Thumbnail Image

Conformal maps from a 2-torus to the 4-sphere

Bohle, Christoph; Leschke, Katrin; Pedit, Franz; Pinkall, Ulrich

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus, the kernel bundle can be extended to the compactification of the spectral curve and we obtain a linear 2-torus worth of algebraic curves in projective 3-space. The original conformal immersion of the 2-torus is recovered as the orbit under this family of the point at infinity on the spectral curve projected to the 4-sphere via the twistor fibration.
Published in: Journal fĂĽr die reine und angewandte Mathematik (Crelles Journal), 10.1515/crelle.2011.156, De Gruyter
  • Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
  • This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.