I am reading the book A Course in Minimal Surfaces by Colding and Minicozzi. In chapter 2, corollary 2.9 and theorem 2.10, the authors claimed that if $\Sigma^2\subset\mathbb{R}^3$ is a complete, stable, 2-sided minimal surface, then the universal cover of $\Sigma$ is also stable.
It is written that the proposition 1.39 implied so. Here is the proposition:
Proposition 1.39: If $\Sigma$ is a complete noncompact minimal hypersurface with trivial normal bundle, then the following are equivalent:
(a) $\lambda_1(\Omega,L)\ge 0$ for every bounded domain $\Omega\subset\Sigma$,
(b) $\lambda_1(\Omega,L)>0$ for every bounded domain $\Omega\subset\Sigma$,
(c) There exists a positive function $u$ with $Lu=0$.
Here $L$ is the stability operator $L\eta=\Delta_{\Sigma}\eta+|A|^2\eta+Ric_M(N,N)\eta$, and the $\lambda_1$ is defined as $$\lambda_1(\Omega,L)=\inf\left\{-\int_\Omega\eta L\eta\ \bigg|\ \eta\in C_0^{\infty}(\Omega) \text{ and }\int_{\Omega}\eta^2=1\right\}.$$
It was also proved that if there is a positive function $u$ on $\Omega$ with $Lu=0$, then $\Omega$ is stable.
However, I really cannot see how this proposition implies that the universal cover of $\Sigma^2$ is stable.
Any hint will be appreciated. Thank you in advance!
Lift your function $u$ from $\Sigma$ to its universal cover. As the covering map is a local isometry, you know that $Lu=0$ on the universal cover and your Proposition 1.39 then implies the universal cover is stable.