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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 \(S^3\).

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

Citations:

Zbl 0725.58010
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