Constrained Willmore tori in the 4-sphere. (English) Zbl 1217.53059
It is proved that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere \(S^4\) is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in \(\mathbb{R}^4\cong S^4\setminus\{\infty\}\) for some point \(\infty\in S^4\) at infinity. This implies that all constrained Willmore tori in \(S^4\) can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of N. J. Hitchin’s approach [J. Differ. Geom. 31, No. 3, 627–710 (1990; Zbl 0725.58010)] to the study of harmonic tori in \(S^3\).
Reviewer: Kaarin Riives (Tartu)
MSC:
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
53C43 | Differential geometric aspects of harmonic maps |
53A30 | Conformal differential geometry (MSC2010) |
49Q10 | Optimization of shapes other than minimal surfaces |