Surfaces in fourdimensional hyperkähler manifolds whose twistor lifts are harmonic sections
Author:
Kazuyuki Hasegawa
Journal:
Proc. Amer. Math. Soc. 139 (2011), 309317
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 selfdual Einstein manifolds whose twistor lifts are harmonic sections. We apply our main theorem to the case of fourdimensional hyperkähler manifolds. As a corollary, we prove that a surface of genus zero in fourdimensional 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 threedimensional 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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Additional Information
Kazuyuki Hasegawa
Affiliation:
Faculty of Teacher Education, Institute of Human and Social Sciences, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 9201192, Japan
Email:
kazuhase@staff.kanazawau.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299392010106573
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 GrantinAid 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.
