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Moduli for pointed convex domains

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Summary

A moduli space for the class of pointed strictly linearly convex domains in ℂn 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\(\bar \partial _b \) complex of the boundary of the indicatrix.

The paper is related to one of László Lempert, Holomorphic invariants, normal forms and moduli space of convex domains. Ann. Math128, 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada

    J. Bland

  2. Department of Mathematics, University of Washington, 98195, Seattle, Washington, USA

    T. Duchamp

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  1. J. Bland
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  2. T. Duchamp
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Additional information

Oblatum 26-IX-1989 & 22-III-1990

Partially supported by an NSERC grant.

The second author wishes to thank the University of Toronto and the Mathematical Sciences Research Institute at Berkeley, where portions of the paper were written.

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Bland, J., Duchamp, T. Moduli for pointed convex domains. Invent. math. 104, 61–112 (1991). https://doi.org/10.1007/BF01245067

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  • DOI: https://doi.org/10.1007/BF01245067

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