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On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle
Author(s):
Qun
He;
Yi-Bing
Shen
Journal:
Proc. Amer. Math. Soc.
134
(2006),
871-880.
MSC (2000):
Primary 53C60;
Secondary 53B40
Posted:
July 19, 2005
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Additional information
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  in a Randers space is just that for the Riemannian metric  , and therefore the Bernstein type theorem in the special Randers space of dimension  is true. Moreover, a Bernstein type theorem in the  -dimensional Minkowski space is established by considering the volume form induced from the projective sphere bundle.
References:
-
- [1]
- D. Bao, S.S. Chern, A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. of Math., 143 (1996), 1-20. MR 1381986 (97d:53071)
- [2]
- D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, GTM 200, Springer-Verlag, 2000. MR 1747675 (2001g:53130)
- [3]
- S.S. Chern, Riemannian geometry as a special case of Finsler geometry, Contemporary Math., 196, Amer. Math. Soc., Providence, 1996, 51-58. MR 1403576 (98e:53026)
- [4]
- M. Dajczer, Submanifolds and Isometric Immersions, Math. Lect. Ser., No. 13, Publish or Perish, Inc., Houston, Texas, 1990. MR 1075013 (92i:53049)
- [5]
- P. Dazord, Tores Finslériens sans points conjugués, Bull. Soc. Math. France, 99(1971), 171-192. MR 0309037 (46:8148)
- [6]
- M. do Carmo, C.K. Peng, Stable complete minimal surfaces in
 are planes, Bull. AMS, 1(1979), 903-905. MR 0546314 (80j:53012) - [7]
- Q. He, Y.B. Shen, On mean curvature of Finsler submanifolds, Preprint.
- [8]
- H. Rund, The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. MR 0105726 (21:4462)
- [9]
- Z. Shen, On Finsler geometry of submanifolds, Math. Ann., 311(1998), 549-576. MR 1637939 (99j:53095)
- [10]
- -, Lectures on Finsler geometry, World Sci., 2001, Singapore. MR 1845637 (2002f:53032)
- [11]
- L. Simon, Equations of mean curvature type in
 independent variables, Pac. J. Math., 69(1977), 245-268. MR 0454854 (56:13099) - [12]
- J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88(1968), 62-105. MR 0233295 (38:1617)
- [13]
- M. Souza, J. Spruck, K. Tenenblat, A Bernstein type theorem on a Randers space, Math. Ann., 329(2004), 291-305. MR 2060364 (2005c:53095)
- [14]
- M. Souza, K. Tenenblat, Minimal surfaces of rotation in a Finsler space with a Randers metric, Math.Ann., 325 (2003), 625-642. MR 1974561 (2003m:53131)
- [15]
- Y.B. Shen, Y. Zhang, The second variation of harmonic maps between Finsler manifolds, Science in China, Ser.A, 47 (2004), 39-51. MR 2054666 (2005b:58021)
- [16]
- Y.B. Shen, X.H. Zhu, On stable complete minimal hypersurfaces in
 , Amer. J. Math., 120(1998), 103-116. MR 1600268 (99c:58040) - [17]
- A.C. Thompson, Minkowski Geometry, Encyclopedia of Math. and its Appl., 63, Cambridge Univ. Press, Cambridge, 1996. MR 1406315 (97f:52001)
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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:
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
Posted:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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