manuscripta mathematica

, Volume 130, Issue 3, pp 311–352 | Cite as

The spectral curve of a quaternionic holomorphic line bundle over a 2-torus



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.

Mathematics Subject Classification (2000)

58J50 35Pxx 37K25 32G13 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Mathematisches Institut der Universität TübingenTübingenGermany
  3. 3.Department of MathematicsUniversity of MassachusettsAmherstUSA

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