arXiv:2611.08562v1 Announce Type: new Abstract: In this paper, we prove that, for suitable values of $\delta$, every complete connected $\delta$-Simons stable special Lagrangian immersion in ${\mathbb C}^n$, $n\ge3$, is flat. We also prove that a $\delta$-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 $\delta>(n-2)^2/(4n)$, and that this coefficient is optimal. At equality, we characterize the $\delta$-Simons stable cones. We further establish regularity and compactness for sequences of proper $\delta$-Simons stable special Lagrangian immersions when $n\ge3$ and $\delta>(n-2)/(n+1)$, 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 $\delta$, we also prove properness, Euclidean volume growth, and finite total curvature for complete connected immersions of finite $\delta$-index.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page