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Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant: \begin{equation} \Delta u = A > 0. \end{equation} Moreover, \begin{equation} 0 < u \leq 1 \text{ on $M$} \quad \text{and} \quad u \equiv 1 \text{ on $\partial M$}. \end{equation}

Continuity forces $u$ to attain its infimum somewhere in the interior of $M$.

Can $u$ have any other critical values? Is the critical set $\lvert \nabla u \rvert^{-1}(\{ 0 \})$ connected?

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    $\begingroup$ I would guess that if your manifold with boundary is two Euclidean balls joined by a thin neck, then the critical set has a least three critical points. There might be some hope for proving what you want for convex domains using ideas such as in this paper of Korevaar. $\endgroup$
    – RBega2
    Commented Mar 15, 2023 at 1:35
  • $\begingroup$ @RBega2 That's very cool, thanks for the suggestion! I haven't yet figured out whether I can solve my problem using the article, but I think I should be able to exclude thin necks, so perhaps there's a chance. $\endgroup$
    – Leo Moos
    Commented Mar 15, 2023 at 14:43

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