AUTHOR=Zhou Jundong 

TITLE=Rigidity of Complete Minimal Submanifolds in Spheres

JOURNAL=Frontiers in Physics

VOLUME=Volume 8 - 2020

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.571250

DOI=10.3389/fphy.2020.571250

ISSN=2296-424X

ABSTRACT=Let $M$ be an $n$-dimensional complete minimal submanifold in an
$(n+p)$-dimensional sphere $S^{n+p}$ and $h$ be the second
fundamental form of $M$. In this paper, it is shown that $M$ is
totally geodesic if the $L^2$ norm of $|h|$ on any geodesic ball of
$M$ has less than quadratic growth and  $L^n$ norm of $|h|$ on $M$
is less than a fixed constant. Farther, deleting the condition of
the $L^2$ norm of $|h|$, we show that $M$ is totally geodesic if
$L^n$ norm of $|h|$ on $M$ is less than a fixed constant.
Furthermore, we provide  a sufficient condition such that a complete
stable minimal hypersurface is totally geodesic.