Theorem (Maximum Principle). Let $h$ be a harmonic function on a domain $D$ in $C$.
(a) If $h$ attains a local maximum in $D$, then $h$ is constant.
(b) Suppose that $D$ is bounded and $h$ extends continuously to the boundary $\partial D$ of $D$. If $h\leq 0$ on $\partial D$, then $h\leq 0$ on $\overline{D}$.
The above is a very well known result.
Can anyone please suggest a reference of maximal principle for harmonic function on compact Riemannian manifold with boundary? I only need it for dimension 2 manifold.