| ABSTRACT:The size and the curveness of the surfaces in space, namely the volumes and curvatures of the geometrical objects is the most basic goal of study in differential geometry. The equalities or inequalities among volumes and curvatures also determine the properties of geometric elements. In differential geometry the study of the hypersurfaces in high dimensional Euclidean space is very active. Especially when E.Cartan established the moving frames method, the study of the differential geometry enter into a new era. More and more geometers are interested in the study of the curvatures of hypersurfaces in recent years. The second fundamental form of hypersurfaces reflect the degree of deviation from the surface near any point on the surface to the tangent plane of the point, namely it reflect the shape of the surfaces. By properly estimating the second fundamental form of the surface, we can understand its shape. P.Berard have proved when the second fundamental form of stable complete minimal hypersurfaces in R4(x:M∈R3→R4) satisfies some condition, then it is a hyperplane in R4; On the basis of the above theorem, Haizhong Li and Guoxin Wei estimate the second fundamental form of the stable complete minimal hypersurfaces in R4in more finer way. They conclusion is also true.We promote the result of Haizhog Li and Guoxin Wei to higher dimension in the weak sense. |
