Question on Laplace-Beltrami operator and the maximum principle

Question

So I have this problem on my mind:

Let $\Gamma$ be a $2d-$compact manifold and suppose that $u$ is a solution of:

$-\Delta u=0 \;\forall x\in \Gamma$ ( $\Delta $ here denotes the Laplace-Beltrami operator )

such that $\min_{\Gamma}u=0$ (suppose $u$ is enough smooth to attain its minimum)

My question: Can we deduce from the above that $u=0$?

I intuitively believe the answer is positive and the key role here would play some sort of maximum principle. However I don't know how to proceed.

Could somebody help me through this? Or in case the answer is negative to provide me with a counterexample? Any help is appreciated.

Thanks in advance!

Answer 1

The answer is affirmative. Indeed, we have the energy integral $$ \int_\Gamma |\nabla u|^2\, dS = -\int_\Gamma \Delta u u\, dS, $$ which vanishes, by the harmonicity assumption. Thus, $u$ is constant. Since $\min_\Gamma u=0$, it must be that $u=0$.