A Classification of Complete Self-shrinkers
Abstract
Let X:Mn→Rn+1 be an n-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere Sn(√n) and the standard generalized cylinder Sk(√k)× Rn−k for 2≤ k≤ n−1 are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \cite{CLW}: R>0, S<1, and supM S=1, where R and S denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on S are given.
Related
- Complete self-shrinkers with constant norm of the second fundamental form in RR4
- Rigidity theorems of complete self-shrinkers in the Euclidean space
- Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
- Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and S≤ 1
- Foliation by bi-rotational self shrinkers in Euclidean space
BibTeX
@misc{cheng2026classification,
title = {{A Classification of Complete Self-shrinkers}},
author = {Qing-Ming Cheng and Fengjiang Li and Guoxin Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12794},
eprint = {2609.12794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12794},
}
RIS
TY - UNPB
TI - A Classification of Complete Self-shrinkers
AU - Qing-Ming Cheng
AU - Fengjiang Li
AU - Guoxin Wei
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12794
UR - https://arxiv.org/abs/2609.12794
AB - Let X:M^n→R^n+1 be an n-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere S^n(√n) and the standard generalized cylinder S^k(√k)× R^n−k for 2≤ k≤ n−1 are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \cite{CLW}: R>0, S<1, and sup_M S=1, where R and S denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on S are given.
ER -