Pinched Problems Of Closed Hypersurfaces With Constant Mean Curvature In Unit Spheres

Constant mean curvature closed hypersurfaces in unit spheres are important in submanifold geometry.In 1968,mathematician Chern made a well-known conjecture,this great conjecture had not been fully resolved for nearly half a century.However,many mathematicians have studied this conjecture and obtained some results.This article focuses on issues related to Chern's conjecture.First,we consider Mn is a closed hypersurface with constant mean curvature H in Sn+1 and S be the square of the length of the second fundamental form of Mn.Respectively,without assuming or assuming that S is constant,the display expression of the squeeze constant is obtained,and two important pinch theorems are proved,then we can prove that M must be Clifford torus.These conclusions axe generalizations of the main results in[6]and[21].Secondly,we consider a closed minimal hypersurface in S5.In the case of S,f3 are constants,we get an important estimate' of f4 at the extreme point.The estimate will play a positive role in resolving the 4-dimensional Chern's conjecture.