MODULI FOR POINTED CONVEX DOMAINS

A moduli space for the class of pointed strictly linearly convex domains in C(n) is obtained. It is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the CR structure of the boundary of the indicatrix. These deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix. A careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex Hamiltonian vector fields. The proof of this fact is based on the Folland-Stein estimates for the partial-differential-equation(b) complex of the boundary of the indicatrix. The paper is related to one of Laszlo Lempert, Holomorphic invariants, normal forms and moduli space of convex domains. Ann. Math 128, 47-78 (1988), where other modular data for pointed convex domains were constructed. A method of recovering Lempert's modular data from the deformation moduli is given.