Strict stability of calibrated cones
Bryan Dimler, Jooho Lee
TL;DR
The paper addresses the strict stability of calibrated cones with isolated singularities, focusing on special Lagrangian and coassociative cones. It develops a spectral framework for the Jacobi (stability) operator on cones, employing separated-variable analysis, weighted Rayleigh quotients, and homogeneous Jacobi fields to determine stability. The main contributions are: (i) proving strict stability for all special Lagrangian cones when $n\ge5$, and for $n=4$ with simply connected link; (ii) proving strict stability for all coassociative cones in $\mathbb{R}^7$; and (iii) exhibiting complex cones that are stable but not strictly stable. These results enhance deformation and regularity theory for calibrated submanifolds and provide concrete stability criteria in higher codimension settings.
Abstract
We study the strict stability of calibrated cones with an isolated singularity. For special Lagrangian cones and coassociative cones, we prove the strict stability. In the complex case, we give non-strictly stable examples.