On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle
HTML articles powered by AMS MathViewer
- by Qun He and Yi-Bing Shen PDF
- Proc. Amer. Math. Soc. 134 (2006), 871-880 Request permission
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.References
- David Bao and S. S. Chern, A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. of Math. (2) 143 (1996), no. 2, 233–252. MR 1381986, DOI 10.2307/2118643
- D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675, DOI 10.1007/978-1-4612-1268-3
- Shiing-shen Chern, Riemannian geometry as a special case of Finsler geometry, Finsler geometry (Seattle, WA, 1995) Contemp. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 51–58. MR 1403576, DOI 10.1090/conm/196/02429
- Marcos Dajczer, Submanifolds and isometric immersions, Mathematics Lecture Series, vol. 13, Publish or Perish, Inc., Houston, TX, 1990. Based on the notes prepared by Mauricio Antonucci, Gilvan Oliveira, Paulo Lima-Filho and Rui Tojeiro. MR 1075013
- Pierre Dazord, Tores finslériens sans points conjugués, Bull. Soc. Math. France 99 (1971), 171–192; erratum, ibid. 99 (1971), 397 (French). MR 309037, DOI 10.24033/bsmf.1715
- M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $\textbf {R}^{3}$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903–906. MR 546314, DOI 10.1090/S0273-0979-1979-14689-5
- Q. He, Y.B. Shen, On mean curvature of Finsler submanifolds, Preprint.
- Hanno Rund, The differential geometry of Finsler spaces, Die Grundlehren der mathematischen Wissenschaften, Band 101, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959. MR 0105726, DOI 10.1007/978-3-642-51610-8
- Zhongmin Shen, On Finsler geometry of submanifolds, Math. Ann. 311 (1998), no. 3, 549–576. MR 1637939, DOI 10.1007/s002080050200
- Zhongmin Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. MR 1845637, DOI 10.1142/9789812811622
- Leon Simon, Equations of mean curvature type in $2$ independent variables, Pacific J. Math. 69 (1977), no. 1, 245–268. MR 454854, DOI 10.2140/pjm.1977.69.245
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- Marcelo Souza, Joel Spruck, and Keti Tenenblat, A Bernstein type theorem on a Randers space, Math. Ann. 329 (2004), no. 2, 291–305. MR 2060364, DOI 10.1007/s00208-003-0500-3
- Marcelo Souza and Keti Tenenblat, Minimal surfaces of rotation in Finsler space with a Randers metric, Math. Ann. 325 (2003), no. 4, 625–642. MR 1974561, DOI 10.1007/s00208-002-0392-7
- Yibing Shen and Yan Zhang, Second variation of harmonic maps between Finsler manifolds, Sci. China Ser. A 47 (2004), no. 1, 39–51. MR 2054666, DOI 10.1360/03ys0040
- Yi-Bing Shen and Xiao-Hua Zhu, On stable complete minimal hypersurfaces in $\textbf {R}^{n+1}$, Amer. J. Math. 120 (1998), no. 1, 103–116. MR 1600268, DOI 10.1353/ajm.1998.0005
- A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 871-880
- MSC (2000): Primary 53C60; Secondary 53B40
- DOI: https://doi.org/10.1090/S0002-9939-05-08017-2
- MathSciNet review: 2180905