On Chern's conjecture for minimal hypersurfaces and rigidity of self-shrinkers

Using the generalized Yau parameter method and the Sylvester theory, we verify that if M is a compact minimal hypersurface in Sn+1 whose squared length of the second fundamental form satisfies 0≤|A|2−n≤n22, then |A|2≡n and M is a Clifford torus. Moreover, we prove that if M is a complete self-shrinker with polynomial volume growth in Rn+1, and if the squared length of the second fundamental form of M satisfies 0≤|A|2−1≤121, then |A|2≡1 and M is a round sphere or a cylinder. Our results improve the rigidity theorems due to Ding and Xin [21], [22].