The Rigidity Theorems Of ξ-Submanifolds And λ-Spacelike Hypersurfaces

The study of self-shrinker has been extensive recently,especially on the rigidity prob-lem.As generalizations of self-shrinker,in this dissertation we introduce for the first time the ξ-submanifolds and λ-spacelike hypersurfaces,mainly paying attention to the rigidity theorems for the ξ-submanifolds as well for the generalized λ-spacelike hypersurfaces.In Chapter 1,we introduce the definition of ξ-submanifold and study the Lagrangian ξ-submanifold in C2.As the result,we obtain the following theorem:Theorem 1.1.Let x:M2→C2 be a compact orientable Lagrangian ξ-submanifold. Assume that |h|2+|H-ξ|2≤|ξ|2+4.Then|h|2+|H-ξ|2≡|ξ|2+4 and x(M2)=T2 is a topological torus.Furthermore if〈H,ξ〉is constant and one of the following four conditions holds:(1)|h|2≥2,(2)|H|2≥2,(3)|h|2≥〈H,H-ξ〉,(4)〈H,ξ〉≥0, then,up to a holomorphic isometry on C2,x(M2)=S1(a)×S1(b)is a standard torus. where a and b are positive numbers satisfying a2+b2≥2a2b2.In Chapter 2,we introduce the definition of generalized λ-hypersurface weighted by a function s,prove the following theorem:Theorem 2.2.Let x:Mn→R1n+1 be a complete space-like λ-hypersurface with weight s=∈a. Suppose that〈x,x〉does not change its sign and where the differential operator L is defined by(2.12).Then,either x is totally umbilical and thus isometric to one of the following two hypersurfaces:(1) the hyperbolic space Hn(c)(?)R1n+1 with a sectional curvature c<0;(2) the Euclidean space Rn(?)R1n+1 or,there exists some p∈Mn such that,at pTheorem 2.3. Let x:Mn→R1n+1 be a complete space-like λ-hypersurface with s=〈x,x〉.Suppose that where the differential operator L is defined by(2.22).Then,either x is totally umbilical and thus isometric to one of the following two embedded hypersurfaces:(1) the hyperbolic space Hn(c)(?)R1n+1 with c<0;(2) the Euclidean space Rn(?)R1n+1 or,there exists some p∈Mn such that...