Let $\, \overline {\! M}$ be a compact Riemannian manifold with sectional curvature $K_{\, \overline {\! M}}$
satisfying $1/5 K_{\, \overline {\! M}}\le 1$ (resp. $2\le K_{\, \overline {\! M}}10$), which can be
isometrically immersed as a hypersurface in the Euclidean space (resp. the unit Euclidean sphere). Then there exist no stable compact minimal submanifolds in $\, \overline {\! M}$. This extends Shen and Xu's result for ${1\over 4}$-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson–Simons conjecture.
@article{10_4064_cm96_2_6,
author = {Ze-Jun Hu and Guo-Xin Wei},
title = {On the nonexistence of stable minimal submanifolds
and the {Lawson{\textendash}Simons} conjecture},
journal = {Colloquium Mathematicum},
pages = {213--223},
year = {2003},
volume = {96},
number = {2},
doi = {10.4064/cm96-2-6},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm96-2-6/}
}
TY - JOUR
AU - Ze-Jun Hu
AU - Guo-Xin Wei
TI - On the nonexistence of stable minimal submanifolds
and the Lawson–Simons conjecture
JO - Colloquium Mathematicum
PY - 2003
SP - 213
EP - 223
VL - 96
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4064/cm96-2-6/
DO - 10.4064/cm96-2-6
LA - en
ID - 10_4064_cm96_2_6
ER -
%0 Journal Article
%A Ze-Jun Hu
%A Guo-Xin Wei
%T On the nonexistence of stable minimal submanifolds
and the Lawson–Simons conjecture
%J Colloquium Mathematicum
%D 2003
%P 213-223
%V 96
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4064/cm96-2-6/
%R 10.4064/cm96-2-6
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Ze-Jun Hu; Guo-Xin Wei. On the nonexistence of stable minimal submanifolds
and the Lawson–Simons conjecture. Colloquium Mathematicum, Tome 96 (2003) no. 2, pp. 213-223. doi: 10.4064/cm96-2-6
