Conformal maps from a 2-torus to the 4-sphere. (English) Zbl 1261.53057
The authors study the moduli space of conformal immersions, \(f: M \to S^4\), of a given Riemannian surface \(M\) into the \(4\)-sphere \(S^4\), up to Möbius equivalence. This study focuses on the \(2\)-torus case, \(M=T^2\), and the map \(f: T^2 \to S^4\) is a conformal immersion with degree zero normal bundle. The authors associate to such a conformal immersion \(f: T^2 \to S^4\), the moduli space of generalized Darboux transforms, which has the structure of a Riemannian surface, the spectral curve. This allows the authors to describe conformal immersions of \(T^2\) into \(S^4\) in terms of some algebro-geometric data.
Reviewer: Ioan Bucataru (Iaşi)
MSC:
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
53C40 | Global submanifolds |
53A30 | Conformal differential geometry (MSC2010) |
37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |
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