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Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note that I assume $n$ is the dimension of $M$).

Let $\omega:(M,g) \to (\mathbb{R}^n,e)$ be the harmonic function corresponding to the Dirichlet problem imposed by $f$, i.e $\omega|_{\partial M}=f|_{\partial M}$.

Is it true that $\omega$ must be an immersion? Does anything change if we assume $f$ is (globally) injective?

Remark. Note that since $f$ is an immersion, it is locally injective, hence $f|_{\partial M}$ is locally injective. (In particular naive "counter-examples" like taking $\,f|_{\partial M}$ to be constant so $\omega$ is constant do not work). In fact we have $\text{rank}(d\omega_p)\ge \text{rank}\big(d(\omega|_{\partial M})_p\big)= \text{rank}\big(d(f|_{\partial M})_p\big)=n-1$ for every $p \in \partial M$.

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Let $M$ be the closed unit disk in the plane. Let $f$ be a diffeomorphism of $M$ onto a smooth Jordan region $D$ in the plane. If $D$ is not convex, we can easily arrange that the average $$\int_{\partial{M}}f(z)ds$$ does not belong to $\overline{D}$. This implies that the harmonic extension $F$ of $f$ maps some interior points of the unit disk $M$ to points outside $D$. Let $z_0$ be some point in the interior of $M$ such that $F(z_0)\in\partial F(M)\backslash \overline{D}$. Evidently, $F$ is not an immersion at this point.

Remark. It is a classical theorem of Rado, Kneser and Choquet that if $D$ is convex then $F$ is a diffeomorphism inside $M$. I have not looked in their papers, but I am sure this example must be there.

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