A global pinching theorem of minimal hypersurfaces in the sphere

Abstract:

Let ${M^n} \subset {S^{n + 1}}(1)$ be a compact embedded minimal hypersurface in the sphere $(n \geq 3)$, and $\sigma$ the square of the length of the second fundamental form of ${M^n}$. Suppose ${M^n}$ has nonnegative Ricci curvature. Then there is a constant $A(n)$, depending only on $n$, such that if $||\sigma |{|_{n/2}} < A(n)$, then ${M^n}$ must be totally geodesic. Here $||\sigma |{|_K} = {(\int _M {{\sigma ^K}} )^{1/K}}$. It is related to the results of J. Simons [6] and S. T. Yau [9] about the minimal hypersurfaces in the sphere. For the case $n = 2$, we also have a similar discussion.