The Umbilical Characteristics Of Hypersurfaces In Conformal Flat Manifolds

In this paper,the rigid classification problems of hypersurfaces in conformal flat Riemannian manifolds are studied.Under the condition that the squared length of the second fundamental form of hypersurfaces has a positive upper bound,we ob-tained the gap theorem or rigid theorem of the complete hypersurface with constant mean curvature,or constalt scalar curvature,or be a linear Weingarten hypersurface in conformal flat Riemannian manifolds.The specific contents obtained in this paper include the following four sections:The first section is theoretical preparation,it mainly introduced the basic for-mulas of submanifold geometry,the Omori-Yau maximum principle and the algebra inequalities used in the proofe of the main theorems.In the second section,we study the umbilical characteristics of the complete hy-persurface with constant mean curvature in conformal flat Riemannian manifold-s.Applying the Laplace operator to the total umbilical tensor on the hypersurface,using the generalized Okumura type inequality and combining with the Omori-Yau maximum principle,we obtain the umbilical results of this type of hypersurface.In the third section,we study the umbilical characteristics of the complete hypersurface with constant scalar curvature in conformal flat Riemannian man-ifolds.Applying the famous Cheng-Yau box operator to the mean curvature of the hypersurface,estimating the results by using the generalized Okurmura type in-equality,under the condition of the squared length of the second fundamental form bounded from above,then the rigid classification theorem is obtained,which similar to the case of constant mean curvature.In the fourth section,we study the umbilical characteristics of the linear Wein-garten complete hypersurface in conformal flat Riemannian manifolds.Another new elliptic difEerential operator is defined based on the Cheng-Yau box operator.Appling it to the mean curvature of the hypersurface,combining with the gener-alized Okumura type inequality and maximum principle,we finally obtain the rigid classification and gap theorem.