RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

• Seo, Keom-Kyo
• Published : 2008.07.31
• 123 12

Abstract

Let $M^n$ be a complete immersed super stable minimal submanifold in $\mathbb{R}^{n+p}$ with fiat normal bundle. We prove that if M has finite total $L^2$ norm of its second fundamental form, then M is an affine n-plane. We also prove that any complete immersed super stable minimal submanifold with flat normal bundle has only one end.

Keywords

minimal submanifolds;Bernstein type theorem;flat normal bundle

References

1. H. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in $R^{n+1}$, Math. Res. Lett. 4 (1997), 637-644 https://doi.org/10.4310/MRL.1997.v4.n5.a2
2. S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer-Verlag 1970, 59-75
3. M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $R^{3}$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 903-906 https://doi.org/10.1090/S0273-0979-1979-14689-5
4. M. do Carmo, Stable complete minimal hypersurfaces, Proc. Beijing Symp. Differential Equations and Differential Geometry 3 (1980), 1349-1358
5. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211 https://doi.org/10.1002/cpa.3160330206
6. J. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure. Appl. Math. 26 (1973), 361-379 https://doi.org/10.1002/cpa.3160260305
7. R. Schoen and S.-T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature, Comment. Math. Helv. 51 (1976), 333-341 https://doi.org/10.1007/BF02568161
8. Y. Shen and X. Zhu, On stable complete minimal hypersurfaces in $R^{n+1}$, Amer. J. Math. 120 (1998), 103-116 https://doi.org/10.1353/ajm.1998.0005
9. K. Smoczyk, G. Wang, and Y. Xin, Bernstein type theorems with flat normal bundle, Calc. Var. PDE. 26 (2006), 57-67 https://doi.org/10.1007/s00526-005-0359-0
10. J. Spruck, On stable complete minimal hypersurfaces in $R^{n+1}$, Amer. J. Math. 120 (1998), 103-116 https://doi.org/10.1353/ajm.1998.0005
11. C. Terng, Submanifolds with flat normal bundle, Math. Ann. 277 (1987), 95-111 https://doi.org/10.1007/BF01457280
12. Q. Wang, On minimal submanifolds in an Euclidean space, Math. Nachr. 261/262 (2003), 176-180 https://doi.org/10.1002/mana.200310120
13. Y. Xin, Bernstein type theorems without graphic condition, Asian J. Math. 9 (2005), 31-44 https://doi.org/10.4310/AJM.2005.v9.n1.a3
14. P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), 1051-1063 https://doi.org/10.2307/2159628

Cited by

1. L2 harmonic 1-forms on minimal submanifolds in hyperbolic space vol.371, pp.2, 2010, https://doi.org/10.1016/j.jmaa.2010.05.048
2. Rigidity of minimal submanifolds with flat normal bundle vol.120, pp.4, 2010, https://doi.org/10.1007/s12044-010-0039-7
3. ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE vol.46, pp.6, 2009, https://doi.org/10.4134/BKMS.2009.46.6.1213
4. Bernstein type theorems for complete submanifolds in space forms vol.285, pp.2-3, 2012, https://doi.org/10.1002/mana.201000039