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 ${\mathbb C}^n$, $n\ge3$, is flat. We also prove that a $δ$-Simons stable special Lagrangian cone $C\subset{\mathbb C}^n$ with $C\cap{\mathbb S}^{2n-1}$ 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 $n\ge3$ and $δ>(n-2)/(n+1)$, under suitable restrictions on the initial singular sets, assuming integral current extensions $T_j$ with $\partial T_j=0$ and uniformly bounded mass on compact subsets. In dimension three, we allow initial singular sets $S_j$ satisfying $\mathcal H^1(S_j)=0$. For suitable values of $δ$, we also prove properness, Euclidean volume growth, and finite total curvature for complete connected immersions of finite $δ$-index.