Let us consider a compact Riemann manifold $(M,g)$ and a function $u$ on $(M,g)$. $u$ also satisfies that $|\triangle_g u|\leq \varepsilon$ where $\varepsilon$ is a small constant. Can we obtain that $\operatorname{\mathop{osc}}(u)\leq C_\varepsilon$ for some small constant $C_\varepsilon$ dependent only on $\varepsilon$ and the geometry of $(M,g)$?
If this is not true, how to construct a counterexample?