Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-09-01T14:52:46.408Z Has data issue: false hasContentIssue false
  • English
  • Français

The Poincaré Dual of a Geodesic Algebraic Curve in a Quotient of the 2-Ball

Published online by Cambridge University Press:  20 November 2018

Stephen S. Kudla  and
John J. Millson
Stephen S. Kudla
Affiliation:
University of Maryland, College Park, Maryland
John J. Millson
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C 2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D 1 in the inverse image of N under p. The choice of D 1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ 1. Let π : D → D 1 be the fiber bundle obtained by exponentiating the normal bundle of D 1 in D. Let μ be the volume form of D 1.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 485 - 499
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Berger, M., Ganduchonand, P., Mazet, E., Le spectre d'une variétériemannienne, Lecture Notes in Mathematics 194 (Springer-Verlag, New York).Google Scholar
2. Gaffney, M., Asymptotic distributions associated with the Laplacian for forms, Comm. Pure and Appl. Math. 11 (1958), 535–545.Google Scholar
3. Kobayashi, S. and Nomizu, K., Foundations of differential geometry (Interscience Publishers, John Wiley and Sons, New York, 1969).Google Scholar
4. Kudla, S. and Millson, J., Harmonic differentials and closed geodesies on a Riemann surface, to appear in Invent. Math.Google Scholar
5. Kudla, S. and Millson, J., Geodesic cycles and the Weil representation I: Quotients of hyperbolic space and Siegel modular forms, preprint.Google Scholar