The Local Geometry Of Self-shrinkers

Let x:Mn→Rn+P be an n-dimensional submanifold in n+p-dimensional Euclidean space Rn+P. Let x= x⊥+xT, where T is the projection onto the tangent bun del of M and ⊥ is the projection onto the normal bundel of M. A submanifold x:Mn→Rn+P is said to be a self-shrinker in Rn+P if it satisfies x⊥=-H,where H is the mean curvature vector. We mainly study self-shrinker of mean curvature flow in n+p-dimensional Euclidean space Rn+p.In the first part of this paper,we get the self-shrinker equations of graph of funcation. The second part is the main contect of this paper, we discuss the classification of self-shrinker.For arbitrary dimension self-shrinker if xT= 0, then Mn is the minimal submanifold of Sn+p-1(?)Rn+P. We prove that if Mn is compact and H is parallel (▽⊥H= 0)then Mn is the minimal submanifold of Sn+p-1(?)Rn+P. We also prove that if Mn is compact and |x|2≤n(or|x|2≥n), then Mn is the minimal submanifold of Sn+p-1(?)Rn+P. The above results generalize the known related results in the hy-persurface. At last, we get the local intergrable condition of self-shrinker in Rn+1, As application, we prove that if xT|Up≠0, then |x|2 only depends on the integral curve of x⊥|Up.