Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections
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Abstract:
We determine surfaces of genus zero in self-dual Einstein manifolds whose twistor lifts are harmonic sections. We apply our main theorem to the case of four-dimensional hyperkähler manifolds. As a corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is a harmonic section. In particular, if the mean curvature vector field is parallel with respect to the normal connection, then the surface is totally umbilic. Thus, our main theorem is a generalization of Hopf’s theorem for a constant mean curvature surface of genus zero in three-dimensional Euclidean space. Moreover, we can also see that a Lagrangian surface of genus zero in the complex Euclidean plane with conformal Maslov form is the Whitney sphere.References
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Additional Information
- Kazuyuki Hasegawa
- Affiliation: Faculty of Teacher Education, Institute of Human and Social Sciences, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan
- Email: kazuhase@staff.kanazawa-u.ac.jp
- Received by editor(s): October 12, 2009
- Received by editor(s) in revised form: April 1, 2010
- Published electronically: August 20, 2010
- Additional Notes: This work is partially supported by the Grant-in-Aid for Young Scientists (B) No. 20740046, the Ministry of Education, Culture, Sports, Science and Technology, Japan
- Communicated by: Jon G. Wolfson
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 309-317
- MSC (2000): Primary 53C42, 58E20
- DOI: https://doi.org/10.1090/S0002-9939-2010-10657-3
- MathSciNet review: 2729093