PreprintCC0arXiv
Rigidity theorems of complete self-shrinkers in the Euclidean space
Abstract
We classify complete Lagrangian self-shrinking surfaces in ℝ4 with constant squared norm of the mean curvature vector, without assuming an upper bound for the second fundamental form. Their images are a plane through the origin, the round cylinder S1(1)× ℝ, or the Clifford torus S1(1)× S1(1). More generally, a complete self-shrinking surface in ℝ4 with constant |H⃗|2 is either a plane, a round cylinder, or a complete minimal immersion into S3(√2). The proof uses the function |x|2−|H⃗|2, the Bochner formula, and completeness.
Related
- Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space
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- Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
- The rigidity theorems for complete self-expander of mean curvature flow
- Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
BibTeX
@misc{li2026rigidity,
title = {{Rigidity theorems of complete self-shrinkers in the Euclidean space}},
author = {Zhi Li and Guoxin Wei and Cheng Xing},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29842},
eprint = {2610.29842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29842},
note = {CC0},
}
RIS
TY - UNPB
TI - Rigidity theorems of complete self-shrinkers in the Euclidean space
AU - Zhi Li
AU - Guoxin Wei
AU - Cheng Xing
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29842
UR - https://arxiv.org/abs/2610.29842
AB - We classify complete Lagrangian self-shrinking surfaces in ℝ^4 with constant squared norm of the mean curvature vector, without assuming an upper bound for the second fundamental form. Their images are a plane through the origin, the round cylinder S^1(1)× ℝ, or the Clifford torus S^1(1)× S^1(1). More generally, a complete self-shrinking surface in ℝ^4 with constant |H⃗|^2 is either a plane, a round cylinder, or a complete minimal immersion into S^3(√2). The proof uses the function |x|^2−|H⃗|^2, the Bochner formula, and completeness.
ER -