The Classification Theorems Of Lagrangian ξ-submanifolds And ξ-translators In C~2

In this paper,we aim to study classification problems of complete Lagrangian ξsubmanifolds and complete Lagrangian ξ-translators.As is known,both self-shrinkers and translators are very important to the mean curvature flow since they characterize,respectively,the Type Ⅰ and Type Ⅱ singularities of the solutions.Due to this,these two kinds of solitons have been extensively studied,and a number of important and interesting results have been obtained,including some classification theorems,rigidity theorems and Bernstein-type theorems.As the natural generalization of the concept of self-shrinkers,Qing-Ming Cheng and Guoxin Wei introduced the concept of λ-hypersurfaces and studied the variation problem and the stability property of λ-hypersurfaces.A few years ago,Xingxiao Li et al.introduced the concept of ξ-submanifolds,which naturally generalize self-shrinkers and λ-hypersurfaces with higher codimension,and systematically investigated the variational characterization and the W-stability of the general ξ-submanifolds in Rm+p.In this paper,we naturally generalize the concept of translators to ξ-translators and study classification problems of Lagrangian ξ-submanifolds and Lagrangian ξ-translators in the complex 2-plane C2,respectively.In particular,we divide this paper into three chapters.The details are as follows:In the first chapter,we introduce the definition of ξ-translators and introduce the background of ξ-submanifolds,ξ-translators and Lagrangian submanifolds.We also explain the importance of Lagrangian ξ-submanifolds and Lagrangian ξ-translators.In the second chapter,we study classification problems of complete Lagrangian ξsubmanifolds and complete Lagrangian ξ-translators in the complex 2-plane C2.As the result,we first prove a classification theorem of all complete Lagrangian ξ-submanifolds in C2 of constant square norm of the second fundamental form(see Theorem 2.2).Then,the main idea of the proof also allows us to obtain a similar classification theorem for complete Lagrangian ξ-translators.In fact,we first prove that the plane is the only complete Lagrangian translator in C2 with constant square norm of the second fundamental form(see Theorem 2.3).On the basis of this,we can prove a more general classification theorem for complete Lagrangian ξ-translators in C2(see Theorem 2.4).In the third chapter,we study classification problems of complete space-like Lagrangian ξ-submanifolds and complete space-like Lagrangiant-translators in R24,the pseudo-Euclidean 4-spaces of signature 2 endowed with the canonical complex structure.As the result,we first prove a classification theorem for all complete space-like Lagrangianξ-submanifolds in R24 of constant square norm of the second fundamental form(see Theorem 3.1).Correspondingly,we obtain a classification theorem for complete space-like Lagrangian ξ-translators in R24(see Theorem 3.3)by proving a more general Bernsteintype theorem for complete space-like translators in the pseudo-Euclidean space Rpm+p(see Theorem 3.2),which is also of independent significance.