can the gradient of a harmonic function =0 at some interior point of a manifolds with two ends?

Question

M is a complete noncompact Riemannian manifold with two ends. There exists a nonconstant bounded harmonic function f defined on the whole M. Then is it possible that $|\nabla f|=0$ at some interior point at M? Please give examples if it is possible.

Answer 1

$f(x,u)=xy$ on open unit disk in the Euclidean plane.

One can also prove existence of such functions on each complete 2-ended Riemannian manifold which is not an annulus. However such proofs (the ones I know) will be non-constructive, they are based on Dirichlet method.

It follows from the paper

P. Li and L. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), p. 359-383.

that given a complete Riemannian manifold $M$ with two ends $E_1, E_2$, there exists a proper harmonic function $f: M\to (0,1)$ which converges to $0$ on the end $E_1$ and to $1$ on $E_2$. If $f$ does not have critical points, then $M$ is diffeomorphic to the product $N\times (0,1)$. Thus, to construct an example, take a manifold with 2 ends which is not diffeomorphic to a product of another manifold with an interval. For instance, 2-dimensional torus with 2 disks removed will do the job.