Characterizations And Stability Of ξ-Submanifolds

We aim to give more characterizations of the ξ-submanifolds basing on the self-shrinker and λ-hypersurfaces. The present dissertation consists of three parts. In part one,we provide two characterizations of ξ-submanifolds in Rm+p; In part two, we investigate the stability of ξ-submanifolds in Rm+p; In part, three, we research the rigidity of ξ—submanifolds in C2.The dissertation is divided into three chapters. The organizing structure is as follows:Chapter One is an introduction consisting of two sections and presents the back-ground and the main contents.Chapter Two firstly gives two characterizations of ξ-submanifolds, one establishs the equivalence between ξ-submanifolds and the submanifolds with parallel mean curvaturevector in the Gaussian space (Rm+p,e-|x|2/m<·,·>) (theorem 1.2), the other proves that ξ-submanifolds are the critical points of two weighted volume functionalsVξ andVξ (theorem 1.3). Then, we systematically research the (W-)stability properties of ξ-submanifolds by calculating the second variation formulas of the weighted volume functionals. As the main result, it is proved that if x : Mm→Rm+p be a complete and properly immersedξ-submanifold with flat normal bundle, then as a critical point of Vw under VP-variations(the conditional critical point), x is W-stable if and only if x(Mm) is an m-plane.(theorem 1.4).Chapter Three introduces the rigidity of ξ-submanifolds in C2. Basing on the rigidity theorems of self-shrinker in C2 generalizes to the ξ-submanifolds. It is proved that x(M2)is either a Lagrangian surface or a plane if ∫M|h|2e-|x|2/2dVM<∞ and the K?hler augle θsatisfies some additional conditions(theorem 1.6 and 1.7).