manuscripta mathematica

, Volume 130, Issue 3, pp 311-352

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

  • Christoph BohleAffiliated withInstitut für Mathematik, Technische Universität Berlin Email author 
  • , Franz PeditAffiliated withMathematisches Institut der Universität TübingenDepartment of Mathematics, University of Massachusetts
  • , Ulrich PinkallAffiliated withInstitut für Mathematik, Technische Universität Berlin

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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