scalar curvature of minimal hypersurfaces in a sphere

We first extend the well-known scalar curvature pinching theorem due to Peng-Terng, and prove that if $M$ a closed minimal hypersurface in $S^{n+1}$ $(n=6,7)$, then there exists a positive constant $\delta(n)$ depending only on $n$ such that if $n\leq S\leq n+\delta(n)$, then $S \equiv n$, i.e., $M$ is one of the Clifford torus $S^{k}(\sqrt{\frac{k}{n}})\times S^{n-k}(\sqrt{\frac{n-k}{n}}), k=1,2,...,n-1$. Secondly, we point out a mistake in Ogiue and Sun’s paper in which they claimed that they had solved the open problem proposed by Peng and Terng.

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Published 1 January 2007