Conformal maps from a 2-torus to the 4-sphere
-
Christoph Bohle
,
Katrin Leschke
Abstract
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.
©[2012] by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Conformal maps from a 2-torus to the 4-sphere
- Sofic random processes
- Overlap properties of geometric expanders
- Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8
- A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group
- Spin representations of Weyl groups and the Springer correspondence
- Higgs bundles over the good reduction of a quaternionic Shimura curve
Articles in the same Issue
- Conformal maps from a 2-torus to the 4-sphere
- Sofic random processes
- Overlap properties of geometric expanders
- Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8
- A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group
- Spin representations of Weyl groups and the Springer correspondence
- Higgs bundles over the good reduction of a quaternionic Shimura curve
