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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections


Author: Kazuyuki Hasegawa
Journal: Proc. Amer. Math. Soc. 139 (2011), 309-317
MSC (2000): Primary 53C42, 58E20
Published electronically: August 20, 2010
MathSciNet review: 2729093
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10657-3
PII: S 0002-9939(2010)10657-3
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.