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Assume that $M^n(n\geq3)$ is a complete hypersurface in $\mathbb{R}^{n+1}$ with zero scalar curvature. Assume that $B, H, g$ is the second fundamental form, the mean curvature and the induced metric of $M$, respectively. We prove that $M$ is a hyperplane if $$-P_1(\nabla H,\nabla|H|)\leq-\delta|H||\nabla H|^2$$ for some positive constant $\delta$, where $P_1=nHg-B$ which denotes the first order Newton transformation, and $$\left(\int_M|H|^ndv\right)^\frac{1}{n}<\alpha$$ for some small enough positive constant $\alpha$ which depends only on $n$ and $\delta$. We also derive similar result for complete hypersurfaces in $\mathbb{S}^{n+1}$ with constant scalar curvature $R=n(n-1)$.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v56n3.23.02}, url = {http://global-sci.org/intro/article_detail/jms/21873.html} }Assume that $M^n(n\geq3)$ is a complete hypersurface in $\mathbb{R}^{n+1}$ with zero scalar curvature. Assume that $B, H, g$ is the second fundamental form, the mean curvature and the induced metric of $M$, respectively. We prove that $M$ is a hyperplane if $$-P_1(\nabla H,\nabla|H|)\leq-\delta|H||\nabla H|^2$$ for some positive constant $\delta$, where $P_1=nHg-B$ which denotes the first order Newton transformation, and $$\left(\int_M|H|^ndv\right)^\frac{1}{n}<\alpha$$ for some small enough positive constant $\alpha$ which depends only on $n$ and $\delta$. We also derive similar result for complete hypersurfaces in $\mathbb{S}^{n+1}$ with constant scalar curvature $R=n(n-1)$.