The Bonnet problem for harmonic maps to the three-sphere
-
Bart Dioos
and
Joeri Van der Veken
Abstract
In previous work [15] the authors defined transforms for non-conformal harmonic maps from a Riemann surface into the 3-sphere. An observation from that study was that two invariants, a real and a complex function, determine a non-conformal harmonic map up to isometries of the 3-sphere. We now show that if the first invariant of a harmonic map and its transformed map are the same, then these maps are either congruent or the harmonic map belongs to a particular 1-parameter family. Inspired by this result we discuss the Bonnet problem for non-conformal harmonic maps: to what extent is a harmonic map determined by its first invariant?
Acknowledgements
The authors would like to thank Makoto Sakaki for his helpful comments and remarks.
Communicated by: P. Eberlein
Funding: This work is partially supported by the Belgian Interuniversity Attraction Pole P07/18 and project 3E160361 of the KU Leuven Research Fund. J. Van der Veken is supported by the Excellence Of Science project G0H4518N of the Belgian government.
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Articles in the same Issue
- Frontmatter
- A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
- On mixed Lp John ellipsoids
- Valuations on convex functions and convex sets and Monge–Ampère operators
- Weierstrass points on Kummer extensions
- The Bonnet problem for harmonic maps to the three-sphere
- Maximal arcs in projective planes of order 16 and related designs
- On the decomposition of the small diagonal of a K3 surface
- Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
- Higgs bundles and fundamental group schemes
- A relation between the curvature ellipse and the curvature parabola
- Non-orientable three-submanifolds of G2-manifolds
- The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
- A class of analytic pairs of conjugate functions in dimension three
Articles in the same Issue
- Frontmatter
- A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions
- On mixed Lp John ellipsoids
- Valuations on convex functions and convex sets and Monge–Ampère operators
- Weierstrass points on Kummer extensions
- The Bonnet problem for harmonic maps to the three-sphere
- Maximal arcs in projective planes of order 16 and related designs
- On the decomposition of the small diagonal of a K3 surface
- Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples
- Higgs bundles and fundamental group schemes
- A relation between the curvature ellipse and the curvature parabola
- Non-orientable three-submanifolds of G2-manifolds
- The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group
- A class of analytic pairs of conjugate functions in dimension three
