Global pinching theorems for minimal submanifolds in spheres

Kairen Cai. "Global pinching theorems for minimal submanifolds in spheres." Colloquium Mathematicae 96.2 (2003): 225-234. <http://eudml.org/doc/284521>.

@article{KairenCai2003,
abstract = {Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^\{n+p\}(1)$. By using the Sobolev inequalities of P. Li to get $L_\{p\}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and $||σ||_\{p\}$ the mean curvature and the $L_\{p\}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if $||σ||_\{n/2\} < C$, then M is a minimal submanifold in the sphere $S^\{n+p-1\}(1+H²)$ with sectional curvature 1+H².},
author = {Kairen Cai},
journal = {Colloquium Mathematicae},
keywords = {Sobolev inequality; mean curvature; minimal submanifold},
language = {eng},
number = {2},
pages = {225-234},
title = {Global pinching theorems for minimal submanifolds in spheres},
url = {http://eudml.org/doc/284521},
volume = {96},
year = {2003},
}

TY - JOUR
AU - Kairen Cai
TI - Global pinching theorems for minimal submanifolds in spheres
JO - Colloquium Mathematicae
PY - 2003
VL - 96
IS - 2
SP - 225
EP - 234
AB - Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n+p}(1)$. By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and $||σ||_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if $||σ||_{n/2} < C$, then M is a minimal submanifold in the sphere $S^{n+p-1}(1+H²)$ with sectional curvature 1+H².
LA - eng
KW - Sobolev inequality; mean curvature; minimal submanifold
UR - http://eudml.org/doc/284521
ER -