In the picture below, why is $\lambda(g_{ij})$ the first eigenvalue of $-4\Delta+R$?
The first eigenvalue is strange to me. After reading the wiki about it, I find
$$\lambda_1=\inf\limits_{u\ne 0}\frac{\int_\Omega|\nabla u|^2}{\int_\Omega|u|^2}$$
where $u$ is the solution to $$ \begin{cases} \Delta u + \lambda u = 0 & \text{in $\Omega$} \\ \left. u\right|_{\partial \Omega} = 0 \end{cases} $$
But seemingly, it is not the first eigenvalue in pictures 1,2 and 3. Also, why is the first eigenfunction $u_0>0$ in picture 3 bigger than $0$?
Pictures 1, 2 and 3 are from the 203rd page of this paper.
