So I'm trying to understand the proof for the following theorem from Lan-Hsuan Huang: "Trapped Surfaces, Topology of Black Holes, and the Positive Mass Theorem":
Any orientable locally outermost closed minimal surface $\Sigma$ in a Riemannian 3-manifold $(M; g)$ with nonnegative scalar curvature must be a topological sphere.
The proof starts with finding an outward normal variation $\Sigma_t$ with first order variation $X = e^u \nu$ and mean curvature $H_t$ such that $\partial_t H_t |_{t=0} = \lambda e^u$ for some constant $\lambda$. Further calculations lead then to the following inequality.
$\lambda \leq -\Delta_\Sigma u + K_\Sigma$
$\Delta_\Sigma$ denotes the Laplace-Beltrami Operator.
Integration yields: $0 \leq \lambda * Area(\Sigma) \leq 2\pi \chi(\Sigma)$.
I know that $\int_\Sigma -\Delta_\Sigma u + K_\Sigma = 2\pi \chi(\Sigma)$ because of the Gauss Bonnet Theorem. But then I don't understand why $\int_\Sigma -\Delta_\Sigma u = \int_{\partial\Sigma} k_g$.