# Maximum principle of harmonic function on compact manifold

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.