PreprintCC BY 4.0arXiv
Complete self-shrinkers with constant norm of the second fundamental form in RR4
Abstract
Let X:M3→RR4 be a complete self-shrinker with constant squared norm S of the second fundamental form. We prove that S=0 or S=1, and that X(M) is isometric to a hyperplane or a standard product SSSk(√k)×RR3−k, 1≤ k≤3. This proves a conjecture proposed by Cheng and Wei for three-dimensional hypersurfaces.
Related
- The second gap for self-shrinkers with constant norm of the second fundamental form
- Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form
- Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and S≤ 1
- Rigidity theorems of complete self-shrinkers in the Euclidean space
- Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
BibTeX
@misc{fu2026complete,
title = {{Complete self-shrinkers with constant norm of the second fundamental form in \(\RR^4\)}},
author = {Yu Fu and Dan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24964},
eprint = {2610.24964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24964},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete self-shrinkers with constant norm of the second fundamental form in RR^4
AU - Yu Fu
AU - Dan Yang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24964
UR - https://arxiv.org/abs/2610.24964
AB - Let X:M^3→RR^4 be a complete self-shrinker with constant squared norm S of the second fundamental form. We prove that S=0 or S=1, and that X(M) is isometric to a hyperplane or a standard product SSS^k(√k)×RR^3−k, 1≤ k≤3. This proves a conjecture proposed by Cheng and Wei for three-dimensional hypersurfaces.
ER -