Let $M$ be an oriented closed smooth manifold, and let $C=M\times[0,1]$, the cylinder over $M$. Let $g$ be an arbitrary Riemannian metric on $C$ (in particular, $g$ may look nothing like a product metric). Let $f:C\to\mathbb{R}$ be the unique $g$-harmonic function satisfying $$f|_{M\times\{i\}}=i,\quad i=0,1.$$
**Question:** Is it necessarily true that $f$ has no critical points?

I believe that in the simplest case $M=S^1$ the answer is positive. This is related to classification of annuli and can be shown by arguments regarding holomorphic maps between Riemann surfaces. See this question and the accepted answer therein (by Mohan Ramachandran) for more details.

Also, the answer is clearly positive for any $M$, provided that $g$ is the product metric of some Riemannian metrics on $M$ and $[0,1]$. Hence, this should continue to hold for metrics that are "close" to a product metric as well. It thus seems natural to ask *how close to a product metric* a Riemannian metric needs to be (and what that even means) in order for the answer to remain "yes".

Any reference or insight are appreciated.