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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.

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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.

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