Variational characterizations of $$\xi $$-submanifolds in the Eulicdean space $${{\mathbb {R}}}^{m+p}$$

Abstract

ξ-submanifold in the Euclidean space R m+p is a natural extension of the concept of selfshrinker to the mean curvature flow in R m+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 R m+p is a ξ-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space (R m+p , e − |x| 2 m ·, · ); Then, two weighted volume functionals V ξ andV ξ 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.2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25.

A Bernstein-type theorem for -submanifolds with flat normal bundle in the Euclidean spaces

“…The λ -hypersurfaces can be characterized as having constant weighted mean curvatureH = e − |X| 2 2nH , whereH is the mean curvature of M n in Gaussian metric space. As Li and Li [10] pointed out, if self-shrinkers and λ -hypersurfaces take the places of minimal submanifolds and hypersurfaces with constant mean curvature, respectively, then ξ -submanifolds are expected to take the place of submanifolds with parallel mean curvature vector. Therefore, the research of ξ -submanifolds is interesting and significant.…”

“…In 2016, Li and Chang [9] derived a rigidity theorem for Lagrangian ξ -submanifolds in the complex 2-plane C 2 . Li and Li [10] gave some characterizations for ξ -submanifolds. They showed that a submanifold in R m+p is a ξ -submanifold if and only if its modified mean curvatureH = e − |X| 2 2nH is parallel when it is viewed as a submanifold in the Gaussian space (R n+p , e − |x| 2 n δ AB ) .…”

“…Typical examples of ξ -submanifolds include the ξ -curves, the standard spheres centered at the origin, submanifolds in a sphere with parallel mean curvature vector, and so on (cf. [10]). For convenience, here we give the details of the n -planes not necessarily passing through the origin.…”

“…They showed that a submanifold in R m+p is a ξ -submanifold if and only if its modified mean curvatureH = e − |X| 2 2nH is parallel when it is viewed as a submanifold in the Gaussian space (R n+p , e − |x| 2 n δ AB ) . In [10], the authors also proved that any complete and properly immersed ξ -submanifold with a normal bundle must be an n -plane if it is W-stable.…”

“…10) where we use ∇ ⊥ ξ = 0 . Hence, it follows from (3.10) that∑ ∧ • • • ∧ e k−1 ∧ ν α ∧ e k+1 ∧ • • • ∧ e n , ζ⟩…”