https://neckpinch.com/authors/zhaoping-li/New arXiv papers by Zhaoping Li, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1612.09024/arxiv:1612.09024Thu, 29 Dec 2016 00:00:00 GMTXingxiao Li, Zhaoping Li — ξ-submanifold in the Euclidean space bbr^(m+p) is a natural extension of the concept of self-shrinker to the mean curvature flow in bbr^(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 bbr^(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 (bbr^(m+p),e^(−fr|x|²m)lagl·,·ragl); Then, two weighted volume functionals V_ξ and bar V_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.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness