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Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
Abstract
Let M be an n-dimensional complete connected self-shrinker in ℝn+p. Denote by Ric and H the Ricci curvature tensor and the mean curvature vector of M, respectively. We prove that if the self-shrinker M satisfies Ric≥((n−2)/n2+εn)|H|2g, then M is either a linear subspace or a round shrinking sphere, where εn is an explicit positive constant equal to 1/3n2+O(1/n5). Moreover, we obtain a rigidity theorem that is sharp in codimension two for the self-shrinker satisfying Ric≥(n−2)/n2|H|2g.
Related
- Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and S≤ 1
- Rigidity theorems of complete self-shrinkers in the Euclidean space
- Four-dimensional shrinkers with pinched curvatures
- Classification of quadratically pinched translators in higher codimension
- Complete Kähler Manifolds with pinched Ricci curvature
BibTeX
@misc{li2026rigidity,
title = {{Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension}},
author = {Juan Li and Zhiyuan Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06001},
eprint = {2609.06001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06001},
}
RIS
TY - UNPB
TI - Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
AU - Juan Li
AU - Zhiyuan Xu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06001
UR - https://arxiv.org/abs/2609.06001
AB - Let M be an n-dimensional complete connected self-shrinker in ℝ^n+p. Denote by Ric and H the Ricci curvature tensor and the mean curvature vector of M, respectively. We prove that if the self-shrinker M satisfies Ric≥((n−2)/n^2+ε_n)|H|^2g, then M is either a linear subspace or a round shrinking sphere, where ε_n is an explicit positive constant equal to 1/3n^2+O(1/n^5). Moreover, we obtain a rigidity theorem that is sharp in codimension two for the self-shrinker satisfying Ric≥(n−2)/n^2|H|^2g.
ER -