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In $\mathbb R^{n\geq3}$, spheres can be characterized by single layer potentials having constant eigenfunction. More precisely we have the following:

Theorem (H. Shahgholian) Let $\Omega\subset \mathbb R^{n\geq3}$ be a bounded smooth domain and for some $c>0$

$$\int_{\partial\Omega}\frac{1}{\|x-y\|^{n-2}}ds_x=\frac{c}{\|y\|^{n-2}},\,\,\forall y\not\in\bar{\Omega}$$ Then $\partial\Omega$ is a sphere centered at the origin. (here $ds$ represents the surface area).

Are there generalization of this statement for Riemannian manifolds and geodesic balls?

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    $\begingroup$ Have you tried working this out for spherically symmetric Riemannian manifolds with $\Omega$ some geodesic ball about the center? What are the results? $\endgroup$
    – Willie Wong
    Commented Feb 10, 2020 at 19:11
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    $\begingroup$ @WillieWong That seems like a great point to start. Currently, I am trying to see if the proof in Shagholian's can be adapted to $\mathbb S^n$. $\endgroup$
    – BigM
    Commented Feb 12, 2020 at 15:34

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