TY - JOUR
T1 - Remarks on Gap Theorems for Complete Hypersurfaces with Constant Scalar Curvature
AU - Bai , Jin-Chuan
AU - Luo , Yong
JO - Journal of Mathematical Study
VL - 3
SP - 279
EP - 290
PY - 2023
DA - 2023/07
SN - 56
DO - http://doi.org/10.4208/jms.v56n3.23.02
UR - https://global-sci.org/intro/article_detail/jms/21873.html
KW - hypersurfaces, constant scalar curvature, gap theorem.
AB - <p style="text-align: justify;">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 $$\left(\int_M|H|^ndv\right)^\frac{1}{n}&lt;\alpha$$ 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 $R=n(n-1)$.</p>