Prove
Let $I (u,v) = \langle u, \mathcal{L}v \rangle_{L^2 (\Sigma) } + \langle u , \mathcal{B} v \rangle_{L^2(\partial \Sigma)} $
then $\rho$ is frist eigenfunction of $\mathcal{I}$ if
\begin{align} \mathcal{I}( \rho, \phi) = \sigma_1 \langle \rho, \phi \rangle_{L^2 (\partial \Sigma) } \end{align} $\forall \phi \in C^{\infty}( \Sigma )$. This is equivalent to saying that
$\mathcal{L}\rho = 0$ $ \text{on}$ $ \Sigma $, $ \mathcal{B}\rho=\sigma_1 \rho $ $ \text{on } \partial \Sigma,$
for $\mathcal{L} = \Delta + f $ and $\mathcal{B} = \partial_{\nu} + g$. $f,g \in C^{\infty}(\Sigma)$
furthermore,
\begin{eqnarray} \sigma_1 =\inf_{\phi \in \mathcal{C}^{\infty}(\Sigma) \setminus \{0\}} \dfrac{\mathcal{I}(\phi, \phi)}{\int _{\partial \Sigma} \phi^2} \end{eqnarray}