The gap theorems for some extremal submanifolds in a unit sphere

Xi Guo and Lan Wu

Communications in Mathematics (2015)

  • Volume: 23, Issue: 1, page 85-93
  • ISSN: 1804-1388

Abstract

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Let M be an n -dimensional submanifold in the unit sphere S n + p , we call M a k -extremal submanifold if it is a critical point of the functional ∫ M ρ 2 k d v . In this paper, we can study gap phenomenon for these submanifolds.

How to cite

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Wu, Xi Guo and Lan. "The gap theorems for some extremal submanifolds in a unit sphere." Communications in Mathematics 23.1 (2015): 85-93. <http://eudml.org/doc/271655>.

@article{Wu2015,
abstract = {Let $M$ be an $n$-dimensional submanifold in the unit sphere $S^\{n+p\}$, we call $M$ a $k$-extremal submanifold if it is a critical point of the functional $\int _M\rho ^\{2k\}\,\mathrm \{d\}v $. In this paper, we can study gap phenomenon for these submanifolds.},
author = {Wu, Xi Guo and Lan},
journal = {Communications in Mathematics},
keywords = {Extremal functional; Mean curvature; Totally umbilical},
language = {eng},
number = {1},
pages = {85-93},
publisher = {University of Ostrava},
title = {The gap theorems for some extremal submanifolds in a unit sphere},
url = {http://eudml.org/doc/271655},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Wu, Xi Guo and Lan
TI - The gap theorems for some extremal submanifolds in a unit sphere
JO - Communications in Mathematics
PY - 2015
PB - University of Ostrava
VL - 23
IS - 1
SP - 85
EP - 93
AB - Let $M$ be an $n$-dimensional submanifold in the unit sphere $S^{n+p}$, we call $M$ a $k$-extremal submanifold if it is a critical point of the functional $\int _M\rho ^{2k}\,\mathrm {d}v $. In this paper, we can study gap phenomenon for these submanifolds.
LA - eng
KW - Extremal functional; Mean curvature; Totally umbilical
UR - http://eudml.org/doc/271655
ER -

References

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  3. Kenmotsu, K., 10.2748/tmj/1178242819, Tohoku. Math. J., 22, 1970, 240-248, (1970) Zbl0202.21003MR0268791DOI10.2748/tmj/1178242819
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  6. Li., H., 10.4310/MRL.2002.v9.n6.a6, Math. Res. Letters, 9, 2002, 771-790, (2002) MR1906077DOI10.4310/MRL.2002.v9.n6.a6
  7. Simons., J., 10.2307/1970556, Ann. of Math., 88, 1968, 62-105, (1968) MR0233295DOI10.2307/1970556
  8. Xu, H.-W., Yang., D., The gap phenomenon for extremal submanifolds in a Sphere, Differential Geom and its Applications, 29, 2011, 26-34, (2011) MR2784286
  9. Wu., L., A class of variational problems for submanifolds in a space form, Houston J. Math., 35, 2009, 435-450, (2009) MR2519540

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