The linearized minimal surfaces problem

Abstract

We characterize the kernel of the linearization $R$ of the minimal surface problem about the Euclidean metric in a bounded smooth domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, $n\ge 2$, with the background minimal surfaces being the Euclidean planes. We show that, in the whole-space Euclidean decomposition, the kernel consists of potential fields and TT fields. For bounded domains, a similar phenomenon appears with additional boundary coupling conditions; in particular, the TT part may be coupled to a harmonic conformal component.