I learned that on Euclidean space, if for $p \in \mathbb{N}$ $$\Delta^p u=0$$ and $u$ is bounded, then $u$ is a constant.
I want to know if this is also true for closed manifold, for the case $p=1$, this can be directly proved by here, but it seems that this method can not be applied to the poly-harmonic case, I wonder if there is any result for the bounded poly-harmonic function on closed manifold.
