AUTHOR=Zhou Jundong 

TITLE=Rigidity of Complete Minimal Submanifolds in Spheres

JOURNAL=Frontiers in Physics

VOLUME=8

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=<p>Let <italic>M</italic> be an <italic>n</italic>-dimensional complete minimal submanifold in an (<italic>n</italic> + <italic>p</italic>)-dimensional sphere 𝕊<sup><italic>n</italic>+<italic>p</italic></sup>, and let <italic>h</italic> be the second fundamental form of <italic>M</italic>. In this paper, it is shown that <italic>M</italic> is totally geodesic if the <italic>L</italic><sup>2</sup> norm of |<italic>h</italic>| on any geodesic ball of <italic>M</italic> is of less than quadratic growth and the <italic>L</italic><sup><italic>n</italic></sup> norm of |<italic>h</italic>| on <italic>M</italic> is less than a fixed constant. Further, under only the latter condition, we prove that <italic>M</italic> is totally geodesic. Moreover, we provide a sufficient condition for a complete stable minimal hypersurface to be totally geodesic.</p>