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Article
The spectral curve of a quaternionic holomorphic line bundle over a 2-torus
MANUSCRIPTA MATHEMATICA
  • C Bohle
  • F Pedit, University of Massachusetts - Amherst
  • U Pinkall
Publication Date
2009
Abstract
A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and asymptotic behavior of the spectral curve. If this curve has finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or the Willmore energy of an immersion from a 2-torus into the 4-sphere is given by the residue of a specific meromorphic differential on the curve. Also, the kernel bundle of the Dirac type operator evaluated over points on the 2-torus linearizes in the Jacobian of the spectral curve. Those results are presented in a geometric and self contained manner.
Comments

This is the pre-published version harvested from arXiv. The published version is located at http://www.springerlink.com/content/yp03237467850493/

Pages
311-352
Citation Information
C Bohle, F Pedit and U Pinkall. "The spectral curve of a quaternionic holomorphic line bundle over a 2-torus" MANUSCRIPTA MATHEMATICA Vol. 130 Iss. 3 (2009)
Available at: http://works.bepress.com/franz_pedit/3/