2016/12/29 by Xingxiao Li, Li, Xingxiao, Li Zhaoping +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #Primary 53A30 #Secondary 53B25 #math.DG #msc:53A30 #msc:53B25
paper · pdf · doi:10.48550/arxiv.1612.09024
23 pages; submitted in a journal
arxiv created 2016/12/29 · openalex publication_date 2016/12/29 · arxiv updated 2016/12/30 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
ξ-submanifold in the Euclidean space \bbrm+p is a natural extension of the concept of self-shrinker to the mean curvature flow in \bbrm+p. It is also a generalization of the λ-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for ξ-submanifolds are established. First, it is shown that a submanifold in \bbrm+p is a ξ-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space (\bbrm+p,e^-\fr|x|2m\lagl⋅,⋅\ragl); Then, two weighted volume functionals Vξ and Vξ are introduced and it is proved that ξ-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the (W-)stability properties for ξ-submanifolds are systematically studied. In particular, it is proved that m-planes are the only properly immersed, complete W-stable ξ-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.