Let $\gamma$ be a compact and sufficiently smooth manifold/surface without boundary, and we consider the Poisson Equation on $\gamma$: $$-\Delta_\gamma u = f$$ with $\int_\gamma f = 0$, and $\Delta_\gamma$ is the Laplace-Beltrami operator on $\gamma$.
I wonder if the following estimate in $L^\infty$ norm holds:
$$|u(x)-\bar{u}| \le C|f|_\infty$$
where $\bar{u}$ is the mean value of $u$. In the case of $\gamma=\Omega\subset \mathbb{R}^n$, we can use comparison principle to obtain the estimate(add boundary value on right hand side), but in the above case, $\int_\gamma f = 0$ should always be satisfied, implying that $f$ may change its sign and I don't know how to get the desired estimate.
My question would be: do we have a modified comparison principle? Does the $L^\infty$ estimate holds? If it doesn't hold, can we modify the right hand side to control the $L^\infty$ norm of $u$? Thanks!