Rigidity, Regularity and Compactness of Special Lagrangian Submanifolds under Simons Stability
Abstract
In this paper, we prove that, for suitable values of δ, every complete connected δ-Simons stable special Lagrangian immersion in ℂn, n≥3, is flat. We also prove that a δ-Simons stable special Lagrangian cone C⊂ℂn with C∩𝕊2n−1 smooth and compact is a finite union of planes if δ>(n−2)2/(4n), and that this coefficient is optimal. At equality, we characterize the δ-Simons stable cones. We further establish regularity and compactness for sequences of proper δ-Simons stable special Lagrangian immersions when n≥3 and δ>(n−2)/(n+1), under suitable restrictions on the initial singular sets, assuming integral current extensions Tj with ∂ Tj=0 and uniformly bounded mass on compact subsets. In dimension three, we allow initial singular sets Sj satisfying ℋ1(Sj)=0. For suitable values of δ, we also prove properness, Euclidean volume growth, and finite total curvature for complete connected immersions of finite δ-index.
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BibTeX
@misc{li2026rigidity,
title = {{Rigidity, Regularity and Compactness of Special Lagrangian Submanifolds under Simons Stability}},
author = {Caiyan Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08562},
eprint = {2611.08562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08562},
}
RIS
TY - UNPB
TI - Rigidity, Regularity and Compactness of Special Lagrangian Submanifolds under Simons Stability
AU - Caiyan Li
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08562
UR - https://arxiv.org/abs/2611.08562
AB - In this paper, we prove that, for suitable values of δ, every complete connected δ-Simons stable special Lagrangian immersion in ℂ^n, n≥3, is flat. We also prove that a δ-Simons stable special Lagrangian cone C⊂ℂ^n with C∩𝕊^2n−1 smooth and compact is a finite union of planes if δ>(n−2)^2/(4n), and that this coefficient is optimal. At equality, we characterize the δ-Simons stable cones. We further establish regularity and compactness for sequences of proper δ-Simons stable special Lagrangian immersions when n≥3 and δ>(n−2)/(n+1), under suitable restrictions on the initial singular sets, assuming integral current extensions T_j with ∂ T_j=0 and uniformly bounded mass on compact subsets. In dimension three, we allow initial singular sets S_j satisfying ℋ^1(S_j)=0. For suitable values of δ, we also prove properness, Euclidean volume growth, and finite total curvature for complete connected immersions of finite δ-index.
ER -