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Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in ${ℝ}^{n+p}$ with flat normal bundle. We prove that if the second fundamental form A of M satisfies ${\int }_{M}i{\left|A\right|}^{\alpha }<\infty$, where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and ${\int }_M{\left|A\right|}^d<\infty$, 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 ℝ⁷ are considered.