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

 

On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle


Authors: Qun He and Yi-Bing Shen
Journal: Proc. Amer. Math. Soc. 134 (2006), 871-880
MSC (2000): Primary 53C60; Secondary 53B40
Published electronically: July 19, 2005
MathSciNet review: 2180905
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Abstract: By using the volume form induced from the projective sphere bundle of the Finsler manifold, we study the Finsler minimal submanifolds. It is proved that such a volume form for the Randers metric $F=\alpha +\beta$ in a Randers space is just that for the Riemannian metric $\alpha$, and therefore the Bernstein type theorem in the special Randers space of dimension $\leq 8$ is true. Moreover, a Bernstein type theorem in the $3$-dimensional Minkowski space is established by considering the volume form induced from the projective sphere bundle.


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Additional Information

Qun He
Affiliation: Department of Applied Mathematics, Tongji University, Shanghai 200092, People’s Republic of China
Email: hequn@mail.tongji.edu.cn

Yi-Bing Shen
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310028, People’s Republic of China
Email: yibingshen@zju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08017-2
PII: S 0002-9939(05)08017-2
Keywords: Finsler volume form, minimal surface, Randers space, Minkowski space
Received by editor(s): June 4, 2004
Received by editor(s) in revised form: October 13, 2004
Published electronically: July 19, 2005
Additional Notes: The first author was supported in part by NNSFC (no.10471105).
The second author was supported in part by NNSFC (no.10271106).
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.