Global Rigidity Theorems For 2-dimensional Translating Solitons In The Euclidean Space

The translating soliton is a submanifold satisfying H=V⊥ in Euclidean space,where H is the mean curvature vector of the submanifold and V is a unit constant vector in Euclidean space.The translating soliton corresponds to a special solution of mean curvature flow and is the Type-II singularity model of mean curvature flow.Its geometric classification is very important for singularity analysis of mean curvature flow.Xin[32]proved a global rigidity theorem for n(≥3)-dimensional complete translating solition with respect to the second fundamental form.Wang-Xu-Zhao[24]proved a global rigidity theorem for n(≥3)-dimensional complete translating solition with respect to the trace-free second fundamental form.In this paper,we mainly study the global rigidity of 2-dimensional complete translating solitons in Euclidean space.We extend the theorems due to Xin[32]and WangXu-Zhao[24]to 2-dimensional translating solitions,and proved the following theoremsTheorem A:Let ∑ be a 2-dimensional complete translating solition of the mean curvature flow in R2+p.There is an absolute constant C1>0 such that if (?) then ∑ is a linear sub space of R2+p.Theorem B:Let ∑ be a 2-dimensional complete translating solition of the mean curvature flow in R2+p.There is an absolute constant C2>0 such that if (?) then ∑ is a linear sub space of R2+p.Remark:In the above theorems,C1>C2 for p≥2.