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On stable currents and positively curved hypersurfaces
Author(s):
Yi-Bing
Shen;
Qun
He
Journal:
Proc. Amer. Math. Soc.
129
(2001),
237-246.
MSC (1991):
Primary 53C42;
Secondary 58A25
Posted:
June 7, 2000
MathSciNet review:
1784025
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Abstract:
We establish a nonexistence theorem for stable currents (or stable varifolds) in complete  -pinched hypersurfaces of a real space form with nonnegative constant sectional curvature. This is a partial positive answer to the well-known conjecture of Lawson and Simons.
References:
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- Aminov,J., On the instability of a minimal surface in an
 -dimensional Riemannian space of positive curvature, Math. USSR Sb., 29 (1976), 359-375. - [Ho]
- Howard,R., The nonexistence of stable submanifolds, varifolds, and harmonic maps in sufficiently pinched simply connected Riemannian manifolds, Michigan Math. J., 32 (1985), 321-334. MR 87h:58040
- [HW1]
- Howard,R. and Wei,S.W., On the existence and non-existence of stable submanifolds and currents in positively curved manifolds and the topology of submanifolds, preprint.
- [HW2]
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- [HPS]
- Hu,H.S., Pan,Y.L. and Shen,Y.B., On harmonic maps and a pinching theorem for positively curved hypersurfaces, Proc. Amer. Math. Soc., 99 (1987), 182-186. MR 87m:58042
- [LS]
- Lawson,H.B. and Simons,J., On stable currents and their applications to global problems in real and complex geometry, Ann. of Math., 98 (1973), 427-450. MR 48:2881
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- [SP]
- Shen,Y.B. and Pan,Y.L., On harmonic maps from ellipsoids, Acta Math. Scientia, 6 (1986), 71-75 (Chinese).
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Additional Information:
Yi-Bing
Shen
Affiliation:
Department of Mathematics, West-Brook Campus, Zhejiang University, Hangzhou 310028, People's Republic of China
Email:
ybshen@dial.zju.edu.cn
Qun
He
Affiliation:
Department of Mathematics, West-Brook Campus, Zhejiang University, Hangzhou 310028, People's Republic of China
DOI:
10.1090/S0002-9939-00-05753-1
PII:
S 0002-9939(00)05753-1
Keywords:
Stable currents,
$\delta $-pinched manifolds,
$\delta $-pinched hypersurfaces
Received by editor(s):
March 15, 1999
Posted:
June 7, 2000
Additional Notes:
This research was supported in part by NNSFC and NSFZP
Communicated by:
Jozef Dodziuk
Copyright of article:
Copyright
2000,
American Mathematical Society
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