manuscripta mathematica

, Volume 130, Issue 3, pp 311–352

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


DOI: 10.1007/s00229-009-0288-x

Cite this article as:
Bohle, C., Pedit, F. & Pinkall, U. manuscripta math. (2009) 130: 311. doi:10.1007/s00229-009-0288-x


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)


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