{"paper":{"title":"Variational characterizations of $\\xi$-submanifolds in the Eulicdean space $\\bbr^{m+p}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Xingxiao Li, Zhaoping Li","submitted_at":"2016-12-29T02:28:33Z","abstract_excerpt":"$\\xi$-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 $\\lambda$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $\\xi$-submanifolds are established. First, it is shown that a submanifold in $\\bbr^{m+p}$ is a $\\xi$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $(\\bbr^{m+p},e^{-\\fr{|x|^2}{m}}\\lagl\\cdot,\\cdot\\ragl)$; Then, two weight"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09024","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}