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Abstract

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.