On stable currents in positively pinched curved hypersurfaces

Volume 98 / 2003

Jintang Li Colloquium Mathematicum 98 (2003), 79-86 MSC: Primary 53C42; Secondary 58A25. DOI: 10.4064/cm98-1-6


Let $M^n\, (n\geq 3)$ be an $n$-dimensional complete hypersurface in a real space form $N(c)$ $(c\geq 0)$. We prove that if the sectional curvature $K_M$ of $M$ satisfies the following pinching condition: $c+\delta < K_M\leq c+1,$ where $\delta ={1\over 5}$ for $n\geq 4$ and $\delta ={1\over 4}$ for $n=3$, then there are no stable currents (or stable varifolds) in $M$. This is a positive answer to the well-known conjecture of Lawson and Simons.


  • Jintang LiDepartment of Mathematics
    Xiammen University
    361005 Xiammen
    Fujian, P.R. China

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