Remarks on Gap Theorems for Complete Hypersurfaces with Constant Scalar Curvature (Journal of Mathematical Study, 2023)
Assume that $M^n(n≥3)$ 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 $$-P_1( abla H, abla|H|)≤-δ|H|| abla H|^2$$ for some positive constant $δ$, where $P_1=nHg-B$ which denotes the first order Newton transformation, and $$\left(\int_M|H|^ndv\right)^\frac{1}{n}