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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.

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I found a paper online which may be helpful for your question. The link is: https://projecteuclid.org/download/pdf_1/euclid.ojm/1353054807

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