Year: 2026
Type: preprint
Abstract: We classify complete Lagrangian self-shrinking surfaces in $\mathbb{R}^{4}$ 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 $S^1(1)\times \mathbb{R}$, or the Clifford torus $S^1(1)\times S^1(1)$. More generally, a complete self-shrinking surface in $\mathbb{R}^{4}$ with constant $|\vec{H}|^2$ is either a plane, a round cylinder, or a complete minimal immersion into $S^3(\sqrt2)$. The proof uses the function $|x|^2-|\vec{H}|^2$, the Bochner formula, and completeness.
Source: arXiv (Cornell University)
Authors: Li, Zhi, Wei, Guoxin, Xing, Cheng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2778054725, G2923938946
OpenAlex ID: W7223199122