Given a minimal surface $\Sigma$ in $R^3$ with associated normal field $N$, I am told that each of the components of $N$ is a Jacobi field, meaning that $Lu=0$ where L is the stability (Jacobi) operator and $u$ is the function $u(x) = \langle N(x), e_i \rangle$, with $e_1 = (1,0,0)$, etc.
Supposedly this follows from looking at a one parameter family of isometries and the second variation formula, as given in e.g. Minicozzi and Colding's course on minimal surfaces, which I am studying from.
I am just exploring this material for the first time, I guess I'm missing something obvious but I just can't see the conclusion. As in the reference, the motivation is in obtaining a proof that any minimal graph is stable.
Intuition and rigor both welcome!