发表机构
Chongqing University of Technology; Osaka Metropolitan University; South China Normal University（重庆理工大学; 大阪公立大学; 华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Chern猜想，在中间问题（若S>n则S>n+1/2 n）上取得进展：对4≤n≤13给出肯定解，并对一般n证明S>1.4556737869 n，优于Chen的结果。
AI 中文摘要
设 $X: M^n\to \mathbb S^{n+1}(1)$ 为具有常数量曲率的闭极小超曲面。Chern猜想指出，对于具有常数量曲率的闭极小超曲面，若 $S>n$，则 $S\geq 2n$，其中 $S$ 为第二基本形式的模长平方。作为部分结果，Peng-Terng \cite{pt1, pt2}、Yang-Cheng \cite{yc1, yc2, yc3} 和 Suh-Yang \cite{sy} 得到：若 $S>n$，则 $S>n+\dfrac{3}{n}$。最近，Chen \cite{c} 证明了若 $S>n$，则 $S>n+\frac{26}{59}n$。由于解决Chern猜想是一个困难问题，作为解决该猜想的中间步骤，人们希望证明若 $S>n$，则 $S>n+\frac{1}{2}n$。在本文中，对于 $4\leq n\leq 13$，我们给出了这一中间问题的肯定解。对于一般的 $n$，我们证明：对于具有常数量曲率的完备极小超曲面，若 $S>n$，则 \\[ S>\frac{10000000}{6869671}\\,n >1.4556737869\\,n>n+\dfrac{221}{485}n \\]，这优于Chen \cite{c} 的结果。
英文摘要
Let $X: M^n\to \mathbb S^{n+1}(1)$ be a closed minimal hypersurface with constant scalar curvature. Chern conjecture says that a closed minimal hypersurface with constant scalar curvature, $S\geq 2n$ if $S>n$, where $S$ is the squared norm of the second fundamental form. As a partial result, Peng-Terng \cite{pt1, pt2}, Yang-Cheng \cite{yc1, yc2, yc3} and Suh-Yang \cite{sy} obtained that if $S>n$, then $S>n+ \dfrac{3}n$. Recently, Chen \cite{c} has proved $S>n+\frac{26}{59}n$ if $S>n$. Since resolving Chern conjecture is a hard problem, as a midway for resolving Chern conjecture, one wants to prove $S>n+\frac12 n$ if $S>n$. In this paper, for $4\leq n\leq 13$, we give a positive solution for this midway problem. For general $n$, we prove that for a complete minimal hypersurface with constant scalar curvature, \[ S>\frac{10000000}{6869671}\,n >1.4556737869\,n>n+\dfrac{221}{485}n \] if $S>n$, which is better than the result of Chen \cite{c}.
