Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu¹

¹ Department of Mathematics Nanchang University 330031 Nanchang, P.R. China

Annales Polonici Mathematici, Tome 116 (2016) no. 2, pp. 145-154

Résumé

Let $M^n$ $(n\geq 3)$ be an $n$-dimensional complete super stable minimal submanifold in $\mathbb {R}^{n+p}$ with flat normal bundle. We prove that if the second fundamental form $A$ of $M$ satisfies $\int _M|A|^\alpha \infty $, where $\alpha \in [2(1-\sqrt {2/n}), 2(1+\sqrt {2/n})]$, then $M$ is an affine $n$-dimensional plane. In particular, if $n\leq 8$ and $ \int _{M}|A|^d\infty $, $d=1,3,$ then $M$ is an affine $n$-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^\alpha $-norm curvature in $\mathbb {R}^{7}$ are considered.

Détail

DOI : 10.4064/ap3743-12-2015

Keywords: geq n dimensional complete super stable minimal submanifold mathbb flat normal bundle prove second fundamental form satisfies int alpha infty where alpha sqrt sqrt affine n dimensional plane particular leq int infty affine n dimensional plane moreover complete strongly stable hypersurfaces constant mean curvature finite alpha norm curvature mathbb considered