometry of one dimensional holomorphic foliations

Giuseppe Tomassini, Sergio Venturini|ArXiv.org|Jul 29, 2009

Geometry and complex manifolds8 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the existence of a $C^k$ solution $u$ to the complex Monge-Ampère equation $({\rm d}{{\rm d}^{c}}u)^{n+1} = 0$ that defines a real hypersurface $V$ in a complex manifold $M$, where $u$ serves as a defining equation for $V$ and the associated Monge-Ampère foliation induces the Reeb foliation on $V$. The key result is that such a solution exists if and only if the integral curves of the Reeb vector field $\xi_0$ are real analytic, achieved by solving a Cauchy problem for infinitesimal symmetries of CR distributions.

ABSTRACT

Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θbe a contact form for (V,HT(V)), ξ_0 the Reeb vector field determined by θand assume that ξ_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of ξ_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Ampère equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Monge-Ampère 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 CR distributions of codimension one which is of independent interest (cf. Theorem 3.1).

Motivation & Objective

• To determine the conditions under which a real hypersurface $V$ in a complex manifold $M$ admits a $C^k$ defining function $u$ solving the complex Monge-Ampère equation $({\rm d}{{\rm d}^{c}}u)^{n+1} = 0$.
• To establish a correspondence between the analyticity of the Reeb vector field's integral curves and the existence of such a solution $u$.
• To show that the Monge-Ampère foliation induced by $u$ coincides with the foliation defined by the Reeb vector field on $V$.
• To develop and apply a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one, which is of independent interest.

Proposed method

• The authors define a calibrated foliation $(\xi, u)$, where $\xi$ is a vector field satisfying $[\xi, J\xi] = 0$, ${{\rm d}^{c}}u(\xi) = 0$, and ${\rm d}u(\xi) = 1$, ensuring $\xi$ generates a holomorphic foliation by Riemann surfaces.
• They use the theory of generalized analytic functions to show that the set $Z$ of points where $\xi$ is an infinitesimal symmetry of ${{\rm Ker}}\,{\rm d}u \cap {{\rm Ker}}\,{{\rm d}^{c}}u$ intersects each leaf in a discrete or full subset.
• A Cauchy problem is solved for infinitesimal symmetries of CR distributions of codimension one, proving that if $\xi_0$ has real analytic integral curves, then a unique calibrated foliation $(\xi, u)$ exists near $V$ with $u|_V = 0$ and $\xi|_V = \xi_0$.
• The solution $u$ is shown to satisfy $({\rm d}{{\rm d}^{c}}u)^{n+1} = 0$ via a determinant identity involving the Levi form and the vanishing of $\xi \mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}{\rm d}{{\rm d}^{c}}u$.
• The converse is proven by showing that if $u$ is a $C^k$ solution to the Monge-Ampère equation, then the associated vector field $\xi = \xi_u$ satisfies $[\xi, J\xi] = 0$, and its integral curves are real analytic by Proposition 3.1.

Experimental results

Research questions

• RQ1 Under what conditions does a real hypersurface $V$ in a complex manifold $M$ admit a $C^k$ defining function $u$ solving the complex Monge-Ampère equation $({\rm d}{{\rm d}^{c}}u)^{n+1} = 0$?
• RQ2 When does the Monge-Ampère foliation induced by $u$ coincide with the foliation generated by the Reeb vector field $\xi_0$ on $V$?
• RQ3 What is the role of the analyticity of the integral curves of $\xi_0$ in the existence of such a solution $u$?
• RQ4 How can a Cauchy problem for infinitesimal symmetries of CR distributions be solved to construct $u$?

Key findings

• A $C^k$ solution $u$ to the complex Monge-Ampère equation $({\rm d}{{\rm d}^{c}}u)^{n+1} = 0$ exists in a neighborhood $M_0$ of $V$ if and only if the integral curves of the Reeb vector field $\xi_0$ are real analytic.
• The solution $u$ is a defining function for $V$, with $u|_V = 0$ and ${\rm d}u \neq 0$ near $V$, and satisfies ${{\rm d}^{c}}u|_{T(V)} = \theta$, where $\theta$ is the contact form on $V$.
• The Monge-Ampère foliation associated to $u$ induces on $V$ the same foliation as the Reeb vector field $\xi_0$.
• The converse holds: if a $C^k$ solution $u$ exists, then the integral curves of the associated vector field $\xi_u$ are real analytic, implying $\xi_u|_V = \xi_0$.
• The proof relies on solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one, which is shown to be of independent interest.