# Characterization of geodesic balls

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?

• Have you tried working this out for spherically symmetric Riemannian manifolds with $\Omega$ some geodesic ball about the center? What are the results? Feb 10 '20 at 19:11
• @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 $\mathbb S^n$.
– BigM
Feb 12 '20 at 15:34