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](https://mathoverflow.net/questions/253392/uniformization-for-annuli-with-boundary?newreg=ee6a2d35e50c40bf9db9be5a5db91b26) 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.



Can a harmonic function on a topological cylinder have critical points?
If $M$ has dimension at least $2$ the harmonic function $f$ can have critical points. This can be quickly deduced from Calabi's characterisation of harmonic $1$-forms.


**Example.** Suppose $M=S^2$. Then it is not hard to construct a Morse function $f$ on $M\times [0,1]$ satisfying the following properties 


1) $f(M,\varepsilon)=\varepsilon$, $f(M,1-\varepsilon)=1-\varepsilon$ for any small positive $\varepsilon$.


2) $f$ has exactly two critical points $x\_1,\,x\_2$ in $M\times (0,1)$, $x\_1$ is of index $1$ and $x\_2$ is of index $2$.


3) All level sets $f=c$ are connected. These are spheres for $c\notin [f(x\_1),f(x\_2)]$ and tori for $c\in (f(x\_1),f(x\_2))$.


Now, for such a function $f$ its differential $df$ is a $1$-form that is well defined on $S^2\times S^1$, where we identify two boundaries of $S^2\times [0,1]$ by the obvious map.


Finally, using Calabi's theorem characterising $1$-forms that are harmonic for some metric, we can find a metric on $S^2\times S^1$, for which $df$ is harmonic. QED


Here is the formulation of Calabi's theorem


**Theorem.** Let $M$ be a closed smooth manifold. A closed $1$-form on $M$ having Morse-type zeros is *intrinsically harmonic* if and only if it is *transitive*.


**Remarks.** i) Calabi's theorem can be found, for example in Section 9 of Farber's book, 
Topology of closed $1$-forms.


The original paper of Calabi is called: "An intrinsic characterisation of harmonic 1 forms", it's here: <https://www.jstor.org/stable/j.ctt13x10qw>


ii) A $1$-form on $M$ is called *intrinsically harmonic* if there is a metric for which it is harmonic.


iii) The transitivity of $df$ on $S^2\times S^1$ follows immediately from Property 3) of $f$. 


iv) A $1$-form $\alpha$ on $M$ is called *transitive* if for any point $p\in M$ with $\alpha(p)\ne 0$ there is a loop $S^1\to M$ passing through $p$, so that the restriction of $\alpha$ to the loop never vanishes on it.


