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Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form
Abstract
Let X:M3→R4 be a complete, connected self-shrinker without boundary, and suppose that the squared norm S of its second fundamental form is constant. We prove that S=0 or S=1. Accordingly, the image is a hyperplane through the origin or a generalized cylinder 𝕊k(√k)×R3−k, 1≤ k≤3, and the immersion is a Riemannian covering onto its image.
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BibTeX
@misc{guan2026rigidity,
title = {{Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form}},
author = {Shanlin Guan and Yuhang Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18947},
eprint = {2610.18947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18947},
}
RIS
TY - UNPB
TI - Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form
AU - Shanlin Guan
AU - Yuhang Zhao
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18947
UR - https://arxiv.org/abs/2610.18947
AB - Let X:M^3→R^4 be a complete, connected self-shrinker without boundary, and suppose that the squared norm S of its second fundamental form is constant. We prove that S=0 or S=1. Accordingly, the image is a hyperplane through the origin or a generalized cylinder 𝕊^k(√k)×R^3−k, 1≤ k≤3, and the immersion is a Riemannian covering onto its image.
ER -